$~{}~{}+c\left(x_{\sigma}\partial_{\mu}-x_{\mu}\partial_{\sigma}\right)$ | $+f\left(x_{\mu}^{2}+x_{\nu}^{2}\left(1-\frac{c^{2}}{b^{2}}\right)+\left(\frac{2a}{b}+\frac{2c}{b}x_{\sigma}\right)x_{\nu},x_{\sigma}-\frac{c}{b}x_{\nu}\right)$
$so\left(3\right)$ linear combination | $~{}F\left(R,b\tan\theta\sin\phi+c\cos\phi-aM_{1}\right)$
$a\partial_{\mu}+b\theta_{\left(\nu\sigma\right)}\partial_{\theta_{\left(\nu\sigma\right)}}+cR\partial_{R}$ | $\frac{1}{r_{\left(\nu\sigma\right)}^{2}}f\left(\theta_{\left(\nu\sigma\right)}-\frac{b}{c}\ln r_{\left(\nu\sigma\right)},\frac{a+cx_{\mu}}{cr_{\left(\nu\sigma\right)}}\right)$
$\left(a\partial_{\mu}+b\partial_{\nu}+c\partial_{\sigma}+lR\partial_{R}\right)$ | $\frac{1}{\left(a+lx_{\mu}\right)^{2}}f\left(\frac{b+lx_{\nu}}{l\left(a+lx_{\mu}\right)},\frac{c+lx_{\sigma}}{l\left(a+lx_{\mu}\right)}\right)$

As we have seen in section 7.3.2, the Lie point symmetries of the Klein Gordon
equation are generated from the conformal group of the space; therefore, we
have to consider the admitted CKVs in addition to the HV and the KVs.

###### The two dimensional case

We recall that the conformal algebra of a two dimensional space is infinite
dimensional [101] and in coordinates with line element $ds^{2}=2dwdz$ are
given by the vectors
$X=F\left(w\right)\partial_{w}+G\left(z\right)\partial_{z},$ with conformal
factor $\psi=\frac{1}{2}\left(F_{,w}+G_{,z}\right)$. In the coordinates
$\left(w,z\right),$ the 2D Klein Gordon Klein Gordon equation (7.45) is

$u_{wz}+V\left(w,z\right)u=0.$

The Lie symmetry condition (7.21) becomes

$\left(FV\right)_{,w}+\left(GV\right)_{,z}=0$ (7.46)

from which follows that there are infinite many potentials for which the 2D
Klein Gordon equation admits Lie point symmetries.

###### The three dimensional case

The 3D Euclidian space admits the three special CKVs

$K_{C}^{\mu}=\frac{1}{2}\left(\left(x^{\mu}\right)^{2}-\left(\left(x^{\sigma}\right)^{2}+\left(x^{\nu}\right)^{2}\right)\right)\partial_{i}+x^{\mu}x^{\nu}\partial_{x}+x^{\mu}x^{\nu}\partial_{z}~{},~{}\mu,\sigma,\nu=1,2,3$

with corresponding conformal factor $\psi_{C}=x^{\mu}$.

From the symmetry condition (7.21) of theorem 7.3.3, it follows that a special
CKV generates the Lie point symmetry
$X=K_{C}^{\mu}-\frac{1}{2}x^{\mu}u\partial_{u}$ for the 3D Klein Gordon (7.45)
only for the potential

$V\left(x,y,z\right)=\frac{1}{\left(x^{\sigma}\right)^{2}}F\left(\frac{x^{\nu}}{x^{\sigma}},\frac{\delta_{\kappa\lambda}x^{\kappa}x^{\lambda}}{x^{\sigma}}\right).$

One is possible to continue with the linear combinations and deduce all cases
that the 3D Klein Gordon equation admits a Lie symmetry. These results hold
for both the Klein Gordon and the conformal Klein Gordon equation. It
particular one can show that the results remain still valid for the conformal
Klein Gordon equation provided the metric defining the conformal Laplacian is
conformally flat.

####  7.6 The Klein Gordon equation in a spherically symmetric space-time

In this section, we consider the Lie point symmetries of the Klein Gordon
equation in a non-flat space and in particular in the static spherically
symmetric empty space-time; that is the exterior Schwarzschild solution given
by the metric ($\tau$ is the radial coordinate)

$ds^{2}=-a^{2}\left(\tau\right)dt^{2}+d\tau^{2}+b^{2}\left(\tau\right)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (7.47)

The Lagrangian of Einstein field equations for this space-time is [102, 103]

$L=2ab^{\prime 2}+4ba^{\prime}b^{\prime}+2a$ (7.48)

where $"^{\prime}"~{}$means derivative with respect to the radius $\tau$. If
we see the Lagrangian (7.48) as a dynamical system in the space of variables
$\left\\{a,b\right\\}$, then this system is ”autonomous”; hence, admits the
Noether symmetry $\partial_{\tau}$ with corresponding Noether integral the
”Hamiltonian” $h=$constant. ( $h$ _is not_ the ”energy” because the coordinate
is the radial distance not the time)

$h=2ab^{\prime 2}+4ba^{\prime}b^{\prime}-2a.$

It can be shown directly that $h=\frac{2}{a}G_{1}^{1},$ where $G_{1}^{1}$ is
the Einstein tensor. Because the space is empty, from Einstein’s equations
follows that $h=0.$ The Euler-Lagrange equations are

$a^{\prime\prime}-\frac{a}{2b^{2}}b^{\prime
2}+\frac{1}{b}a^{\prime}b^{\prime}+\frac{1}{2}\frac{a}{b^{2}}=0$
$b^{\prime\prime}+\frac{1}{2b}b^{\prime 2}-\frac{1}{2b}=0.$

We end up with a system of three equations whose solution will give the
functions $a(\tau),b(\tau).$ It is found that the solution of the system does
not give these functions in the well known closed form. This is due to the
Lagrangian we have considered; Indeed we shall show below that it is possible
to find the solution in closed form by considering a Lagrangian conformally
related to the Lagrangian (7.48).

Applying Theorem 4.3.2, we find that the Lagrangian (7.48) admits the Noether
point symmetry

$X_{1}=2\tau\partial_{\tau}+H$

where $H=-2a\partial_{a}+2b\partial_{b}$ is a non-gradient homothetic vector
of the two dimensional kinetic metric

$d\bar{s}^{2}=2adb^{2}+4bdadb$ (7.49)

defined from the Lagrangian (7.48).

The Klein Gordon equation defined by the metric (7.49) with potential
$V\left(a,b\right)=2a$, is

$-\frac{a}{4b^{2}}u_{aa}+\frac{1}{2b}u_{ab}-\frac{1}{4b^{2}}u_{a}-2au=0$
(7.50)

and admits as Lie symmetries the vectors [103]

$u\partial_{u}~{}~{},~{}b\left(a,b\right)\partial_{u}$
$H=-2a\partial_{a}+2b\partial_{b}~{}~{},~{}~{}X_{2}=\frac{1}{ab}\partial_{a}~{}~{},~{}X_{3}=\frac{a}{2b}\partial_{a}-\partial_{b}$

where the vectors $X_{2},X_{3}$ are proper CKVs of the two dimensional metric
(7.49).

It is possible to find solutions of the Klein Gordon equation (7.50) which are
invariant with respect to one of the admitted Lie symmetries.

For example, let us consider the Lie point symmetry $H_{u}=H-2cu\partial_{u}.$
The zero order invariants of $H_{u}$ are
$w=ab~{},~{}u=a^{c}\Phi\left(w\right).$ Replacing in the PDE we find the
solution 222We have found this by making use of the library SADE [104] of
MAPLE.

$u\left(a,b\right)=a^{c}\left[c_{1}I_{c}^{B}\left(2\sqrt{2}ab\right)+c_{2}K_{c}^{B}\left(2\sqrt{2}ab\right)\right]$

where $I^{B},K^{B}$ are the Bessel modified functions [103]. Working similarly
for the Lie point symmetry $H+eX_{2}-cu\partial_{u}$ we find the solution

$u\left(a,b\right)=\left(a^{2}-eb^{-1}\right)^{\frac{c}{4}}\left[c_{1}I_{-\frac{c}{2}}\left(2\sqrt{2b\left(a^{2}b-e\right)}\right)+c_{2}K_{\frac{c}{2}}^{B}\left(2\sqrt{2b\left(a^{2}b-e\right)}\right)\right].$

One can find more solutions using linear combinations of the Lie symmetries.

Following proposition 7.3.6, we look for a conformal metric for which one of
the CKVs $X_{2},X_{3}$ becomes a KV and write the corresponding conformally
related Lagrangian which admits this CKV as a Noether point symmetry.

We consider the vector $X_{2}$ and the conformally related metric

$ds^{2}=N^{2}\left(4adb^{2}+8bda~{}db\right)$ (7.51)

where $N\left(a,b\right)=g\left(b\right)\sqrt{a}$ and $g\left(b\right)$ is an
arbitrary function of its argument. It is easy to show that the vector $X_{2}$
is a KV of this metric hence a Noether symmetry for the family of conformally
related Lagrangians

$\bar{L}=N^{2}\left[2a\left(\frac{db}{dr}\right)^{2}+4b\left(\frac{da}{dr}\right)\left(\frac{db}{dr}\right)\right]+\frac{2a}{N^{2}}$
(7.52)

where we have considered the coordinate transformation
$d\tau=\frac{dr}{N^{2}\left(a,b\right)}$. The Noether function for this
Noether symmetry is the ”Hamiltonian” of the Lagrangian (7.52). This constant
and the two Lagrange equations for the ”generalized” coordinates $a,b$ provide
a system of differential equations which will give the functions
$a(\tau),b(\tau).$

We consider $g\left(b\right)=g_{0}=$constant, that is from the family of
Lagrangians (7.52) we take the Lagrangian

$L=g_{0}^{2}\left[2a^{2}\left(\frac{db}{dr}\right)^{2}+4ab\left(\frac{da}{dr}\right)\left(\frac{db}{dr}\right)\right]+\frac{2}{g_{0}^{2}}.$
(7.53)

For this Lagrangian we have the following system of equations:
a. The ”Hamiltonian” of the Lagrangian (7.53)

$a^{2}\left(\frac{db}{dr}\right)^{2}+2b^{2}\left(\frac{da}{dr}\right)\left(\frac{db}{dr}\right)-V_{0}=0$
(7.54)

b. The Euler Lagrange equations of (7.53) with respect to the variables
$a,b~{}$

$\frac{d^{2}a}{dr^{2}}+\frac{1}{a^{2}}\left(\frac{da}{dr}\right)^{2}+\frac{2}{b}\left(\frac{da}{dr}\right)\left(\frac{db}{dr}\right)=0$
(7.55)

$\frac{d^{2}b}{dr^{2}}=0$ (7.56)

where we have set $V_{0}=g_{0}^{-4}$. The solution of the system of equations
(7.54)-(7.56) is

$b\left(r\right)=b_{1}r+b_{2}~{}~{},~{}~{}~{}a^{2}\left(r\right)=\frac{V_{0}r+2a_{1}b_{1}}{2b_{1}\left(b_{1}r+b_{2}\right)}.$

Under the linear transformation $b_{1}r=b_{1}R-\frac{b_{2}}{b_{1}},$ and if we
set $V_{0}=2\left(b_{1}\right)^{2}~{},~{}a_{1}=-2m+b_{2}$, we obtain the
exterior Schwarzschild solution in the standard coordinates

$ds^{2}=-\left(1-\frac{2m}{R}\right)dt^{2}+\left(1-\frac{2m}{R}\right)^{-1}dR^{2}+R^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$ (7.57)

The choice of the function $g\left(b\right)$ is essentially a choice of the
coordinate system. Obviously the final solution must always be the exterior
Schwarzschild solution. In order to show this let us consider
$g\left(b\right)=\sqrt{b}$ so that
$d\tau=\left(a\left(r\right)b\left(r\right)\right)^{-1}dr.$ Then, we get the
Lagrangian

$\bar{L}=2a^{2}b\left(\frac{db}{dr}\right)^{2}+4ab^{2}\left(\frac{da}{dr}\right)\left(\frac{db}{dr}\right)+\frac{2}{b}$

and the system of equations

$\frac{d^{2}b}{dr^{2}}+\frac{1}{b}\left(\frac{db}{dr}\right)^{2}=0$
$2a^{2}b\left(\frac{db}{dr}\right)^{2}+4ab^{2}\left(\frac{da}{dr}\right)\left(\frac{db}{dr}\right)-\frac{2}{b}=0$
$\frac{d^{2}a}{dr^{2}}+\frac{1}{a}\left(\frac{da}{dr}\right)+\frac{2}{b}\left(\frac{da}{dr}\right)\left(\frac{db}{dr}\right)-\frac{a}{2b^{2}}\left(\frac{db}{dr}\right)^{2}+\frac{1}{2ab^{4}}=0$

The solution of the system is

$b^{2}\left(r\right)=r~{},~{}a^{2}\left(r\right)=\left(\sqrt{r}\right)^{-1}\left(4\sqrt{r}+a_{1}\right)$

from which follows: $d\tau^{2}=\left(4r+\sqrt{r}a_{1}\right)^{-1}dr^{2}.$
Therefore, for this choice of Lagrangian, the metric is

$ds^{2}=-\left(\frac{4\sqrt{r}+a_{1}}{\sqrt{r}}\right)dt^{2}+\frac{1}{4r+\sqrt{r}a_{1}}dr^{2}+r\left(d\theta+\sin^{2}\phi
d\theta^{2}\right).$ (7.58)

If we make the transformation $r=R^{2}$ , $dt\rightarrow\frac{1}{2}dt$ and
$a_{1}=-8m,$ we retain the metric (7.58) in the standard form (7.51).

Working similarly, we find that $X_{3}$ becomes a KV for the conformal metric
(7.51) if $N_{3}\left(a,b\right)=f\left(a^{2}b\right)\sqrt{a}$ and generates a
Noether point symmetry for the conformal Lagrangian (7.52) (with $N_{3}$ in
the place of $N)\ $ and continue as above.

####  7.7 WKB approximation

In WKB approximation, we search for solutions of the Klein Gordon equation of
the form $u=A_{n}e^{iS\left(x^{k}\right)},$ where $S\left(x^{k}\right)$ has to
satisfy the null Hamilton Jacobi equation [105, 106].

$g^{ij}S_{,i}S_{,j}+\bar{V}\left(x^{k}\right)=0$ (7.59)

where $g^{ij}$ is the metric defining the Yamabe operator and
$\bar{V}\left(x\right)=\frac{n-2}{4\left(n-1\right)}R-V\left(x^{k}\right)$. We
study the symmetries of the PDE (7.59).

We search for Lie point symmetries of the form [1]

$X=\xi^{i}\left(x^{i},S\right)\partial_{i}+\eta\left(x^{i},S\right)\partial_{S}.$

The symmetry condition is

$X^{\left[1\right]}\left(H_{null}\right)=\lambda\left(x^{k}\right)\left(H_{null}\right)~{},~{}mod\left[H_{null}\right]=0~{}$

where $X^{\left[1\right]}$ is the first prolongation and
$\lambda=\lambda\left(x^{i}\right)$. Replacing the first prolongation
$X^{\left[1\right]}$ in the symmetry condition and collecting terms of powers
of $S_{,i}$ we find the following result.

###### Theorem 7.7.1

The Lie point symmetries of the null Hamilton Jacobi equation (7.59) are the
vectors

$X=\xi^{i}\left(x^{k}\right)\partial_{i}+\left(a_{0}S+b_{0}\right)\partial_{S}$
(7.60)

where $\xi^{i}\left(x^{k}\right)$ is a CKV of the metric $g_{ij},$
$a_{0},b_{0}$ are constants and the following condition holds

$V_{k}\xi^{k}+2\psi V-a_{0}V=0.$ (7.61)

Proof . The symmetry condition is

$X^{\left[1\right]}\left(H_{null}\right)=\lambda\left(H_{null}\right)~{},\text{
\ }mod\left[H_{null}\right]=0~{}$

where $X^{\left[1\right]}$ is the first prolongation and we set
$\lambda=\lambda\left(x^{i}\right)$. Replacing $X^{\left[1\right]},$ we find

$X^{\left[1\right]}H_{null}=\frac{1}{2}g_{,k}^{ij}\xi^{k}S_{i}S_{j}+g^{ij}S_{j}\eta_{i}^{\left[1\right]}+V_{k}\xi^{k}$

where

$g^{ij}u_{j}\eta_{i}^{\left[1\right]}=g^{ij}S_{j}\eta_{,i}+g^{ij}S_{j}S_{i}\eta_{s}-g^{rj}S_{i}S_{j}\xi_{,r}^{j}-g^{ir}S_{r}S_{i}S_{j}\xi_{,u}^{j}.$

We can easily show that $\xi_{,u}^{i}=0$ so that the the symmetry conditions
become:

$\displaystyle g_{,k}^{ij}\xi^{k}-2g^{r(j}\xi_{,r}^{j)}+2g^{ij}\eta_{S}$
$\displaystyle=$ $\displaystyle\lambda g^{ij}$ (7.62) $\displaystyle
V_{k}\xi^{k}$ $\displaystyle=$ $\displaystyle\lambda V$ (7.63)
$\displaystyle\eta_{,i}$ $\displaystyle=$ $\displaystyle 0.$ (7.64)

From (7.62) and (7.64) we have that $\eta=a_{0}S+b_{0}$. Then, the conditions
(7.62),(7.63) become

$\displaystyle L_{\xi}g_{ij}$ $\displaystyle=$
$\displaystyle\left(a_{0}-\lambda\right)g_{ij}$ (7.65) $\displaystyle
V_{k}\xi^{k}$ $\displaystyle=$ $\displaystyle\lambda V.$ (7.66)

Setting $a_{0}-\lambda=2\psi$ or $\lambda=a_{0}-2\psi,$ we see that condition
(7.65) implies that the Lie point symmetries of the null Hamilton Jacobi
equation (7.59) are generated from the CKVs of the the kinematic metric
$g_{ij}$. Finally, condition (7.66) becomes

$V_{k}\xi^{k}+2\psi V-a_{0}V=0.$

Comparing the symmetry condition (7.61) of the null Hamilton Jacobi equation
and the symmetry condition (7.26) of the Yamabe Klein Gordon equation, we have

###### Proposition 7.7.2

If a CKV of the metric which defines the Yamabe operator generates a Lie point
symmetry for the Yamabe Klein Gordon equation (7.24), then the same vector
generates a Lie point symmetry of the null Hamilton Jacobi equation (7.59).
The reverse holds if the CKV generating Lie symmetry for the null Hamilton
Jacobi equation (7.59) satisfies condition (7.61) with $a_{0}=0$.

Proposition 7.7.2 relates the Lie point symmetries of the Hamilton Jacobi
equation with the Lie point symmetries of the Yamabe Klein Gordon when the
constant $a_{0}=0$. The question which arises is what happens to the Lie
symmetries when $a_{0}\neq 0$. The answer is given in the following
proposition

###### Proposition 7.7.3

If a CKV generates Lie point symmetry for the null Hamilton Jacobi equation
(7.59) satisfying condition (7.61) with $a_{0}\neq 0$, then this CKV produces
a Lie point symmetry for the Euler-Lagrange equations for a conformally
related Lagrangian if there exist a conformal factor $N\left(x^{k}\right)$ for
which the CKV becomes KV or HV.

Proof . The Lie point symmetries of the autonomous system

$\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{i}\dot{x}^{j}+V^{,i}=0$ (7.67)

are produced from the special projective algebra of the metric provided the
potential satisfies the condition

$L_{\eta}V^{,i}+d_{0}V^{,i}=0$ (7.68)

where $d_{0}$ is a constant and $L_{\eta}$ denotes Lie derivative with respect
to $\eta^{i}$. Equation (7.67) remains the same for the conformally related
Lagrangian, that is, we have

$x^{\prime\prime i}+\bar{\Gamma}_{jk}^{i}x^{\prime i}x^{\prime
j}+V^{\prime,i}=0.$

Therefore, the symmetry group will be again the special projective group of
the conformal metric. The special projective group is not preserved under a
conformal transformation but the subgroups of KVs and the homothetic group are
preserved. Then these two subgroups are common for both metrics. Subsequetly,
if the CKV of the metric $g_{ij}$ becomes a KV/HV of the conformal metric
$\bar{g}_{ij}=N^{2}g_{ij}$ then it will be a Lie point symmetry of the Euler
Lagrange equations of the conformally related Lagrangian $\bar{L}$. This
symmetry will not be a Noether symmetry of $\bar{L}$ except in the case that
$d_{0}=2\bar{\psi}~{}\ \left(\bar{\psi}=1~{}\text{for HV and }\psi=0\text{ for
KV}\right).$

####  7.8 Conclusion

We have determined the Lie point symmetries of Schrödinger equation and the
Klein Gordon equation in a general Riemannian space. It has been shown that
these symmetries are related to the homothetic algebra and the conformal
algebra of the metric. Furthermore, these symmetries have been related to the
Noether point symmetries of the classical Lagrangian for which the metric
$g_{ij}$ is the kinematic metric. More precisely, for the Schrödinger equation
(7.13) it has been shown that if a KV/HV of the metric $g_{ij}$ produces a Lie
point symmetry of the Schrödinger equation, then it produces a Noether point
symmetry for the Classical Lagrangian in the space with metric $g_{ij}$ and
potential $V(x^{k})$. For the Klein Gordon equation the situation is
different; the Lie point symmetries of the Klein Gordon are generated by
elements of the conformal group of the metric $g_{ij}.$ The KVs and the HV of
this group produce a Noether symmetry of the classical Lagrangian with a
constant gauge function. However the proper CKVs produce a Noether point
symmetry for the conformal Lagrangian if there exists a conformal factor
$N\left(x^{k}\right)$ such that the CKV becomes a KV/HV of $g_{ij}$.

We have applied these general results to three cases of practical interest:
the motion in a central potential, the classification of all potentials in
Euclidian 2D and 3D space for which the Schrödinger equation and the Klein
Gordon equation admit a Lie point symmetry and finally we have considered the
Lie symmetries of the Klein Gordon equations in the static, spherically
symmetric empty spacetime. In the last case, we have demonstrated the role of
Lie symmetries and that of the conformal Lagrangians in the determination of
the closed form solution of Einstein equations.

Furthermore, we investigated the Lie symmetries of the null Hamilton Jacobi
equation and we proved that if a CKV generates a point symmetry for the Klein
Gordon equation, then it also generates a Lie point symmetry for the null
Hamilton Jacobi equation.

The knowledge of the Lie symmetries of the Schrödinger equation and the Klein
Gordon equation in a general Riemannian space makes possible the determination
of solutions of these equations which are invariant under a given Lie
symmetry. In addition, they can be used in Quantum Cosmology [107, 108, 109,
110, 105] to determine the form of solutions of the Wheeler–DeWitt equation
[111] in a given Riemannian space.

###  Chapter 8 The geometric origin of Type II hidden symmetries

####  8.1 Introduction

Lie symmetries assist us in the simplification of differential equations (DEs)
by means of reduction. As it was indicated in Chapter 2, the reduction is
different for ordinary differential equations (ODEs) and partial differential
equations (PDEs). In the case of ODEs, the use of a Lie symmetry reduces the
order of ODE by one while in the case of PDEs, the reduction by a Lie symmetry
reduces by one the number of independent and dependent variables, but not the
order of the PDE. A common characteristic in the reduction of both cases is
that the Lie symmetry which is used for the reduction is not admitted as such
by the reduced DE, it is ”lost”.

It has been found that the reduced equation is possible to admit more Lie
symmetries than the original equation. These new Lie symmetries have been
termed Type II hidden symmetries. Also, if one works in the reverse way and
either increases the order of an ODE or increases the number of independent
and dependent variables of a PDE, then, it is possible that the new (the
‘augmented’) DE admits new point symmetries not admitted by the original DE.
This type of Lie symmetries are called Type I hidden symmetries.

The Type I and Type II hidden symmetries have studied extensively in the
recent years by various authors (see e.g. [112, 99, 14, 100, 113, 114, 115]).
In the following sections, we consider mainly the Type II hidden symmetries as
they are the ones which could be used to reduce further the reduced DE.

The origin of Type II hidden symmetries is different for the ODEs and the
PDEs, although it has been shown recently that they are nearly the same [116].
For the case of ODEs, the inheritance or not of a Lie symmetry, the $X_{2}$
say, by the reduced ODE depends on the commutator of that symmetry with the
symmetry used for the reduction, the $X_{1}$ say. For example, if only two Lie
symmetries $X_{1},X_{2}$ are admitted by the original equation and the
commutator $[X_{1},X_{2}]=cX_{2}$ where $c$ may be zero, then reduction by
$X_{1}$ results in $X_{2}$ being a nonlocal symmetry for the reduced ODE while
reduction by $X_{2}$ results in $X_{1}$ being an inherited Lie symmetry of the
reduced ODE. In the reduction by $X_{1},$ the symmetry $X_{2}$ is a Type I
hidden symmetry of the original equation relative to the reduced equation. In
the case of more than two Lie symmetries the situation is the same, if the Lie
bracket gives a third Lie symmetry, the $X_{3}$ say. Then, the point like
nature of a symmetry is preserved only if reduction is performed using the
normal subgroup and $X_{3}$ has a certain expression [117].

The above scenario is transferred to PDEs as follows. The reduced PDE loses
the symmetry used to reduce the number of variables and it may lose other Lie
symmetries depending on the structure of the associated Lie algebra depending
if the admitted subgroup is normal or not [117]. Similarly, if $X_{1},X_{2}$
are Lie symmetries of the original PDE with commutator $[X_{1},X_{2}]=cX_{2}$
where $c$ may be zero, then reduction by $X_{2}$ results in $X_{1}$ being a
symmetry of the reduced PDE while reduction by $X_{1}$ results in an
expression which has no relevance for the PDE [117].

In addition to that scenario, B. Abraham - Shrauner and K.S. Govinder have
proposed a new potential source for the Type II hidden symmetries [118] based
on the observation that different PDEs with the same variables, which admit
different Lie symmetry algebras, may be reduced to the same target PDE. Based
on that observation, they propose that the target PDE inherits Lie symmetries
from all reduced PDEs, which explains why some of the new symmetries are not
admitted by the specific PDE used for the reduction. In this context arises
the problem of identifying the set of all PDEs which lead to the same reduced
PDE after reduction by a Lie symmetry. In a recent paper [116], it has been
shown that this is also the case with the ODEs; that is, it is shown that
different differential equations which can be reduced to the same equation
provide point sources for each of the Lie symmetries of the reduced equation
even though any particular of the higher order equations may not provide the
full complement of Lie symmetries. Therefore, concerning the ODEs the Lie
symmetries of the reduced equation can be viewed as having two sources.
Firstly, the point and nonlocal symmetries of _a given_ higher order equation
and secondly, the point symmetries of _a variety_ of higher order ODEs.
Finally, in a newer paper [118], it has been shown by a counter example that
Type II hidden symmetries for PDEs can have a nonpoint origin, i.e. they arise
from contact symmetries or even nonlocal symmetries of the original equation.
Other approaches may be found in [119].

In the present Chapter we will study the reduction and the consequent
existence of Type II hidden symmetries of the homogeneous heat equation

$\Delta u-u_{t}=0$ (8.1)

and the Laplace equation

$\Delta u=0$ (8.2)

in certain classes of Riemannian spaces, where

$\Delta=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial
x^{i}}\left(\sqrt{\left|g\right|}g^{ij}\frac{\partial}{\partial x^{j}}\right)$

is the Laplace operator. In a general Riemannian space, the homogeneous heat
equation (8.1) admits three Lie point symmetries and the Laplace equation
(8.2) admits two Lie point symmetries, which are not useful for reduction.
This implies that if we wish to find ‘sound’ reductions of equations (8.1) and
(8.2), we have to consider Riemannian spaces which admit some type of
symmetry(ies) of the metric (these symmetries are not Lie symmetries and are
called collineations). Indeed, as it has been shown in Chapter 6, the Lie
symmetries of the homogeneous heat and the Laplace equation in a Riemannian
space are generated from the elements of the homothetic algebra and the
conformal algebra of the space respectively. Thus, one expects that in spaces
with a nonvoid conformal algebra there will be Lie symmetry vectors which will
allow for the reduction of equations (8.1), (8.2) and the possibility of the
existence of Type II hidden symmetries.

In section 8.3 we reduce the Laplace equation with the extra Lie symmetries
existing in (a) a decomposable space - that is a Riemannian space which admits
a gradient Killing vector (KV) - (b) in a space which admits a gradient
Homothetic vector (HV) and (c) in a space which admits a special Conformal
killing vector (sp.CKV). In section 8.4, we apply the results of the previous
section and find the Type II symmetries of the Laplace equation in four and
three dimensional Minkowski spacetimes. Also we fully recover previous results
[100]. In order to study the reduction of Laplace equation by a non-gradient
HV and a proper CKV we consider two further examples. In section 8.5, we
consider the algebraically special vacuum solution of Einstein’s equations
known as Petrov type III [120] and we make the reduction using the Lie point
symmetry generated by the nongradient HV. Moreover we do the same in a
conformally flat spacetime where the proper CKVs generate Lie symmetries.

In section 8.6, we reduce the homogeneous heat equation (8.1) with the extra
Lie symmetries existing (i) in a space which admits a gradient KV and (ii) in
a space which admits a gradient HV. In section 8.7, we consider the special
cases of the previous section, that is, a decomposable space whose
nondecomposable part is a maximally symmetric space of non-vanishing curvature
and the spatially flat Friedmann Robertson Walker (FRW) space time used in
Cosmology. Finally in section 8.8 we consider the reduction of the homogeneous
heat equation in the Petrov type III spacetime using the Lie symmetry which is
generated by the HV.

We emphasize that all results are derived in a purely geometric manner without
the use of a computer package. However, we have verified them with the
libraries PDEtools and SADE [121, 104] of Maple111www.maplesoft.com.

####  8.2 Lie symmetries of Laplace equation in certain Riemannian spaces

In a general Riemannian space, Laplace equation (8.2) admits the Lie
symmetries

$~{}X_{u}=u\partial_{u}~{},~{}X_{b}=b\left(t,x\right)\partial_{u}$

where $b\left(t,x\right)$ is a solution of Laplace equation. These symmetries
are too general to provide useful reductions and lead to reduced PDEs which
posses Type II hidden symmetries. However if we restrict our considerations to
spaces which admit a conformal algebra (proper or not) then we will have new
Lie symmetries, hence new reductions of Laplace equation eqn (8.2), which
might lead to Type II hidden symmetries.

In the following sections, we consider spaces in which the metric $g_{ij}$ can
be written in a generic form. The spaces we shall consider are: a. Spaces
which admit a gradient KV (decomposable spaces) b. Spaces which admit a
gradient HV and c. Spaces which admit a sp.CKV.

The generic form of the metric for each type is as follows
($A,B,\ldots=1,2,\ldots,n$):

  1. a.

If a $1+n-$dimensional Riemannian space admits a gradient KV, the
$S^{i}=\partial_{z}~{}\left(S=z\right)$ say, then the space is decomposable
along $\partial_{z}$ and the metric is written as (see e.g. [44])

$ds^{2}=dz^{2}+h_{AB}y^{A}y^{B}~{},~{}h_{AB}=h_{AB}\left(y^{C}\right)$

  2. b.

If a $1+n-$dimensional Riemannian space admits a gradient HV, the
$H^{i}=r\partial_{r}~{}\left(H=\frac{1}{2}r^{2}\right),~{}\psi_{H}=1$ say,
then the metric can be written in the generic form [83]

$ds^{2}=dr^{2}+r^{2}h_{AB}dy^{A}dy^{B}~{},~{}~{}h_{AB}=h_{AB}\left(y^{C}\right)$

  3. c.

If a $1+n-$dimensional Riemannian space admits a sp.CKV then admits a gradient
KV and a gradient HV [83, 22] and the metric can be written in the generic
form

$ds^{2}=-dz^{2}+dR^{2}+R^{2}f_{AB}\left(y^{C}\right)dy^{A}dy^{B}$

while the sp.CKV is $C_{S}=\frac{z^{2}+R^{2}}{2}\partial_{z}+zR\partial_{R}$
with conformal factor $\psi_{C_{S}}=z$.

The Riemannian spaces which admit non-gradient proper HV do not have a generic
form for their metric. However, the spaces for which the HV acts simply
transitively are a few and are given together with their homothetic algebra in
[120]. A special class of these spaces are the algebraically special vacuum
space-times known as Petrov type N, II , III, D. In section 8.5.1, we shall
consider the reduction of Laplace equation in the Petrov type III spacetime
whose metric is

$ds^{2}=2d\rho
dv+\frac{3}{2}xd\rho^{2}+\frac{v^{2}}{x^{3}}\left(dx^{2}+dy^{2}\right)$

with the symmetry generated by the non-gradient HV
$H=v\partial_{v}+\rho\partial_{\rho}~{},~{}\psi_{H}=1.$ The reduction of
Laplace equation in the rest of the Petrov types is similar both in the
working method and results and there is no need to consider them explicitly.

Finally we shall consider the conformally flat space

$ds^{2}=e^{2t}\left[dt^{2}-\delta_{AB}y^{A}y^{B}\right]$

which admits a proper CKV which produces a Lie symmetry222According to theorem
6.4.2, the condition for this is that the conformal factor satisfies Laplace
equation and we reduce Laplace equation using this symmetry.

In what follows all spaces are assumed to be of dimension $n>2$.

####  8.3 Reduction of the Laplace equation in certain Riemannian spaces

As we have seen in Chapter 6, the Lie symmetries of Laplace equation (8.2) in
a Riemannian space are generated from the CKVs (not necessarily proper) whose
conformal factor satisfies Laplace equation. This condition is satisfied
trivially by the KVs ($\psi=0),$ the HV ($\psi_{;i}=0)$ and the sp.CKVs
($\psi_{;ij}=0).$ Therefore these vectors (which span a subalgebra of the
conformal group) are among the Lie symmetries of Laplace equation. Concerning
the proper CKVs it is not necessary that their conformal factor satisfies the
Laplace equation, therefore they may not produce Lie symmetries for Laplace
equation.

#####  8.3.1 Riemannian spaces admitting a gradient KV

Without loss of generality we assume the gradient KV to be the $\partial_{z},$
so that the metric has the generic form

$ds^{2}=dz^{2}+h_{AB}dy^{A}dy^{B}~{},~{}h_{AB}=h_{AB}\left(y^{C}\right)$ (8.3)

where $h_{AB}$ $A,B,C=1,...,n$ is the metric of the $n-$ dimensional space.
For the metric (8.3) Laplace equation (8.2) takes the form

$u_{zz}+h^{AB}u_{AB}-\Gamma^{A}u_{B}=0.$ (8.4)

and admits as _extra_ Lie symmetry the gradient KV $\partial_{z}.$

We reduce (8.4) by using the zeroth order invariants $y^{A}~{},~{}w=u~{}$ of
the extra Lie symmetry $\partial_{z}$. Taking these invariants as new
coordinates, equation (8.4) reduces to

${}_{h}\Delta w=0$ (8.5)

which is Laplace equation in the $n~{}$dimensional space with metric $h_{AB}.$
Now we recall the result that the conformal algebra of the $n$ metric $h_{AB}$
and the $1+n$ metric (8.3) are related as follows [44]:

a. The KVs of the $n$ metric are identical with those of the $n+1$ metric

b. The $1+n$ metric admits a HV if and only if the $n$ metric admits one and
if ${}_{n}H^{A}$ is the HV of the $n$ metric then the HV of the $1+n$ metric
is given by the expression

${}_{1+n}H^{\mu}=z\delta_{z}^{\mu}+_{n}H^{A}\delta_{A}^{\mu}\text{
}\mu=x,1,...,n.$ (8.6)

d. The $1+n$ metric admits CKVs if and only if the $n$ metric $h_{AB}$ admits
a gradient CKV (for details see [44]).

Therefore Type II hidden symmetries for (8.2) exist if the $n~{}$metric
$h_{AB}$ admits more symmetries. Specifically, the sp.CKVs of the $h_{AB}$
metric as well as the proper CKVs whose conformal function is a solution of
Laplace equation (8.62) generate Type II hidden symmetries.

#####  8.3.2 Riemannian spaces admitting a gradient HV.

In Riemannian spaces which admit a gradient HV, $H$ say, there exists a
coordinate system in which the metric is written in the form [83]

$ds^{2}=dr^{2}+r^{2}h_{AB}\left(y^{C}\right)dy^{A}dy^{B}$ (8.7)

and the gradient HV is $H=r\partial_{r}.$ In these coordinates the Laplacian
(8.2) takes the form

$u_{rr}+\frac{1}{r^{2}}h^{AB}u_{AB}+\frac{\left(n-1\right)}{r}u_{r}-\frac{1}{r^{2}}\Gamma^{A}u_{A}=0$
(8.8)

and admits the extra Lie symmetry (see Theorem  6.4.2) $H$. We reduce (8.8)
using $H.$

The zeroth order invariants of $H$ are $y^{A}~{},~{}w\left(y^{A}\right)~{}$and
using them it follows easily that the reduced equation is

${}_{h}\Delta u=0$ (8.9)

that is, the Laplacian defined by the metric $h_{AB}$.

It is easy to establish the following results concerning the conformal
algebras of the metrics (8.7) and $h_{AB}$.
1\. The KVs of $h_{AB}$ are also KVs of (8.7).
2\. The HV of (8.7) is independent from that of $h_{AB}$.
3\. The metric (8.7) admits proper CKVs if and only if the $n$ metric $h_{AB}$
admits gradient CKVs. This is because (8.7) is conformally related with the
decomposable metric

$ds^{2\prime}=dr^{2}+h_{AB}\left(y^{C}\right)dy^{A}dy^{B}.$ (8.10)

The above imply, that Type II hidden symmetries we shall have from the HV of
the metric $h_{AB},$ the sp.CKVs and finally from the proper CKVs of $h_{AB}$
whose conformal factor is a solution of Laplace equation (8.9).

#####  8.3.3 Riemannian spaces admitting a sp.CKV

It is known [22], that if an $n=m+1$ dimensional ($n>2)$ Riemannian space
admits sp.CKVs then also admits a gradient HV and as many gradient KVs as the
number of sp.CKVs. In these spaces there exists always a coordinate system in
which the metric is written in the form [83]

$ds^{2}=-dz^{2}+dR^{2}+R^{2}f_{AB}\left(y^{C}\right)dy^{A}dy^{B}$ (8.11)

where $\partial_{z}$ is the gradient KV and $z\partial_{z}+R\partial_{R}$ is
the gradient HV. $f_{AB}\left(y^{C}\right)\,,~{}A,B,C,..=1,2,,...m-1$ is an
$m-1$ dimensional metric. For a general $m-1~{}$ dimensional metric $f_{AB}$
the $n~{}$dimensional metric (8.11) admits the following special Conformal
group

$\displaystyle K_{G}$ $\displaystyle=$
$\displaystyle\partial_{z}~{},~{}H=z\partial_{z}+R\partial_{R}$ $\displaystyle
C_{S}$ $\displaystyle=$
$\displaystyle\frac{z^{2}+R^{2}}{2}\partial_{z}+zR\partial_{R}$

where $K_{G}$ is a gradient KV, $H$ is a gradient HV and $C_{S}$ is a sp.CKV
with conformal factor $\psi_{C_{S}}=z$. In these coordinates Laplace equation
(8.2) takes the form

$-u_{zz}+u_{RR}+\frac{1}{R^{2}}h^{AB}u_{AB}+\frac{\left(m-1\right)}{R}u_{R}-\frac{1}{R^{2}}\Gamma^{A}u_{A}=0.$
(8.12)

From Theorem 6.4.2, we have that the extra Lie symmetries of (8.12) are the
vectors

$\displaystyle X^{1}$ $\displaystyle=$ $\displaystyle K_{G}~{},~{}X^{2}=H$
$\displaystyle X^{3}$ $\displaystyle=$ $\displaystyle C_{S}+2pzu\partial_{u}$

where $2p=\frac{1-m}{2}~{}$ and the non zero commutators are

$\left[X^{1},X^{2}\right]=X^{1}~{},~{}\left[X^{2},X^{3}\right]=X^{3}$
$\left[X^{1},X^{3}\right]=X^{2}+2pX_{u}.$

We consider the reduction of (8.12) with each of the extra Lie symmetries.

###### Reduction with the gradient KV $X^{1}$.

The first order invariants of $X^{1}$ are $R,y^{C}$, $w\left(R,y^{C}\right)$
and by using them we reduce the Laplacian (8.12) to (8.8) which admits the Lie
symmetry $X^{2}$ generated by the HV. This result is expected because
$\left[X^{1},X^{2}\right]=X^{1}$ [100] hence the Lie symmetry $X^{2}$ is
inherited. Therefore, in this reduction the Type II symmetries are generated
from the CKVs of the metric (8.7). It is possible to continue the reduction by
the gradient HV $H$ and then we find the results of section 8.3.2.

###### Reduction with the gradient HV $X^{2}$.

The reduction with a gradient HV has been studied in section 8.3.2. To apply
the results of section 8.3.2 in the present case we have to bring the metric
(8.11) to the form (8.7). For this we consider the transformation

$z=r\sinh\theta~{},~{}R=r\cosh\theta$

which brings (8.11) to

$ds^{2}=dr^{2}+r^{2}\left(-d\theta^{2}+\cosh^{2}\theta
f_{AB}y^{A}y^{B}\right)$ (8.13)

so that the metric $h_{AB}$ of (8.7) is

$ds_{h}^{2}=\left(-d\theta^{2}+\cosh^{2}\theta f_{AB}y^{A}y^{B}\right).$
(8.14)

The reduced equation of (8.12) under the Lie symmetry generated by the
gradient HV is Laplace equation in the space (8.14). For this reduction we do
not have inherited symmetries and there exist Type II hidden symmetries as
stated in section 8.3.2.

###### Reduction with the sp.CKV $X^{3}$.

Before we reduce (8.12) with the symmetry generated by the sp CKV $X^{3}$, it
is best to write the metric (8.11) in new coordinates. We introduce the new
variable $x$ via the relation

$z=\sqrt{\frac{R\left(xR-1\right)}{x}}.$ (8.15)

In the new variables the metric (8.11) becomes

$ds^{2}=-\frac{R}{4x^{3}\left(xR-1\right)}dx^{2}-\frac{2xR-1}{2x^{2}\left(xR-1\right)}dxdR-\frac{1}{4xR\left(xR-1\right)}dR^{2}+R^{2}f_{AB}\left(y^{C}\right)dy^{A}dy^{B}$
(8.16)

the Laplacian (8.12):

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{x^{2}}{R^{2}}u_{xx}-2\frac{x}{R}\left(2xR-1\right)u_{xR}+u_{RR}+\frac{1}{R^{2}}f^{AB}u_{AB}+$
$\displaystyle+\frac{\left(m-1\right)}{R}u_{R}-\frac{x}{R^{2}}\left(m-1\right)\left(2xR-1\right)u_{x}-\frac{1}{R^{2}}\Gamma^{A}u_{A}$

and the Lie symmetry $X^{3}$

$X^{3}=\sqrt{\frac{R\left(xR-1\right)}{x}}R\partial_{R}+2p\sqrt{\frac{R\left(xR-1\right)}{x}}u\partial_{u}.$

The zeroth order invariants of $X^{3}$ are $x,y^{A}~{},~{}w=uR^{-2p}.$ We
choose $x,y^{A}$ to be the independent variables and $w=w\left(x,y^{A}\right)$
the dependent one. By replacing in (8.3.3) we find the reduced equation

$x^{2}w_{xx}+f^{AB}w_{AB}-\Gamma^{A}w_{A}-2p\left(2p+1\right)w=0$ (8.18)

We consider cases.
The case $m\succeq 4$.

If $2p+1\neq 0,~{}m\succeq 4~{}$ then (8.18) becomes

${}_{\left(m\succeq
4\right)}\bar{\Delta}w-2p\left(2p+1\right)V\left(x\right)w=0$ (8.19)

where $V\left(x\right)=x^{\frac{2}{2-m}}$ and ${}_{\left(m\succeq
4\right)}\bar{\Delta}$ is the Laplace operator for the metric

$d\bar{s}_{\left(m\succeq
4\right)}^{2}=\frac{1}{V\left(x\right)}\left(\frac{1}{x^{2}}dx^{2}+f_{AB}dy^{A}dy^{B}\right).$
(8.20)

Equation (8.19) is the Klein Gordon equation in a space with potential
$V\left(x\right)=x^{\frac{2}{2-m}}$ and metric (8.20). Considering the new
transformation $\phi=\int\sqrt{\frac{1}{xV}}dx~{}\ $ or
$x=\left(m-2\right)^{2-m}\phi^{m-2}$ the metric (8.20) is written

$d\bar{s}_{\left(m\succeq
4\right)}^{2}=d\phi^{2}+\phi^{2}\bar{f}_{AB}dy^{A}dy^{B}$ (8.21)

where $\bar{f}_{AB}=\left(m-2\right)^{-2}f_{AB}$ whereas the potential
$V\left(\phi\right)=\frac{\left(2-m\right)^{2}}{\phi^{2}}$ which is the well
known Ermakov potential [69].

This means that the gradient HV $\phi\partial_{\phi}~{},~{}\psi_{\phi}=1$, is
a Lie symmetry of (8.19) which is the Lie symmetry $X^{2}.$ Therefore, if the
metric $\bar{f}_{AB}$ admits proper CKVs which satisfy the conditions of
Theorem 7.3.3, then these vectors generate Type II hidden symmetries of
(8.12).
The case $m=3.$

If $2p+1=0,~{}$then $~{}m=3$ and $f_{AB}$ is a two dimensional metric. In this
case, equation (8.18) becomes

$x^{2}w_{xx}+f^{AB}w_{AB}-\Gamma^{A}w_{A}=0$ (8.22)

or, by multiplying with $x^{2}$

${}_{\left(m=3\right)}\bar{\Delta}w=0$ (8.23)

which is the Laplacian in the three dimensional space with metric

$d\bar{s}_{\left(m=3\right)}^{2}=\frac{1}{x^{4}}dx^{2}+\frac{1}{x^{2}}f_{AB}dy^{A}dy^{B}.$
(8.24)

By making the new transformation $x=\frac{1}{r},\ $ (8.23) is of the form
(8.7) and admits the gradient HV $r\partial_{r}~{}$which gives an inherited
symmetry. We conclude that Type II hidden symmetries of (8.24) will be
generated from the proper CKVs of the metric (8.24) which satisfy the
condition of Theorem 6.4.2.
The case $m=2$.

For $m=2,~{}$ $f_{AB}$ is a one dimensional metric and (8.11) is

$ds^{2}=-dz^{2}+dR^{2}+R^{2}d\theta^{2}$ (8.25)

which is a flat metric333The only three dimensional space who admits sp.CKV is
the flat space, because in that case we also have a gradient HV and a gradient
KV.. In this space, Laplace equation (8.12) admits ten Lie point symmetries,
as many as the elements of the Conformal algebra of the flat 3D space. Six of
these vectors are KVs, one vector is a gradient HV and three are sp.CKVs (see
example 2.7.12). We reduce the Laplace equation with the symmetry $X^{3}$ and
the reduced equation is (8.18) which for $f_{AB}=\delta_{\theta\theta}$
becomes

$x^{2}w_{xx}+w_{\theta\theta}+\frac{1}{4}w=0.$ (8.26)

Equation (8.26) is in the form of (6.13) (see Chapter 6) with
$A^{ij}=diag\left(x^{2},1\right)$ and $B^{i}=0.~{}$ By replacing in the
symmetry conditions (6.21)-(6.25) we find the Lie symmetries

$X=\xi^{i}\partial_{i}+\left(a_{0}w+b\right)\partial_{w}$

where $\xi^{i}$ are the CKVs of the two dimensional space with metric
$A^{ij}$. In this case, all proper CKVs of the two dimensional space $A^{ij}$
generate Type II Lie symmetries. Recall that the conformal algebra of a two
dimensional space is infinite dimensional.

####  8.4 Type II hidden symmetries of the 3D and the 4D wave equation

In this section we apply the general results of the previous section to
specific spaces where the metric is known.

#####  8.4.1 Laplacian in $M^{4}$

The Laplace equation in the four dimensional Minkowski spacetime $M^{4}$

$ds^{2}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}$ (8.27)

is the wave equation [100] in $E^{3}$

$u_{tt}-u_{xx}-u_{yy}-u_{zz}=0.$ (8.28)

The conformal algebra of the metric (8.27) is generated by 15 vectors (see
example 2.7.12). From theorem 6.4.2 we have that the extra Lie point
symmetries of (8.28) are the vectors

$K_{G}^{1}~{},~{}K_{G}^{A}~{},~{}X_{R}^{1A}~{},~{}X_{R}^{AB}~{},~{}H~{},~{}X_{C}^{1}-tu\partial_{u}~{},~{}X_{C}^{A}-y^{A}u\partial_{u}$
(8.29)

where $y^{A}=\left(x,y,z\right).$

The nonzero commutators are

$\displaystyle\left[K_{G}^{I},X_{R}^{IJ}\right]$ $\displaystyle=$
$\displaystyle-K_{G}^{J}~{},~{}\left[K_{G}^{I},~{}H\right]=K_{G}^{I}$
$\displaystyle\left[K_{G}^{I},X_{C}^{I}\right]$ $\displaystyle=$
$\displaystyle H-X_{u}~{},~{}\left[K_{G}^{I},X_{C}^{J}\right]=X_{R}^{IJ}$
$\displaystyle\left[H,X_{C}^{I}\right]$ $\displaystyle=$ $\displaystyle
X_{C}^{I}~{},~{}\left[X_{R}^{IJ},X_{C}^{I}\right]=X_{C}^{J}~{}.$

###### Reduction with a gradient KV

We choose to make reduction of (8.28) with the gradient KV
$K_{G}^{z}=\partial_{z}$. The reduced equation is

$w_{tt}-w_{xx}-w_{yy}=0$ (8.30)

which is Laplace equation in the space $M^{3}.$ The extra Lie symmetries of
(8.30) are

$K_{G}^{1}~{},~{}K_{G}^{x}~{},~{}K_{G}^{y}~{},~{}~{}X_{R}^{1a}~{}\
,~{}X_{R}^{ab}~{},~{}H$ (8.31)

and are inherited symmetries (see also the last commutators). The Type II
symmetries are the vectors

$\bar{X}_{C}^{1}-\frac{1}{2}tu\partial_{u}~{},~{}\bar{X}_{C}^{x}-\frac{1}{2}xu\partial_{u}~{},~{}\bar{X}_{C}^{y}-\frac{1}{2}yu\partial_{u}$
(8.32)

that is the Type II hidden symmetries are generated from the sp.CKVs of
$M^{3}$.

###### Reduction with the gradient HV

In this case it is better to switch to hyperspherical coordinates
$(r,\theta,\phi,\zeta).$

In these coordinates the metric (8.27) is written

$ds^{2}=dr^{2}-r^{2}\left[d\theta^{2}+\cosh^{2}\theta\left(d\phi^{2}+\cosh^{2}\phi
d\zeta^{2}\right)\right]$ (8.33)

and the wave equation (8.28) becomes

$u_{rr}-\frac{1}{r^{2}}\left(u_{\theta\theta}+\frac{u_{\phi\phi}}{\cosh^{2}\theta}+\frac{u_{\zeta\zeta}}{\cosh^{2}\theta\cosh\phi^{2}}\right)+\frac{3}{r}u_{r}-2\frac{\tanh\theta}{r^{2}}u_{\theta}-\frac{\tanh\phi}{r^{2}\cosh^{2}\theta}u_{\phi}=0.$
(8.34)

According to the analysis of section 8.3.2 the reduced equation is (8.9) which
is the Laplacian in the three dimensional space of the variables
$\left(\theta,\phi,\zeta\right):$

$w_{\theta\theta}+\frac{w_{\phi\phi}}{\cosh^{2}\theta}+\frac{w_{\zeta\zeta}}{\cosh^{2}\theta\cosh\phi^{2}}+2\frac{\tanh\theta}{r^{2}}w_{\theta}+\frac{\tanh\phi}{r^{2}\cosh^{2}\theta}w_{\phi}=0.$
(8.35)

This space is a space of constant curvature. The conformal algebra444All
spaces of constant curvature are conformally flat, hence, they admit the same
conformal algebra with the flat space but not the same subalgebras, i.e. the
same conformal factors. of a 3D space of constant non-vanishing curvature
consists of 6 non-gradient KVs and 4 proper CKVs555The rotations and the
sp.CKVs of the flat space are KVs for the space of constant curvature, the
rest are proper gradient CKVs [23]. The conformal factors of the CKVs do not
satisfy the condition ${}_{h}\Delta\psi=0$ (see Theorem 6.4.2); hence, they do
not generate Lie point symmetries for the reduced equation (8.35) whereas for
the same reason the KVs are Lie symmetries of (8.35). Therefore, all point Lie
point symmetries are inherited and we do not have Type II hidden symmetries.

We note that the proper CKVs in a space of constant non-vanishing curvature
are gradient and their conformal factor satisfies the relation [23]

$\psi_{;ab}=C\psi g_{ab}\rightarrow
g^{ab}\psi_{;ab}=nC\psi\rightarrow_{h}\Delta\psi=nC\psi$

which implies that they are Lie symmetries of the conformally invariant
operator but not of the Laplace equation (8.35).

###### Reduction with a sp.CKV

Following the steps of section 8.3.3, we consider the transformation to axi-
symmetric coordinates $\left(t,R,\theta,\phi\right)$ in which (8.28) takes the
form

$u_{tt}-u_{RR}-\frac{1}{R^{2}}\left(u_{\theta\theta}+\frac{u_{\phi\phi}}{\cosh^{2}\theta}\right)-\frac{2}{R}u_{r}-\frac{\tanh\theta}{R^{2}}u_{\theta}=0.$
(8.36)

Applying the transformation (8.15) $t=\sqrt{\frac{R\left(\tau
R-1\right)}{\tau}}$ we find (note that this is the case $m=3$) that (8.36) is
written as (8.3.3) and the reduced equation is the Laplacian
${}_{\left(m=3\right)}\Delta w~{}\ \ $for the 3D metric

$ds^{2}=\frac{1}{\tau^{4}}d\tau^{2}-\frac{1}{\tau^{2}}\left(d\theta^{2}+\cosh^{2}\theta
d\phi^{2}\right).$ (8.37)

The metric (8.37) under the coordinate transformation $\tau=\frac{1}{T}$ is
written

$ds^{2}=dT^{2}-T^{2}\left(d\theta^{2}+\cosh^{2}\theta d\phi^{2}\right)$ (8.38)

which is the flat 3D Lorentzian metric, which does not admit proper CKVs. This
implies that the Lie symmetries of the reduced equation are generated from the
KVs/HV/sp.CKVs of the flat $M^{3}$ metric and all are inherited. Therefore we
do not have Type II hidden symmetries for the reduction with a sp.CKV.

As we shall show in the next section this is not the case for the reduction of
Laplace equation in $M^{3}.$

#####  8.4.2 Laplacian in $M^{3}$

We consider the reduction of Laplace equation in the 3d Minkowski
$M^{3}~{}$spacetime [100], i.e. the wave equation in $E^{2}$

$u_{tt}-u_{xx}-u_{yy}=0.$ (8.39)

As we have seen in section 8.4.1 the extra Lie point symmetries of (8.39) are
the ten vectors (8.31) and (8.32).

###### Reduction with a gradient KV

We choose the vector $\partial_{y}$ and the reduction gives the reduced
equation

$w_{tt}-w_{xx}=0$ (8.40)

which is the one dimensional wave equation. The 2d space $(t,x)$ has an
infinite number of CKVs therefore (8.40) has infinite Lie point symmetries
[1]. From these symmetries the KVs and the HV are inherited symmetries and the
CKVs are Type II symmetries.

###### Reduction with the gradient HV

In order to do the reduction with the gradient HV we introduce spherical
coordinates $(r,\theta,\phi)$ and find that (8.39) becomes

$u_{rr}-\frac{1}{r^{2}}\left(u_{\theta\theta}+\frac{u_{\phi\phi}}{\cosh^{2}\theta}\right)+\frac{2}{r}u_{r}-\frac{\tanh\theta}{r^{2}}u_{\theta}=0.$
(8.41)

According to the results of section 8.3.2 the reduced equation is equation
(8.9) which is the Laplace equation in the 2d space of the variables
$\left(\phi,\theta\right)$

$w_{\theta\theta}+\frac{w_{\phi\phi}}{\cosh^{2}\theta}+\tanh\theta
w_{\theta}=0.$ (8.42)

By making the transformation $\theta=\ln\left(\tan\frac{\rho}{2}\right)$
equation (8.42) becomes

$\sin\left(x\right)^{2}\left(w_{\rho\rho}+w_{\phi\phi}\right)=0$ (8.43)

which is the wave equation (8.40) with $t=\rho~{},~{}x=-i\phi$.

We obtain the results, concerning the Lie symmetries of (8.43) from section
8.4.2 with the difference that the Lie symmetry due to the HV of the two
dimensional metric is not inherited but it is a Type II hidden symmetry.

The reduction of the wave equation in the 4D and in 3D Minkowski space has
been done previously by Abraham-Shrauner et. all [100] and our results
coincide with theirs. For example in the 3d case equation (17) of [100] is our
equation (8.42) in other variables. However there are two differences (a) in
the case of the 2D space they do not obtain that the Lie symmetries are
infinite and (b) they use algebraic computing programs to find the Lie
symmetry generators whereas our approach is geometric and general and does not
need algebraic computing programs to find the complete answer.

The reduction with a sp.CKV has been considered in section 8.3.3.

####  8.5 The Laplace equation in the Petrov type III and in the FRW-like
spacetime

To complete our analysis, we have to reduce the Laplace equation using a non
gradient homothetic vector and a proper (i.e. non special) CKV. In order to do
this, we consider the reduction of Laplace equation in the Petrov type III
spacetime and in the FRW-like spacetime.

#####  8.5.1 The Laplace equation in the Petrov Type III spacetime

In this section we consider the reduction of Laplace equation in spaces which
do not admit gradient KVs or a gradient HV. As it has been mentioned in
section 8.2 we shall consider the algebraically special solutions of Einstein
equations, that is the Petrov type D,N,II and III. In fact we restrict our
discussion to Petrov Type III because both the method of work and the results
are the same for all Petrov types.

The metric of the Petrov type III space-time is

$ds^{2}=2d\rho
dv+\frac{3}{2}xd\rho^{2}+\frac{v^{2}}{x^{3}}\left(dx^{2}+dy^{2}\right)$ (8.44)

with conformal algebra

$\displaystyle K^{1}$ $\displaystyle=$
$\displaystyle\partial_{\rho}~{},~{}K^{2}=\partial_{y}~{},~{}K^{3}=v\partial_{v}-\rho\partial_{\rho}+2x\partial_{x}+2y\partial_{y}$
$\displaystyle H$ $\displaystyle=$ $\displaystyle
v\partial_{v}+\rho\partial_{\rho}~{},~{}\psi=1$

where $K^{1-4}$ are KVs and $H$ is a non-gradient HV. (The space does not
admit proper CKVs).

In this space-time Laplace equation (8.2) takes the form

$-\frac{3}{2}xu_{vv}+2u_{v\rho}+\frac{x^{3}}{v^{2}}\left(u_{xx}+u_{yy}\right)-3\frac{x}{v}u_{v}+\frac{2}{v}u_{\rho}=0.$
(8.45)

From Theorem 6.4.2 we have that the extra Lie point symmetries are the vectors

$X_{1-3}=K_{1-3}~{},~{}~{}X_{4}=H$

with nonzero commutators:

$\left[X_{2,},X_{3}\right]=2X_{2}$
$\left[X_{3},X_{1}\right]=X_{1}~{},~{}~{}\left[X_{1},X_{4}\right]=X_{1}.~{}$

We use $X_{4}~{}$to reduce the PDE because this is the Lie symmetry generated
by the nongradient HV.

The zero order invariants of $X_{4}$ are $\sigma=\frac{\rho}{v},x,y,w$. We
choose $\sigma,x,y$ as the independent variables and
$w=w\left(\sigma,x,y\right)$ as the dependent variable and we find the reduced
equation

$-\sigma\left(\frac{3}{2}x\sigma+2\right)w_{\sigma\sigma}+x^{3}\left(w_{xx}+w_{yy}\right)=0.$
(8.46)

Equation (8.46) can be written

${}_{III}\Delta^{\ast}w-\left(\frac{3x\sigma}{2}+1\right)w_{\sigma}-\frac{3x^{3}\sigma}{2\left(3x\sigma+4\right)}w_{x}=0$
(8.47)

where ${}_{III}\Delta^{\ast}$ is the Laplacian for the metric

$ds^{2}=-\frac{1}{\sigma\left(\frac{3}{2}x\sigma+2\right)}d\sigma^{2}+\frac{1}{x^{3}}\left(dx^{2}+dy^{2}\right).$
(8.48)

The Lie symmetries of (8.47) will be generated from the conformal algebra of
(8.48) with some extra conditions (see equations (6.21)-(6.25)). Finally, we
find that equation (8.47) admits as Lie point symmetries the vectors
$\partial_{y}~{},~{}x\partial_{x}+y\partial_{x}-\sigma\partial_{\sigma}$ which
are inherited symmetries. Therefore we do not have Type II hidden symmetries.

#####  8.5.2 The Laplace equation in the $n~{}$dimensional FRW-like spacetime.

We consider the $n$ dimensional FRW-like space $\left(n>2\right)$ with metric

$ds^{2}=e^{2t}\left[dt^{2}-\left(\delta_{AB}dy^{A}dy^{B}\right)\right]$ (8.49)

where $\delta_{AB}$ is the $n-1$ dimensional Euclidian metric. The reduction
of Laplace equation in this space (for $n=4$) has been studied previously in
[122]. The metric (8.49) is conformally flat hence admits the same CKVs with
the flat space but with different subalgebras. More precisely the space admits

a. $\left(n-1\right)+\frac{\left(n-2\right)\left(n-3\right)}{2}~{}$KVs the
$K_{G}^{A},~{}X_{R}^{AB}~{}$

b. $1$ gradient HV the $K_{G}^{1}=\partial_{t}~{}~{}$
the rest vectors being proper CKVs [45]. In this space Laplace equation (8.2)
becomes

$e^{-2t}\left[u_{tt}-\left(\delta^{AB}u_{AB}\right)+\left(n-2\right)u_{t}\right]=0$
(8.50)

and the extra Lie symmetries are

$K_{G}^{A},~{}X_{R}^{AB}~{},~{}K_{G}^{1}~{},~{}X_{R}^{1A}-2pY^{a}u\partial_{u}$

where $2p=\frac{2-n}{2}$. The algebra of the Lie point symmetries is the same
with that of section 7.55. We consider the reduction with a proper CKV.

###### Reduction with a proper CKV

We may take any of the vectors $X_{R}^{1A}$ (because as one can see in the
Appendix there is a symmetry between the coordinates $y^{A}$). We choose the
vector

$X_{R}^{1x}=x\partial_{t}+t\partial_{x}+2pxu\partial_{u}.$

whose zero order invariants are

$R=t^{2}-x^{2}~{},~{}y^{C}~{},~{}w=e^{-2pt}u.$

We take the dependent variable to be the $w=w\left(R,y^{C}\right)$ and find
the reduced equation

$4Rw_{RR}-\delta^{ab}w_{ab}+4w_{R}-4p^{2}w=0$ (8.51)

where $a=1,\ldots,n-2$. We consider cases.
Case $n>3.$

For $n>3$ equation (8.51) is

${}_{C}\Delta w-4p^{2}f\left(R\right)w=0$ (8.52)

where ${}_{C}\Delta$ is the Laplace operator for the $\left(n-1\right)$
dimensional metric

$ds_{C}^{2}=\frac{1}{f\left(R\right)}\left(\frac{1}{4R}dR^{2}-\delta_{ab}dy^{a}dy^{b}\right)$
(8.53)

and $f\left(R\right)=R^{-\frac{1}{n-3}}$. The metric (8.53) is conformally
flat hence we know its conformal algebra. Application of theorem 7.3.3 gives
that the Lie point symmetries of (8.52) are the vectors

$\displaystyle X_{u}$ $\displaystyle=$ $\displaystyle
u\partial_{u}~{},~{}X_{b}=b\partial_{u}$ $\displaystyle X_{K}^{a}$
$\displaystyle=$
$\displaystyle\partial_{y^{a}}~{},~{}X_{R}^{ab}~{}=y^{b}\partial_{a}-y^{a}\partial_{b}.$

These are inherited symmetries (this result agrees with the commutators). We
conclude that for this reduction we do not have Type II hidden symmetries.

Case $n=3$.

For $n=3$ the reduced equation is a two dimensional equation $\left(\text{that
is }\delta_{AB}=\delta_{yy}\right)$

$4Rw_{RR}-w_{yy}+4w_{R}-\frac{1}{4}w=0$ (8.54)

and admits as Lie point symmetry the KV $\partial_{y}$ which is an inherited
symmetry. Hence, we do not have Type II hidden symmetries.

We conclude that the reduction of Laplace equation in an $n~{}$dimensional FRW
like space with the proper CKV does not produce Type II hidden symmetries and
in fact the inherited symmetries of the reduced equation are the KVs of the
flat metric.

###### Reduction with the gradient HV

The gradient HV $K_{G}^{1}=\partial_{t}$ is a Lie symmetry of the Laplacian
(8.50) hence we consider the reduction by this vector. The zero order
invariants are $y^{A},w$ and lead to the reduced equation

$\delta^{AB}u_{AB}=0$ (8.55)

which is Laplace equation in the flat space $E^{n-1}$. We consider again
cases.
Case $n>3$

In this case the Lie symmetries of (8.55) are given by the vectors

$K_{G}^{A}~{},~{}X_{R}^{AB}~{},~{}_{n-1}H~{},X_{C}^{A}-y^{A}u\partial_{u}.$
(8.56)

From these the $K_{G}^{A},X_{R}^{AB}$ are inherited symmetries and the rest \-
which are produced by the HV and the sp.CKVs of the space $E^{n-1}$ \- are
Type II hidden symmetries.

If $n=3,$ the reduced equation (8.55) is the Laplacian in $E^{2},$ hence,
admits infinite Lie symmetries. Type II hidden symmetries are generated from
the HV and the CKVs of $E^{2}$.

In the following sections, we study the reduction of the homogeneous heat
equation (8.1) in certain Riemannian spaces.

####  8.6 Reduction of the homogeneous heat equation in certain Riemannian
spaces

In a general Riemannian space with metric $g_{ij}$ the heat conduction
equation with flux is

$\Delta u-u_{t}=q$ (8.57)

where $\Delta$ is the Laplace operator
$\Delta=\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x^{i}}\left(\sqrt{g}g^{ij}\frac{\partial}{\partial x^{j}}\right)$ and
$q=q(t,x,u)$. Equation (8.57) can also be written

$g^{ij}u_{ij}-\Gamma^{i}u_{i}-u_{t}=q$ (8.58)

where $\Gamma^{i}=\Gamma_{jk}^{i}g^{jk}$ and $\Gamma_{jk}^{i}$ are the
Christofell Symbols of the metric $g_{ij}$.

For $q=0$, equation (8.58) admits the Lie point symmetries

$X_{t}=\partial_{t}~{}\
,~{}X_{u}=u\partial_{u}~{},~{}X_{b}=b\left(t,x\right)\partial_{u}$ (8.59)

where $b\left(t,x\right)$ is a solution of the heat equation. These symmetries
are too general to provide sound reductions and consequently reduced PDEs
which can give Type II hidden symmetries. However, in Chapter 6 it has been
shown that there is a close relation between the Lie point symmetries of the
heat equation and the collineations of the metric. Specifically it has been
shown that the Lie point symmetries of the heat equation are generated from
the HV and the KVs of $g_{ij}.$ This implies that if we want to have new Lie
point symmetries which will allow for sound reductions of the heat equation
eqn (8.58) we have to restrict our considerations to spaces which admit a
homothetic algebra. Our intention is to keep the discussion as general as
possible therefore we consider spaces in which the metric $g_{ij}$ can be
written in generic form. The spaces we shall consider are:

a. Spaces which admit a gradient KV (8.3).

b. Spaces which admit a gradient HV (8.7).

c. Space which admits a nongradient HV acting simply transitive, i.e. Petrov
Type III (8.44).

In what follows all spaces are of dimension $n\succeq 2$. The case $n=1$
although relatively trivial for our approach in general it is not so and has
been studied for example in [123, 124].

#####  8.6.1 The heat equation in a space which admits a gradient KV

In the $1+n$ decomposable space with line element (8.3) the heat equation
(8.1) takes the form

$u_{zz}+h^{AB}u_{AB}-\Gamma^{A}u_{B}-u_{t}=0.$ (8.60)

Application of Theorem 6.5.2 gives that (8.60) admits the following _extra_
Lie point symmetries generated by the gradient KV $\partial_{x}:$

$X_{1}=\partial_{z}~{},~{}X_{2}=t\partial_{z}-\frac{1}{2}zu\partial_{u}$

with nonvanishing commutators

$\left[X_{t},X_{2}\right]=X_{1}~{},~{}\left[X_{2},X_{1}\right]=\frac{1}{2}X_{u}.$
(8.61)

We reduce (8.60) using the zero order invariants of the extra Lie point
symmetries $X_{1},X_{2}$.

###### Reduction by $X_{1}$

The zero order invariants of $X_{1}$ are

$\tau=t~{},~{}y^{A}~{},~{}w=u.$

Taking these invariants as new coordinates eqn (8.60) reduces to

${}_{h}\Delta w-w_{t}=0$ (8.62)

where ${}_{h}\Delta$ is the Laplace operator in the $n-$dimensional space with
metric $h_{AB}:$

${}_{h}\Delta w=h^{AB}w_{AB}-\Gamma^{A}w_{B}.$ (8.63)

Equation (8.62) is the homogeneous heat eqn (8.1) in the $n~{}$ dimensional
space with metric $h_{AB}$. According to the Theorem 6.5.2 (see Chapter 6),
the Lie symmetries of this equation are the homothetic algebra of $h_{AB}$. As
it has been already mentioned the homothetic algebras of the $n$ and the $1+n$
metrics are related as follows [44]:

a. The KVs of the $n-$ metric are identical with those of the $1+n$ metric.

b. The $1+n$ metric admits a HV if the $n$ metric admits one and if
${}_{n}H^{A}$ is the HV of the $n$ \- metric then the HV of the $1+n$ metric
is given by the expression

${}_{1+n}H^{\mu}=x\delta_{z}^{\mu}+_{n}H^{A}\delta_{A}^{\mu}\qquad\quad\mu=z,1,...,n.$
(8.64)

The above imply that equation (8.62) inherits all symmetries which are
generated from the KVs/HV of the $n-$metric $h_{AB}.$ Hence we do not have
Type II symmetries in this reduction.

###### Reduction by $X_{2}$

The zero order invariants of $X_{2}$ are

$\tau=t~{},~{}y^{A}~{},~{}w=ue^{\frac{z^{2}}{4t}}.$

Taking these invariants as new coordinates eqn (8.60) reduces to

$h^{AB}w_{AB}-\Gamma^{A}w_{B}-w_{\tau}-\frac{1}{2\tau}w=0$ (8.65)

or

${}_{h}\Delta w-w_{\tau}=\frac{1}{2\tau}w.$

This is the nonhomogeneous heat equation with flux
$q\left(\tau,y^{A},w\right)=\frac{1}{2\tau}w$. Application of Theorem 6.5.1
gives the following result666For details see Appendix 8.A..

###### Proposition 8.6.1

The Lie point symmetries of the heat equation (8.65) in an $n-$dimensional
Riemannian space with metric $h_{AB}$ are constructed form the homothetic
algebra of the metric as follows:

a. $Y^{i}$ is a HV/KV.
The Lie symmetry is

$X=\left(2c_{2}\psi\tau+c_{1}\right)\partial_{\tau}+c_{2}Y^{i}\partial_{i}+\left[\left(-\frac{c_{1}}{2\tau}+a_{0}\right)w+b\left(\tau,x\right)\right]\partial_{w}$
(8.66)

b. $Y^{i}=S_{J}^{,i}$ is a gradient HV/KV (the index $J$ counts gradient KVs).
The Lie symmetry is

$X=\left(\psi T_{0}\tau^{2}\right)\partial_{\tau}+T_{0}\tau
S_{J}^{,i}\partial_{i}-\left(\frac{1}{2}T_{0}S_{J}+T_{0}\psi\tau\right)w\partial_{w}$
(8.67)

where $b\left(\tau,x\right)$ is a solution of the heat equation (8.65).

We infer that for this reduction we have the Type II hidden symmetry
$\partial_{\tau}-\frac{1}{2t}w\partial_{w}.$ The rest of the Lie point
symmetries are inherited.

#####  8.6.2 The heat equation in a space which admits a gradient homothetic
vector

For the spacetime with line element

$ds^{2}=dr^{2}+r^{2}h_{AB}dy^{A}dy^{B}$

the homogeneous heat equation becomes

$u_{rr}+\frac{1}{r^{2}}h^{AB}u_{AB}+\frac{\left(n-1\right)}{r}u_{r}-\frac{1}{r^{2}}\Gamma^{A}u_{A}-u_{t}=0$
(8.68)

where $\Gamma^{A}=\Gamma_{BC}^{A}h^{BC}$ and $\Gamma_{BC}^{A}$ are the
connection coefficients of the Riemannian metric $h_{AB}$ ($A,B,C=1,2,...,n).$
Application of Theorem 6.5.2 gives that the heat equation (8.68) admits the
following _extra_ Lie point symmetries generated by the gradient homothetic
vector

$~{}\bar{X}_{1}=2t\partial_{t}+r\partial_{r}~{}\
,~{}~{}\bar{X}_{2}=t^{2}\partial_{t}+tr\partial_{r}-\left(\frac{1}{4}r^{2}+\frac{n}{2}t\right)u\partial_{u}~{}\,$
(8.69)

with nonzero commutators

$\left[X_{t},\bar{X}_{1}\right]=2X_{t}~{}~{},~{}~{}\left[\bar{X}_{1},\bar{X}_{2}\right]=2X_{t}$
$\left[X_{t},\bar{X}_{2}\right]=\bar{X}_{1}-\frac{n}{2}X_{u}.$

We consider again the reduction of (8.68) using the zero order invariants of
these extra Lie point symmetries.

###### Reduction by $\bar{X}_{1}$

The zero order invariants of $\bar{X}_{1}$ are

$\phi=\frac{r}{\sqrt{t}}~{},~{}w=u,~{}y^{A}.$

We choose $w=w\left(\phi,y^{A}\right)$ as the dependent variable.

Replacing in (8.68) we find the reduced PDE

$w_{\phi\phi}+\frac{1}{\phi^{2}}h^{AB}w_{AB}+\frac{\left(n-1\right)}{\phi}w_{\phi}+\frac{\phi}{2}w_{\phi}-\frac{1}{\phi^{2}}\Gamma^{A}w_{A}=0.$
(8.70)

Consider a nonvanishing function $N^{2}\left(\phi\right)$ and divide (8.70)
with $N^{2}\left(\phi\right)$ to get:

$\frac{1}{N^{2}}w_{\phi\phi}+\frac{1}{\phi^{2}N^{2}}h^{AB}w_{AB}+\frac{\left(n-1\right)}{{\phi}N^{2}}w_{\phi}+\frac{\phi}{2N^{2}}w_{\phi}-\frac{1}{\phi^{2}N^{2}}\Gamma^{A}w_{A}=0$
(8.71)

It follows that (for $n>2$) equation (8.71) can be written as

${}_{\bar{g}}\Delta w=0$ (8.72)

where ${}_{\bar{g}}\Delta$ is the Laplace operator if
$N^{2}\left(\phi\right)=\exp\left(\frac{\phi^{2}}{2\left(n-2\right)}\right)$
and $\bar{g}_{ij}$ is the conformally related metric

$d\bar{s}^{2}=\exp\left(\frac{\phi^{2}}{2\left(n-2\right)}\right)\left(d\phi^{2}+\phi^{2}h_{AB}dy^{A}dy^{B}\right).$
(8.73)

According to Theorem 6.4.2 the Lie point symmetries of (8.72) are the CKVs of
the metric (8.73) whose conformal factor satisfies the condition
${}_{\bar{g}}\Delta\psi=0.$ Therefore, Type II hidden symmetries will be
generated from the proper CKVs. The existence and the number of these vectors
depends mainly on the $n$ metric $h_{AB}.$

###### Reduction by $\bar{X}_{2}$

For $\bar{X}_{2}$ the zero order invariants are

$\phi=\frac{r}{t}~{},~{}w=ut^{\frac{n}{2}}e^{\frac{r^{2}}{4t}},~{}y^{A}.$

We choose $w=w\left(\phi,y^{A}\right)$ as the dependent variable and we have
the reduced equation

${}_{g}\Delta w=0$ (8.74)

where

${}_{g}\Delta
w=w_{\phi\phi}+\frac{\left(n-1\right)}{\phi}w_{\phi}+\frac{1}{\phi^{2}}h^{AB}w_{AB}-\frac{1}{\phi^{2}}\Gamma^{A}w_{A}.$
(8.75)

Equation (8.74) is the Laplace equation in the space $\left(\phi,y^{A}\right)$
with metric

$ds^{2}=d\phi^{2}+\phi^{2}h_{AB}dy^{A}dy^{B}.$ (8.76)

The Lie point symmetries of Laplace equation (8.74) are given in Theorem
6.4.2. As in the last case the existence and the number of these vectors
depends mainly on the $n$ metric $h_{AB}.$

We note that both vectors $\bar{X}_{1},\bar{X}_{2}$ are generated form the
gradient HV and in both cases the heat equation is reduced to Laplace
equation. This gives the following

###### Proposition 8.6.2

The reduction of the heat equation (8.1) in a space with metric (8.7)
$\left(n>2\right)~{}$by means of the Lie symmetries generated by the gradient
HV leads to Laplace equation $\Delta u=0$, where $\Delta$ is the Laplace
operator for the metric (8.73) if the reduction is done by $\bar{X}_{1}$ and
for the metric (8.76) if the reduction is done by $\bar{X}_{2}$.

####  8.7 Applications of the reduction of the homogeneous heat equation

In this section, we consider applications of the general results of section
8.6 in various spacetimes.

#####  8.7.1 The heat equation in a $1+n$ decomposable space

Consider the $1+n$ decomposable space

$ds^{2}=dx^{2}+N^{-2}\left(y^{C}\right)\delta_{AB}y^{A}y^{B}$ (8.77)

where $N\left(y^{C}\right)=\left(1+\frac{K}{4}y^{C}y_{C}\right),$ that is, the
$n$ space is a space of constant non vanishing ($K\neq 0)$ curvature. The
space (8.77) does not admit proper HV, however, admits
$\frac{n\left(n-1\right)}{2}+n~{}$nongradient KVs and $1$ gradient KV as
follows [44]

$\displaystyle 1~{}\text{gradient KV}\text{:~{}}$ $\displaystyle\partial_{x}$
$\displaystyle n~{}\text{nongradient KVs}$ : $\displaystyle\text{\
}K_{V}=\frac{1}{N}\left[\left(2N-1\right)\delta_{I}^{i}+\frac{K}{2}Nx_{I}x^{i}\right]\partial_{i}$
$\displaystyle\frac{n\left(n-1\right)}{2}\text{nongradient KVs}$ :
$\displaystyle\text{\ }X_{IJ}=\delta_{[I}^{j}\delta_{J]}^{i}\partial_{i}$

In a space with metric (8.77), the homogeneous heat equation takes the form

$u_{xx}+N^{2}\left(y^{C}\right)\delta^{AB}u_{AB}-\frac{N}{2}Ky^{A}u_{A}-u_{t}=0$
(8.78)

Applying Theorem 6.5.2 we find that equation (8.78) admits the extra Lie point
symmetries

$\partial_{x}~{},~{}t\partial_{x}-\frac{1}{2}xu\partial_{x}~{},K_{V}~{},X_{IJ}.$
(8.79)

The Lie point symmetries which are generated by the gradient KV are777Here the
algebra is the one given in section 8.6.1 and a separate algebra is the
algebra of the KVs of the space of constant curvature. More specifically the
KVs $K_{V}~{},X_{IJ}$ commute with all other symmetries but not between
themselves $\partial_{x}~{},~{}t\partial_{x}-\frac{1}{2}xu\partial_{x}$.

Reduction of (8.78) by means of the gradient KV $\partial_{x}$ results in the
special form of equation (8.62)

$\frac{1}{N^{2}\left(y^{C}\right)}\delta^{AB}u_{AB}-\frac{N}{2}Ky^{A}u_{A}-u_{t}=0.$
(8.80)

This is the homogeneous heat equation in an $n$\- dimensional space of
constant curvature. The Lie point symmetries of this equation have been
determined in section 6.5.1 and are inherited symmetries. Hence, in this case,
we do not have Type II hidden symmetries.

Reduction of (8.78) with the Lie symmetry
$~{}t\partial_{x}-\frac{1}{2}xu\partial_{x}$ gives that the reduced equation
(8.65) is

$N^{2}\left(y^{C}\right)\delta^{AB}w_{AB}-\frac{N}{2}Ky^{A}w_{A}-w_{\tau}=\frac{1}{2\tau}w~{},~{}w=ue^{\frac{x^{2}}{4t}}$
(8.81)

which is the heat equation with flux. By Proposition 8.6.1, the Lie point
symmetries of (8.81) are:

$X=c_{1}\partial_{\tau}+\left(K_{V}~{}+X_{IJ}\right)+\left[\left(-\frac{c_{1}}{2\tau}+a_{0}\right)w+b\left(\tau,y^{C}\right)\right]\partial_{w}$
(8.82)

where $c_{1},a_{0}$ are constants. From section 8.6.1 we have that Type II
hidden symmetry is the one defined by the constant $c_{1}$.

#####  8.7.2 FRW space-time with a gradient HV

Consider the spatially flat FRW metric

$ds^{2}=d\sigma^{2}-\sigma^{2}\left(dx^{2}+dy^{2}+dz^{2}\right)$ (8.83)

This metric admits the gradient HV [45]

$H=\sigma\partial_{\sigma}~{}~{}\left(\psi_{H}=1\right)$

and six nongradient KVs

$X_{1-3}=\partial_{y^{A}}~{}~{},~{}~{}X_{4-6}=y^{B}\partial_{A}-y^{A}\partial_{B}.$

where $y^{A}=\left(x,y,z\right).~{}$

In this space the heat equation takes the form

$u_{\sigma\sigma}-\frac{1}{\sigma^{2}}\left(u_{xx}+u_{yy}+u_{zz}\right)+\frac{3}{\sigma}u_{\sigma}-u_{t}=0.$
(8.84)

The Lie point symmetries of (8.84) are

$\displaystyle\partial_{t}~{},~{}u\partial_{u}~{},~{}b\left(\tau,y^{A}\right)\partial_{u}~{},~{}X_{1-3}~{},~{}X_{4-6}~{},$
$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle
2t\partial_{t}+\sigma\partial_{\sigma}~{},~{}H_{2}=t^{2}\partial_{t}+t\sigma\partial_{\sigma}-\left(\frac{1}{4}\sigma^{2}+2t\right)u\partial_{u}~{}.$

The Lie point symmetries $H_{1},H_{2}$ are produced by the gradient HV
therefore we use them to reduce (8.84). We note that this case is a special
case of the one we considered in section 8.6.2 for $h_{AB}=\delta_{AB}$.

Reduction by $H_{1}$ gives that (8.84) becomes:

$w_{\phi\phi}-\frac{1}{\phi^{2}}\left(w_{xx}+w_{yy}+w_{zz}\right)+\left(\frac{3}{\phi}+\frac{\phi}{2}\right)w_{\phi}=0$
(8.85)

where $\phi=\frac{r}{\sqrt{\sigma}}~{},~{}w=u.$ This is a special form of
(8.70).

Dividing with $N^{2}\left(\phi\right)=\exp\left(\frac{\phi^{2}}{4}\right)$ we
find that (8.85) is written as

${}_{\bar{g}}\Delta w=0$ (8.86)

where the metric $\bar{g}_{ij}$ is the conformally related metric of (8.83):

$d\bar{s}^{2}=e^{\frac{\phi^{2}}{4}}\left(d\phi^{2}-\phi^{2}\left(dx^{2}+dy^{2}+dz^{2}\right)\right)$
(8.87)

We have that the Lie symmetries of (8.86) are generated from elements of the
conformal algebra of the space whose conformal factors satisfy condition
${}_{\bar{g}}\Delta\psi=0.$ The metric (8.86) is conformally flat therefore
its conformal group is the same with that of the flat space [44, 45], however
with different subgroups. We find that these vectors (i.e. the Lie symmetries)
are the vectors

$~{}X_{1-3}~{},~{}X_{4-6}~{},\partial_{t},\text{
}w\partial_{w}~{},~{}b_{0}\left(\phi,y^{A}\right)\partial_{w}.~{}$ (8.88)

We conclude that there are no Type II symmetries for this reduction.

Using reduction by $H_{2}$ we find that (8.84) reduces to :

$w_{\phi\phi}-\frac{1}{\phi^{2}}\left(w_{xx}+w_{yy}+w_{zz}\right)+\frac{3}{\phi}w_{\phi}=0$
(8.89)

where $\phi=\frac{\tau}{\tau}~{},~{}w=ut^{2}e^{\frac{\tau^{2}}{4t}}.~{}$ This
is a special form of (8.74) which is the Laplace equation. In this case the
results of [90] apply and we infer that the Lie point symmetries of (8.89)
are:

$\displaystyle
X_{1-3}~{},~{}X_{4-6}~{},~{}~{}w\partial_{w}~{},~{}b_{1}\left(\phi,y^{A}\right)\partial_{w}$
$\displaystyle X_{7}$ $\displaystyle=$
$\displaystyle\phi\partial_{\phi}~{},~{}X_{8-10}=\phi
y^{A}\partial_{\phi}+\ln\phi\partial_{A}-y^{A}w\partial_{w}.$ (8.90)

The vector $X_{7}$ is the proper HV of the metric and the vectors $X_{8-10}$
the proper CKVs which are not special CKVs, therefore these vectors are Type
II hidden symmetries. A further analysis of (8.89) can be found in [122].

####  8.8 The Heat equation in Petrov type III spacetime

In this section we consider the special class of Petrov type III spacetime
which admits a nongradient HV which acts simply transitively.

The metric of the Petrov type III space-time is

$ds^{2}=2d\rho
dv+\frac{3}{2}xd\rho^{2}+\frac{v^{2}}{x^{3}}\left(dx^{2}+dy^{2}\right)$ (8.91)

with Homothetic algebra

$\displaystyle K^{1}$ $\displaystyle=$
$\displaystyle\partial_{\rho}~{},~{}K^{2}=\partial_{y}~{},~{}K^{3}=v\partial_{v}-\rho\partial_{\rho}+2x\partial_{x}+2y\partial_{y}$
$\displaystyle H$ $\displaystyle=$ $\displaystyle
v\partial_{v}+\rho\partial_{\rho}~{}\left(\psi_{H}=1\right)$

where $K^{1-4}$ are KVs and $H$ is a nongradient HV.

In this space-time equation (8.1) takes the form:

$-\frac{3}{2}xu_{vv}+2u_{v\rho}+\frac{x^{3}}{v^{2}}\left(u_{xx}+u_{yy}\right)-3\frac{x}{v}u_{v}+\frac{2}{v}u_{\rho}-u_{t}=0.$
(8.92)

From Theorem 6.5.2 we have that the extra Lie point symmetries are the vectors

$X_{1-3}=K_{1-3}~{},~{}~{}X_{4}=2t\partial_{t}+H$

with nonzero commutators:

$\left[X_{t},X_{4}\right]=2X_{t}~{},~{}\left[X_{2,},X_{3}\right]=2X_{2}$
$\left[X_{3},X_{1}\right]=X_{1}~{},~{}~{}\left[X_{1},X_{4}\right]=X_{1}~{}.$

We use $X_{4}$ for reduction because this is the Lie point symmetry generated
by the HV.

The zero order invariants of $X_{4}$ are

$\alpha=\frac{v}{\sqrt{t}}~{},~{}\beta=\frac{\rho}{\sqrt{t}}~{},~{}\gamma=x~{},~{}\delta=y~{},~{}w=u.$

We choose $w=w\left(\alpha,\beta,\gamma,\delta\right)$ as the dependent
variable and we find that the reduced PDE is

${}_{III}\Delta w+\frac{\alpha}{2}w_{\alpha}+\frac{\beta}{2}w_{\beta}=0$
(8.93)

where ${}_{III}\Delta$ is the Laplace operator for metric (8.91).

It is clear that the second order PDE (8.93) admits only the Lie symmetries
$X_{2},X_{3}.$ Therefore, we do not have Type II hidden symmetries and the
symmetries $X_{1},X_{4}~{}$ are Type I hidden symmetries.

####  8.9 Conclusion

Up to now in the literature the study of Type II hidden symmetries has been
done by counter examples or by considering very special PDEs and in low
dimensional flat spaces. In this chapter we have improved this scenario and
have studied the problem of Type II hidden symmetries of second order PDEs
from a geometric of view in $n~{}$ dimensional Riemannian spaces. We have
considered the reduction of the Laplace and of the homogeneous heat equation
and the consequent possibility of existence of Type II hidden symmetries in
some general classes of spaces which admit some kind of symmetry; hence, they
admit nontrivial Lie symmetries .

For the Laplace equation, the conclusion of this study is that the Type II
hidden symmetries are directly related to the transition of the CKVs from the
space where the original equation is defined to the space where the reduced
equation resides. In this sense, we related the Lie point symmetries of PDEs
with the basic collineations of the metric i.e. the CKVs. Concerning the
general results of the reduction of Laplace equation we can summarize them as
follows:

  * •

If we reduce the Laplace equation with a gradient KV the reduced equation is a
Laplace equation in the non-decomposable space. In this case, the Type II
hidden symmetries are generated from the special and the proper CKVs of the
non-decomposable space.

  * •

If we reduce the Laplace equation with a gradient HV the reduced equation is a
Laplace equation for an appropriate metric. In this case, the Type II hidden
symmetries are generated from the HV and the special/proper CKVs.

  * •

If we reduce the Laplace equation with the symmetry generated by a sp.CKV, the
reduced equation is the Klein Gordon equation for an appropriate metric that
inherits the Lie point symmetry generated by the gradient HV. In this case,
the Type II hidden symmetries are generated from the proper CKVs.

We also considered the reduction of Laplace equation in some spaces of
interest in which the metric does not admit the symmetries of the previous
cases. In this context, we showed that the reduction with the non-gradient HV
in the Petrov type III does not give any Type II hidden symmetries. Also, it
is of interest the reduction of Laplace equation (i.e. the wave equation) in
Minkowski spaces $M^{4}$ and $M^{3}$ where we recover the results of [100] in
a straightforward manner without the need of computer software. Finally we
considered an $n-$dimensional flat FRW like space and showed that the
reduction with the proper CKV does not produce Type II hidden symmetries.

Moreover, we applied the zero order invariants of the Lie symmetries in order
to reduce the number of independent variables of the homogeneous heat equation
in certain general classes of Riemannian spaces, which admit some type of
basic symmetry. For each reduction, we determined the origin of Type II hidden
symmetries. The spaces we considered are the spaces which admit a gradient KV,
a gradient HV and finally spacetimes which admits a HV which acts simply and
transitively. For the reduction of the homogeneous heat equation and the
existence of Type II hidden symmetries, we found the following general
geometric results:

  * •

If we reduce the homogeneous heat equation via the symmetries which are
generated by a gradient KV $\left(S^{,i}\right)$ the reduced equation is a
heat equation in the nondecomposable space. In this case we have the Type II
hidden symmetry $\partial_{t}-\frac{1}{2t}w\partial_{w}$ provided we reduce
the heat equation with the symmetry $tS^{,i}-\frac{1}{2}Su\partial_{u}$.

  * •

If we reduce the homogeneous heat equation via the symmetries which are
generated by a gradient HV the reduced equation is Laplace equation for an
appropriate metric. In this case the Type II hidden symmetries are generated
from the proper CKVs.

  * •

In Petrov type III spacetime, the reduction of the homogeneous heat equation
via the symmetry generated from the nongradient HV gives a PDE which inherits
the Lie point symmetries, hence no Type II hidden symmetries are admitted.

The above results can be used in many important space-times and help
facilitate the solution of the heat equation in these space-times.

###  Appendix 8.A Proof of Corollary

Proof of Corollary 8.6.1. Using Theorem 6.5.1 and replacing $q$ we have

For case a)

$\displaystyle-
a_{\tau}w+H\left(b\right)-\frac{1}{2\tau}\left(aw+b\right)+\frac{a}{2\tau}w-\left(\psi
c_{2}w+\frac{1}{2\tau}wc_{1}\right)_{\tau}$ $\displaystyle=$ $\displaystyle
0.$ (8.94) $\displaystyle-
a_{\tau}w+H\left(b\right)-\frac{1}{2\tau}b+\frac{1}{2\tau^{2}}wc_{1}$
$\displaystyle=$ $\displaystyle 0$ (8.95)
$\displaystyle\left[-a_{\tau}+\frac{c_{1}}{2\tau^{2}}\right]w+\left[H\left(b\right)-\frac{1}{2\tau}b\right]$
$\displaystyle=$ $\displaystyle 0$ (8.96)

that is

$a=-\frac{c_{1}}{2\tau}+a_{0}~{},~{}~{}H\left(b\right)-\frac{1}{2\tau}b=0$
(8.97)

For case b)

$0=\left(-\frac{1}{2}T_{,\tau}\psi+\frac{1}{2}T_{,\tau\tau}S-F_{,\tau}\right)w-\left(2\psi
q\int Td\tau\right)_{\tau}-Tq_{,i}S^{,i}.$

then

$0=\left(-\frac{1}{2}T_{,\tau}\psi+\frac{1}{2}T_{,\tau\tau}S-F_{,\tau}\right)w+\frac{\psi}{\tau^{2}}\int
Td\tau~{}w-\frac{\psi}{\tau}Tw$ (8.98)

from here we have

$T_{,\tau\tau}=0\rightarrow~{}T=T_{0}\tau+T_{1}$ (8.99)

and

$F=-T_{0}\psi\tau.$

###  Appendix 8.B The homogeneous heat equation in the Petrov spacetimes

In the following subsections, we study the reduction of the homogeneous heat
equation in the Petrov spacetimes of type N,D and II.

#####  8.B.1 Petrov type N

The metric of the Petrov type N space-time is

$ds^{2}=dx^{2}+x^{2}dy^{2}+2d\rho dv+\ln x^{2}d\rho^{2}$ (8.100)

and has the homothetic algebra [120]

$\displaystyle K^{1}$ $\displaystyle=$
$\displaystyle\partial_{\rho}~{},~{}K^{2}=\partial_{v}~{},~{}K^{3}=\partial_{y}$
$\displaystyle H$ $\displaystyle=$ $\displaystyle
x\partial_{x}+\rho\partial_{\rho}+\left(v-2\rho\right)\partial_{v}~{}~{}\left(\psi_{H}=1\right)$

where $K^{1-3}$ are KVs and $H$ is a nongradient HV.

The heat equation (8.1) in this space-time is

$u_{xx}+\frac{1}{x^{2}}u_{yy}+2u_{\rho v}-2\ln
x^{2}~{}u_{vv}+\frac{1}{x}u_{x}-u_{t}=0.$ (8.101)

Application of Theorem 6.5.2 gives that the extra Lie point symmetries of
(8.101) are

$X_{1-3}=K_{1-3}~{}\ ,~{}X_{4}=2t\partial_{t}+H$

with nonzero commutators

$\left[X_{t},X_{4}\right]=2X_{t}$
$\left[X_{1},X_{4}\right]=X_{1}-2X_{2}~{},~{}\left[X_{2},X_{4}\right]=X_{2}.~{}$

We use $X_{4}$ to reduce the PDE because this is the Lie symmetry generated by
the HV. The zero order invariants of $X_{4}$ are

$\alpha=\frac{x}{\sqrt{t}}~{},~{}\beta=\frac{\rho}{\sqrt{t}}~{},~{}\gamma=\frac{v+\rho\ln\left(t\right)}{\sqrt{t}}~{},~{}\delta=y~{},~{}w=u.$
(8.102)

Choosing $\alpha,\beta,\gamma,\delta$ as the independent variables and
$w=w\left(\alpha,\beta,\gamma,\delta\right)$ as the dependent variable we find
that the reduced PDE is

${}_{N}\Delta w+\left(\frac{1}{2}\alpha w_{\alpha}+\frac{1}{2}\beta
w_{\beta}+\left(\frac{1}{2}\gamma-\beta\right)w_{\gamma}\right)=0.$ (8.103)

where ${}_{N}\Delta$ is the Laplace operator for the metric (8.100).

Equation (8.103) is of the form (6.14) with

$A_{ij}=g_{ij}\left(x^{k}\right),B^{i}\left(x^{k}\right)=\Gamma^{i}+\frac{1}{2}\alpha\delta_{\alpha}^{i}+\frac{1}{2}\beta\delta_{\beta}^{i}+\left(\frac{1}{2}\gamma-\beta\right)\delta_{\gamma}^{i},~{}f\left(x^{k},u\right)=0$

where $g_{ij}$ is the metric (8.100). Replacing in equations (6.21)-(6.25) we
obtain the Lie symmetry conditions for (8.103). Because $A_{ij,u}=0$ it
follows from equation (6.23) that the Lie point symmetries are generated from
the CKVs of the metric (8.100). However taking into consideration the rest of
the symmetry conditions we find that the only Lie symmetry which remains is
the one of the KV $X_{3}.$ We conclude that in this reduction we do not have
Type II hidden symmetries.

#####  8.B.2 Petrov type D

The metric of the Petrov type D space-time is

$ds^{2}=-dx^{2}+x^{-\frac{2}{3}}dy^{2}-x^{\frac{4}{3}}\left(d\rho^{2}+dz^{2}\right)$
(8.104)

with Homothetic algebra

$\displaystyle K^{1}$ $\displaystyle=$
$\displaystyle\partial_{\rho}~{},~{}K^{2}=\partial_{z}~{},~{}K^{3}=\partial_{y}~{},~{}K^{4}=z\partial_{\rho}-\rho\partial_{z}$
$\displaystyle H$ $\displaystyle=$ $\displaystyle
x\partial_{x}+\frac{4}{3}y\partial_{y}+\frac{z}{3}\partial_{z}+\frac{\rho}{3}\partial_{\rho}~{}~{}\left(\psi_{H}=1\right)$

where $K^{1-4}$ are KVs and $H$ is a nongradient HV.

In this space-time the heat equation (8.1) takes the form:

$-u_{xx}+x^{\frac{2}{3}}u_{yy}-x^{-\frac{4}{3}}\left(u_{\rho\rho}+u_{zz}\right)-\frac{1}{x}u_{x}-u_{t}=0.$
(8.105)

From Theorem 6.5.2 we have that the extra Lie point symmetries are the vectors

$X_{1-4}=K_{1-4}~{},~{}~{}X_{5}=2t\partial_{t}+H.$

with nonzero commutators:

$\displaystyle\left[X_{t},X_{5}\right]$ $\displaystyle=$ $\displaystyle
2X_{t}$ $\displaystyle\left[X_{1},X_{5}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{3}X_{1}~{},~{}\left[X_{4},X_{1}\right]=-X_{2}$
$\displaystyle\left[X_{2},X_{4}\right]$ $\displaystyle=$ $\displaystyle
X_{1}~{},~{}\left[X_{2},X_{5}\right]=\frac{1}{3}X_{2}$
$\displaystyle\left[X_{3},X_{5}\right]$ $\displaystyle=$
$\displaystyle\frac{4}{3}X_{3}.$

We use $X_{5}$ to reduce the PDE because this is the Lie symmetry generated by
the HV. The zero order invariants of $X_{5}$ are

$\alpha=\frac{x}{t^{\frac{1}{2}}}~{},~{}\beta=\frac{y}{t^{\frac{2}{3}}}~{},~{}\gamma=\frac{\rho}{t^{\frac{1}{6}}}~{},~{}\delta=\frac{z}{t^{\frac{1}{6}}}~{},~{}w=u.$

We choose $\alpha,\beta,\gamma,\delta$ as the independent variables and
$w=w\left(\alpha,\beta,\gamma,\delta\right)$ as the dependent variable and we
find that the reduced PDE is

${}_{D}\Delta w+\left(\frac{1}{2}aw_{\alpha}+\frac{2}{3}\beta
w_{\beta}+\frac{1}{6}\gamma w_{\gamma}+\frac{1}{6}\delta w_{\delta}\right)=0$
(8.106)

where ${}_{D}\Delta$ is the Laplace operator with metric (8.104).

Working again with the Lie symmetry conditions (6.21)-(6.25) we find that
equation (8.106) admits as Lie point symmetry the vector $X_{4}$ only which is
an inherited symmetry. Hence we do not have Type II hidden symmetries.
Obviously the Lie point symmetries $X_{1-4}$ are Type I hidden symmetries for
equation (8.106) for the reduction by $X_{5}.$

#####  8.B.3 Petrov type II

The metric of the Petrov type II space-time is

$ds^{2}=\rho^{-\frac{1}{2}}\left(d\rho^{2}+dz^{2}\right)-2\rho
dxdy+\rho\ln\rho~{}dy^{2}$ (8.107)

with homothetic algebra

$\displaystyle K^{1}$ $\displaystyle=$
$\displaystyle\partial_{x}~{},~{}K^{2}=\partial_{y}~{},~{}K^{3}=\partial_{z}~{}$
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(x+2y\right)\partial_{x}+\frac{1}{3}y\partial_{y}+\frac{4}{3}z\partial_{z}+\frac{4}{3}\rho\partial_{\rho}~{}\left(\psi_{H}=1\right)$

where $K^{1-4}$ are KVs and $H$ is a nongradient HV.

In this space-time equation (8.1) takes the form:

$\rho^{\frac{1}{2}}\left(u_{\rho\rho}+u_{zz}\right)-\frac{1}{\rho}\varepsilon\ln\rho
u_{xx}-\frac{2}{\rho}u_{xy}+\rho^{-\frac{1}{2}}u_{\rho}-u_{t}=0.$ (8.108)

From Theorem 6.5.2 we have that the extra Lie point symmetries are the vectors

$X_{1-3}=K_{1-3}~{},~{}~{}X_{4}=2t\partial_{t}+H$

with nonzero commutators:

$\left[X_{t},X_{4}\right]=2X_{t}$ $\displaystyle\left[X_{1},X_{4}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{3}X_{1}~{},~{}\left[X_{3},X_{4}\right]=\frac{4}{3}X_{3}$
$\displaystyle\left[X_{2},X_{4}\right]$ $\displaystyle=$
$\displaystyle\frac{2}{3}X_{1}+\frac{1}{3}X_{2}.$

We use $X_{4}$ to reduce the PDE because this is the Lie symmetry generated by
the HV. The zero order invariants of $X_{4}$ are

$\alpha=\frac{\rho}{t^{\frac{2}{3}}}~{},~{}\beta=\frac{z}{t^{\frac{2}{3}}}~{},~{}\gamma=\frac{x-\frac{1}{3}y\ln\left(t\right)}{t^{\frac{1}{6}}}~{},~{}\delta=\frac{y}{t^{\frac{1}{6}}}~{}~{},~{}w=u$

We choose $\alpha,\beta,\gamma,\delta$ as the independent variables and
$w=w\left(\alpha,\beta,\gamma,\delta\right)$ as the dependent variable and we
find that the reduced PDE is

${}_{II}\Delta w+\frac{2}{3}aw_{\alpha}+\frac{2}{3}\beta
w_{\beta}+\left(\frac{1}{3}\delta-\frac{1}{6}\gamma\right)w_{\gamma}+\frac{1}{6}\delta
w_{\delta}=0$ (8.109)

where ${}_{II}\Delta$ is the Laplace operator for metric (8.107).

From the Lie symmetry conditions (6.21)-(6.25) it follows that (8.109) does
not admit any Lie point symmetries. Hence, we do not have Type II hidden
symmetries in this case.

##  Part IV Noether symmetries and theories of gravity

###  Chapter 9 Noether symmetries in Scalar field Cosmology

####  9.1 Introduction

The detailed analysis of the cosmological data indicate that the Universe is
spatially flat and has suffered two acceleration phases. An early acceleration
phase (inflation), which occurred prior to the radiation-dominated era, and a
recently initiated accelerated expansion [125, 126, 127, 128, 129, 130, 131].
The source for the late time cosmic acceleration has been attributed to an
unidentified type of matter, the dark energy. Dark energy contrary to the
ordinary baryonic matter has a negative pressure, i.e. a negative equation of
state parameter, which counteracts the gravitational force and leads to the
observed accelerated expansion.

The simplest dark energy probe is the cosmological constant leading to the
$\Lambda$CDM cosmology [132, 133, 134]. However, it has been shown that
$\Lambda$CDM cosmology suffers from two major drawbacks known as the fine
tuning problem and the coincidence problem [135]. Besides $\Lambda$CDM
cosmology, many other candidates have been proposed in the literature. Most
are based either on the existence of new fields (i.e. a scalar field) or in
some modification of the Einstein Hilbert action (see [136]).

In addition to dark energy and the ordinary baryonic matter, it is believed
that the Universe contains a third type of matter, the dark matter. This type
of matter is assumed to be pressureless (non-relativistic) and interacts very
weakly with the standard baryonic matter. Therefore, its presence is mainly
inferred from gravitational effects on visible matter.

In the following, we consider scalar field cosmology (minimally coupled scalar
field and non minimally coupled scalar field) and we propose a geometric
principle (‘selection rule’) for specifying the potential $V(\phi)$ and the
coupling function $F\left(\phi\right)$ of the scalar field in order to solve
analytically the system of the resulting field equations. We propose that the
scalar field model should be selected by the geometric requirement that the
dynamical system of the field equations admits Noether point symmetries. The
main reason for the consideration of this hypothesis is that the Noether
symmetries provide first integrals, which assist the integrability of the
system. Furthermore, as we saw in chapter 4 the Noether symmetries are
generated from the elements of the homothetic algebra of the kinetic metric of
the Lagrangian of the theory. Therefore, with this assumption we let the
theory to select the potential, i.e. the dark energy model.

The idea to use Noether symmetries in cosmology, either on scalar field models
and on modified theories of gravity is not new and indeed a lot of attention
has been paid in the literature [137, 138, 107, 139, 140, 141, 142, 105, 86,
109, 102, 143, 144, 145, 146, 147, 148, 149]. However our approach is
geometric and more fundamental.

The structure of the present chapter is as follows. In sections 9.2 and 9.3 we
discuss the conformal equivalence of Lagrangians for scalar fields in a
Riemannian space of dimension $4$ and $n$ respectively. In particular we
enunciate a theorem which proves that the field equations for a non-minimally
coupled scalar field are the same at the conformal level with the field
equations of the minimally coupled scalar field. The necessity to preserve
Einstein’s equations in the context of Friedmann–Robertson–Walker (FRW) space-
time leads us to apply, in section 9.4, the current general analysis to the
scalar field (quintessence or phantom) spatially flat FRWcosmologies.

In section 9.5 we apply the Noether symmetry approach in non minimally coupled
scalar field in a spatially flat FRW spacetime and by using the Noether
invariants we determine analytical solutions for the field equations.
Furthermore in sections 9.6 and 9.7 we apply the same procedure for a
minimally coupled scalar field in a spatially flat FRW spacetime and in
Bianchi Class A homogeneous spacetimes.

####  9.2 Conformally equivalent Lagrangians and scalar field Cosmology

In this section we discuss the conformal equivalence of Lagrangians for scalar
fields in a general $V^{4}$ Riemannian space. The field equations in the
scalar field cosmology are derived from two different variational principles.
In the first case the scalar field $\phi$ and the gravitational field are
minimally coupled and the equations of motion follow form the action

$S_{M}=\int d\tau
dx^{3}\sqrt{-g}\left[R+\frac{1}{2}g_{ij}\phi^{;i}\phi^{;j}-V\left(\phi\right)\right].$
(9.1)

In the second case the scalar field $\psi$ (which is different from the
minimally coupled scalar field $\phi)$ interacts with the gravitational field
(non minimal coupling) and the field equations follow from the action

$S_{NM}=\int d\tau
dx^{3}\sqrt{-g}\left[F\left(\psi\right)R+\frac{1}{2}\bar{g}_{ij}\psi^{;i}\psi^{;j}-\bar{V}\left(\psi\right)\right]$
(9.2)

where $F(\psi)$ is the coupling function between the gravitational and the
scalar field $\psi$. Below we state the following proposition.

###### Proposition 9.2.1

The field equations for a non minimally coupled scalar field $\psi$ with
Lagrangian $\bar{L}\left(\tau,x^{k},\dot{x}^{k}\right)$ and coupling function
$F(\psi)$ in the gravitational field $\bar{g}_{ij}$ are the same with the
field equations of the minimally coupled scalar field $\Psi$ for a conformal
Lagrangian $L\left(\tau,x^{k},\dot{x}^{k}\right)$ in the conformal metric
$g_{ij}$ $=N^{-2}\bar{g}_{ij},$ where the conformal function
$N=\frac{1}{\sqrt{-2F\left(\psi\right)}}$ with $F\left(\psi\right)<0.$ The
inverse is also true, that is, to a minimally coupled scalar field it can be
associated a unique non minimally coupled scalar field in a conformal metric
and with a different potential function.

Proof. The action for the non minimally coupled Lagrangian
$\bar{L}\left(\tau,x^{k},x^{\prime k}\right)$ is:

$S_{NM}=\int d\tau
dx^{3}\sqrt{-\bar{g}}\left[F\left(\psi\right)\bar{R}+\frac{\varepsilon}{2}\bar{g}^{ij}\psi_{;i}\psi_{;j}-\bar{V}\left(\psi\right)\right]$
(9.3)

where $\varepsilon=1$ for a real field and $\varepsilon=-1$ for phantom field.
Let $~{}g_{ij}$ be the conformally related metric (this is not a coordinate
transformation!):

$g_{ij}=N^{-2}\bar{g}_{ij}.$

Then the action (9.3) becomes111We use the result tha if $A=(a_{ij})$ is a
$4x4$ matrix the  $\det A=\varepsilon^{ijkl}a_{ij}a_{kl}$ hence
$\bar{g}=\varepsilon^{ijkl}\bar{g}_{ij}\bar{g}_{kl}=N^{4}g.$ (9.4) :

$S_{NM}=\int d\tau
dx^{3}N^{4}\sqrt{-g}\left[F\left(\psi\right)\bar{R}+\frac{\varepsilon}{2}N^{-2}g^{ij}\psi_{;i}\psi_{;j}-\bar{V}\left(\psi\right)\right].$

Replacing [150]

$\bar{R}=N^{-2}R-2(n-1)N^{-3}\Delta_{2}N$

where $\Delta_{2}N=g_{ij}N^{;ij}$ we find:

$\displaystyle S_{NM}$ $\displaystyle=$ $\displaystyle\int d\tau
dx^{3}N^{4}\sqrt{-g}\left[F\left(\psi\right)\left(N^{-2}R-6N^{-3}\Delta_{2}N\right)+\frac{\varepsilon}{2}N^{-2}\Delta_{1}\psi-\bar{V}\left(\psi\right)\right]$
$\displaystyle=$ $\displaystyle\int d\tau
dx^{3}N^{4}\sqrt{-g}\left[F\left(\psi\right)N^{-2}R-6F\left(\psi\right)N^{-3}\Delta_{2}N+\frac{\varepsilon}{2}N^{-2}\Delta_{1}\psi-\bar{V}\left(\psi\right)\right]$
$\displaystyle=$ $\displaystyle\int d\tau
dx^{3}\sqrt{-g}\left[F\left(\psi\right)N^{2}R-6F\left(\psi\right)N\Delta_{2}N+\frac{\varepsilon}{2}N^{2}\Delta_{1}\psi-N^{4}\bar{V}\left(\psi\right)\right].$

Define the conformal function in terms of the coupling function $F(\psi)$ by
the requirement ($F\left(\psi\right)<0)$:

$N=\frac{1}{\sqrt{-2F\left(\psi\right)}}~{}.$ (9.5)

We compute

$N_{;i}=\frac{F_{\psi}\psi_{;i}}{\left(-2F\right)^{\frac{3}{2}}}.$ (9.6)

Then the first term in the integral becomes:

$\int d\tau dx^{3}\sqrt{-g}F\left(\psi\right)N^{2}R=\int d\tau
dx^{3}\sqrt{-g}\left(-\frac{R}{2}\right).$

The second term gives after integration by parts:

$\displaystyle\int d\tau
dx^{3}\sqrt{-g}\left(-6F\left(\psi\right)N\Delta_{2}N\right)$ $\displaystyle=$
$\displaystyle\int d\tau
dx^{3}\sqrt{-g}\left(-6\frac{F}{\sqrt{-2F}}N_{;ij}g^{ij}\right)$
$\displaystyle=$ $\displaystyle\int d\tau
dx^{3}\sqrt{-g}\left(-6\frac{F}{\sqrt{-2F}}\frac{1}{\sqrt{-g}}\left(\sqrt{-g}g^{ij}N_{,k}\right)_{,j}\right)$
$\displaystyle=$ $\displaystyle\int d\tau
dx^{3}\left(-6\frac{F}{\sqrt{-2F}}\left(\sqrt{-g}g^{ij}N_{,k}\right)_{,j}\right)$
$\displaystyle=$ $\displaystyle\int d\tau
dx^{3}\sqrt{-g}\left(3\frac{F_{\psi}}{\sqrt{-2F}}\psi_{;j}N_{;i}g^{ij}\right).$

Replacing $N_{;i}$ from (9.6) we find:

$\int d\tau dx^{3}\sqrt{-g}\left(-6F\left(\psi\right)N\Delta_{2}N\right)=\int
d\tau
dx^{3}\sqrt{-g}\left(3\frac{F_{\psi}^{2}}{\left(-2F\right)^{2}}\psi_{;i}\psi_{;j}g^{ij}\right).$

The third term gives:

$\frac{\varepsilon}{2}N^{2}\Delta_{1}\psi=\frac{\varepsilon}{4F}\psi_{;i}\psi_{;j}g^{ij}$

Collecting all the results we find:

$\displaystyle S_{MN}$ $\displaystyle=$ $\displaystyle\int d\tau
dx^{3}\sqrt{-g}\left[-\frac{R}{2}+3\frac{F_{\psi}^{2}}{\left(-2F\right)^{2}}\psi_{;i}\psi_{;j}g^{ij}-\frac{\varepsilon}{4F}\psi_{;i}\psi_{;j}g^{ij}-\frac{\bar{V}\left(\psi\right)}{4F^{2}}\right]$
$\displaystyle=$ $\displaystyle\int d\tau
dx^{3}\sqrt{-g}\left[-\frac{R}{2}+3\frac{F_{\psi}^{2}}{4F^{2}}\psi_{;i}\psi_{;j}g^{ij}-\frac{\varepsilon}{4F}\psi_{;i}\psi_{;j}g^{ij}-\frac{\bar{V}\left(\psi\right)}{4F^{2}}\right]$

or

$S_{MN}=\int d\tau
dx^{3}\sqrt{-g}\left[-\frac{R}{2}+\frac{\varepsilon}{2}\left(\frac{3\varepsilon
F_{\psi}^{2}-F}{2F^{2}}\right)\psi_{;i}\psi_{;j}g^{ij}-\frac{\bar{V}\left(\psi\right)}{4F^{2}}\right].$
(9.7)

We introduce the scalar field $\Psi$ with the requirement:

$d\Psi=\sqrt{\left(\frac{3\varepsilon F_{\psi}^{2}-F}{2F^{2}}\right)}d\psi$
(9.8)

and the action becomes

$S_{MN}=\int d\tau
dx^{3}\sqrt{-g}\left[-\frac{R}{2}+\frac{\varepsilon}{2}\Psi_{;i}\Psi_{;j}g^{ij}-\frac{\bar{V}\left(\Psi\right)}{4F\left(\Psi\right)^{2}}\right].$
(9.9)

We conclude that the scalar field $\Psi$ is minimally coupled to the
gravitational field. Therefore we have proved that to every non minimally
coupled scalar field we may associate a unique minimally coupled scalar field
in a conformally related space and an appropriate potential. Since all
considerations are reversible, the result is reversible.

In the following section we extend the above proposition to a general $V^{n}$
Riemannian space.

####  9.3 Generalization to dimension $n$

Consider the non minimally coupled scalar field $\psi$ whose field equations
are obtained from the action:

$\displaystyle S_{NM}$ $\displaystyle=$ $\displaystyle\int
dx^{n}N^{n}\sqrt{-g}\left[F\left(\psi\right)\bar{R}+\frac{\varepsilon}{2}N^{-2}g^{ij}\psi_{,i}\psi_{,j}-\bar{V}\left(\psi\right)\right]$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\left[\begin{array}[]{c}F\left(\psi\right)N^{n-2}R-2(n-1)F\left(\psi\right)N^{n-3}\Delta_{2}N+\\\
-F\left(\psi\right)N^{n}(n-1)(n-4)\Delta_{1}N+\frac{\varepsilon}{2}N^{n-2}g^{ij}\psi_{,i}\psi_{,j}-N^{n}\bar{V}\left(\psi\right)\end{array}\right]$

where we have substituted [150]

$\bar{R}=N^{-2}R-2(n-1)N^{-3}\Delta_{2}N-(n-1)(n-4)\Delta_{1}N$

and

$\displaystyle\Delta_{1}N$ $\displaystyle=$ $\displaystyle g_{ij}N^{,i}N^{,j}$
$\displaystyle\Delta_{2}N$ $\displaystyle=$ $\displaystyle g_{ij}N^{;ij}.$

Define the function $N(x^{i})$ in terms of the coupling function $F(\psi)$ by
the requirement:

$N^{n-2}=\frac{1}{-2F}.~{},~{}F=-\frac{N^{2-n}}{2}.$

For each term of the action $S_{NM}$ we have the following:
The first term gives:

$\int dx^{n}\sqrt{-g}\left(F\left(\psi\right)N^{n-2}R\right)=\int
dx^{n}\sqrt{-g}\left(-\frac{R}{2}\right).$

The second term gives:

$\displaystyle\int
dx^{n}\sqrt{-g}\left(-2(n-1)F\left(\psi\right)N^{n-3}N_{;ij}g^{ij}\right)$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\left((n-1)N^{2-n}N^{n-3}N_{;ij}g^{ij}\right)$ $\displaystyle=$
$\displaystyle\int
dx^{n}\sqrt{-g}\left((n-1)N^{-1}\frac{1}{\sqrt{-g}}\left(\sqrt{-g}g^{ij}N_{,k}\right)_{,j}\right)$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\left((n-1)N^{-1}N_{;ij}g^{ij}\right)$ $\displaystyle=$
$\displaystyle\int
dx^{n}\sqrt{-g}\left(-(n-1)\left(N^{-1}\right)_{;j}N_{;i}g^{ij}\right).$

Furthermore we compute

$N=\frac{1}{\left(-2F\right)^{\frac{1}{n-2}}}\rightarrow
N^{-1}=\left(-2F\right)^{\frac{1}{n-2}}$
$N_{;i}N_{;j}^{-1}=-\frac{F_{\psi}^{2}}{\left(n-2\right)^{2}F^{2}}\psi_{;i}\psi_{;j}.$

Replacing we find for the second term:

$\int
dx^{n}\sqrt{-g}\left(-2(n-1)F\left(\psi\right)N^{n-3}N_{;ij}g^{ij}\right)=\int
dx^{n}\sqrt{-g}\left(\frac{(n-1)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{F^{2}}\psi_{;i}\psi_{;j}g^{ij}\right).$

(note that this is the same with the previous expression for $n=4$).
The third term gives:

$\displaystyle\int
dx^{n}\sqrt{-g}\left(F\left(\psi\right)N^{n}(n-1)(n-4)\Delta_{1}N\right)$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\left(-\frac{N^{2-n}}{2}N^{n}(n-1)(n-4)\Delta_{1}N\right)$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\left(-\frac{1}{2}N^{2}(n-1)(n-4)\Delta_{1}N\right)$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\left(-\frac{1}{2}\frac{(n-1)(n-4)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{\left(-2F\right)^{\frac{4}{2-n}}F^{2}}\psi_{;i}\psi_{;j}g^{ij}\right).$

Finally the the fourth term gives:

$\int
dx^{n}\sqrt{-g}\left(\frac{\varepsilon}{2}N^{n-2}g^{ij}\psi_{,i}\psi_{,j}\right)=\int
dx^{n}\sqrt{-g}\left(-\frac{\varepsilon}{4}\frac{1}{F}g^{ij}\psi_{;i}\psi_{;j}\right).$

Collecting the results for the last three terms we find

$\displaystyle\int
dx^{n}\sqrt{-g}\left(\frac{(n-1)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{F^{2}}\psi_{;i}\psi_{;j}g^{ij}-\frac{1}{2}\frac{(n-1)(n-4)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{\left(-2F\right)^{\frac{4}{2-n}}F^{2}}\psi_{;i}\psi_{;j}g^{ij}-\varepsilon\frac{1}{4F}g^{ij}\psi_{;i}\psi_{;j}\right)$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\left(\frac{(n-1)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{F^{2}}-\frac{1}{2}\frac{(n-1)(n-4)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{\left(-2F\right)^{\frac{4}{2-n}}F^{2}}-\varepsilon\frac{1}{4F}\right)\psi_{;i}\psi_{;j}g^{ij}$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\frac{\varepsilon}{2}\left(\frac{2\varepsilon(n-1)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{F^{2}}-\varepsilon\frac{(n-1)(n-4)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{\left(-2F\right)^{\frac{4}{2-n}}F^{2}}-\frac{1}{2F}\right)\psi_{,i}\psi_{,j}g^{ij}.$

We define the new scalar field $\Psi$ with the requirement

$d\Psi=\left(\frac{2\varepsilon(n-1)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{F^{2}}-\varepsilon\frac{(n-1)(n-4)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{\left(-2F\right)^{\frac{4}{2-n}}F^{2}}-\frac{1}{2F}\right)^{\frac{1}{2}}d\psi.$

In terms of $\Psi\ $the action becomes

$\displaystyle\int
dx^{n}\sqrt{-g}\frac{\varepsilon}{2}\left(\frac{2\varepsilon(n-1)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{F^{2}}-\varepsilon\frac{(n-1)(n-4)}{\left(n-2\right)^{2}}\frac{F_{\psi}^{2}}{\left(-2F\right)^{\frac{4}{2-n}}F^{2}}-\frac{1}{2F}\right)\psi_{,i}\psi_{,j}g^{ij}.$
$\displaystyle=$ $\displaystyle\int
dx^{n}\sqrt{-g}\left(\frac{\varepsilon}{2}\Psi_{;i}\Psi_{;j}g^{ij}\right).$

Collecting the above we find the action $S_{M}$ of a minimally coupled scalar
field

$S_{M}=\int
dx^{n}\sqrt{-g}\left(-\frac{R}{2}+\frac{\varepsilon}{2}\Psi_{;i}\Psi_{;j}g^{ij}-\frac{\bar{V}\left(\Psi\right)}{\left(-2F\right)^{\frac{n}{n-2}}}\right).$

We note that the new scalar field $\Psi$ is minimally coupled to the
gravitational field $g_{ij}$ and that the potential of $\Psi$ is
$\frac{\bar{V}\left(\Psi\right)}{\left(-2F\right)^{\frac{n}{n-2}}}$.

The above proof agrees with the one given in the paper of [151]. However it is
obviously simpler, more direct and clear.

####  9.4 Conformal Lagrangians in scalar field cosmology

In this section we apply the conformal transformation in the Lagrangian of the
field equations in a FRW spatially flat spacetime.

We consider the flat FRW $\left(K=0\right)$ spacetime whose metric is

$ds^{2}=-dt^{2}+a^{2}\left(t\right)\delta_{ij}dx^{i}dx^{j}$ (9.11)

where $\delta_{ij}$ is the 3-space metric in Cartesian coordinates. The
Lagrangian of a scalar field $\phi$ minimally coupled to gravity in these
coordinates is

$L_{M}\left(a,\dot{a},\phi,\dot{\phi}\right)=-3a\dot{a}^{2}+\frac{\varepsilon}{2}a^{2}\dot{\phi}^{2}-a^{3}V\left(\phi\right).$
(9.12)

The Lagrangian for a non minimally coupled scalar field $\psi$ is

$L_{NM}\left(a,\dot{a},\psi,\dot{\psi}\right)=6F\left(\psi\right)a\dot{a}^{2}+6F_{\psi}\left(\psi\right)a^{2}\dot{a}\dot{\psi}+\frac{\varepsilon}{2}a^{3}\dot{\psi}^{2}-a^{3}V\left(\psi\right)$
(9.13)

where $\ F(\psi)<0$ is the coupling function and $\varepsilon=\pm 1$ where
$\varepsilon=1$ for real scalar field and $\varepsilon=-1$ for phantom field.
The Hamiltonian for the Lagrangian (9.13) is

$E=6F\left(\psi\right)a\dot{a}^{2}+6F_{\psi}\left(\psi\right)a^{2}\dot{a}\dot{\psi}+\frac{\varepsilon}{2}a^{3}\dot{\psi}^{2}+a^{3}V\left(\psi\right).$
(9.14)

We construct a conformal Lagrangian which corresponds to a minimally coupled
scalar field.

To do that we consider first a change in the scale factor from $a(t)$
$\rightarrow A(t)$ defined by the formula (see [152, 140])

$A\left(t\right)=\sqrt{-2F}a\left(t\right)\text{ }$ (9.15)

Then the Lagrangian (9.13) takes the form:

$L_{NM}\left(A,\dot{A},\psi,\dot{\psi}\right)=\frac{1}{\sqrt{-2F}}\left[-3A\dot{A}^{2}+\frac{\varepsilon}{2}\left(\frac{3\varepsilon
F_{\psi}^{2}-F}{2F^{2}}\right)A^{3}~{}\dot{\psi}^{2}\right]-\frac{A^{3}}{\left(-2F\right)^{\frac{3}{2}}}V\left(\psi\right)$
(9.16)

that is, the cross term $\dot{a}\dot{\psi}$ disappears.

Next we consider the coordinate transformation:

$d\Psi=\sqrt{\left(\frac{3\varepsilon
F_{\psi}^{2}-F}{2F^{2}}\right)}~{}d\psi~{}$ (9.17)

and Lagrangian (9.16) becomes:

$L_{NM}\left(A,A^{\prime},\Psi,\Psi^{\prime}\right)=\frac{1}{\sqrt{-2F}}\left[-3A\dot{A}^{2}+\frac{\varepsilon}{2}A^{3}~{}\dot{\Psi}^{2}\right]~{},~{}\bar{V}\left(\psi\right)=\frac{V\left(\Psi\right)}{\left(-2F\right)^{\frac{3}{2}}}$
(9.18)

The form of the Lagrangian (9.18) is (9.12) hence the previous result applies
and under the conformal transformation

$d\tau=\sqrt{-2F\left(\psi\right)}dt$ (9.19)

the Lagrangian (9.18) becomes:

$L_{M}\left(A,\dot{A},\Psi,\Psi^{\prime}\right)=-3AA^{\prime
2}+\frac{\varepsilon}{2}A^{3}\Psi^{\prime
2}-\frac{A^{3}}{\left(-2F\right)^{\frac{3}{2}}}V\left(\Psi\right)$ (9.20)

where a prime ′ indicates derivative wrt the new coordinate $\tau.$

We note that if in the new coordinates $\tau,x^{i}$ we consider the metric

$d\bar{s}^{2}=-d\tau^{2}+A^{2}\left(\tau\right)\delta_{ij}dx^{i}dx^{j}$ (9.21)

then the term $3AA^{\prime 2}$ equals the Ricci scalar $\bar{R}$ of the
conformally flat metric $d\bar{s}^{2}.$ Therefore the Lagrangian (9.20) can be
seen as the Lagrangian of a scalar field $\Psi$ of potential
$\bar{V}\left(\Psi\right)$ minimally coupled to the gravitational filed
$\bar{g}_{ij}$ in the space with metric $d\bar{s}^{2}.$ Replacing the
coordinate $\tau$ and the quantity $A\left(\tau\right)$ from (9.15), (9.19),
we find:

$d\bar{s}^{2}=\sqrt{-2F}(-dt^{2}+a^{2}\left(t\right)\delta_{ij}dx^{i}dx^{j})=\sqrt{-2F}ds^{2}$
(9.22)

that is, the metric $d\bar{s}^{2}$ is conformally related to the metric
$ds^{2}$ with conformal function $\sqrt{-2F}.$ This means that the non-
minimally coupled scalar field in the gravitational field $ds^{2}$ is
equivalent to a minimally coupled scalar field - with appropriate potential
defined in terms of the conformal function - in the gravitational field
$d\bar{s},$ the resutl is reversible. Equivalently the Lagrangians $L_{M},$
$L_{NM}$ are conformally related and the field equations (the Euler-Lagrange
equations) are invariant under the conformal transformation if the Hamiltonian
constrains $H_{M},~{}H_{NM}$ vanish (see Lemma 7.2.2).

In the following sections we apply the Noether symmetry approach as a
geometric selection rule in order to determine the dark energy models; that
is, we search for dark energy models by requiring the field equations to admit
Noether point symmetries.

####  9.5 Noether point symmetries of a non minimally coupled Scalar field.

Consider a non-minimally coupled scalar field with action

$S_{NM}=\int d\tau
dx^{3}\sqrt{-g}\left[F\left(\psi\right)R+\frac{\varepsilon}{2}\bar{g}_{ij}\psi^{;i}\psi^{;j}-\bar{V}\left(\psi\right)\right]+\int
L_{m}d\tau dx^{3}$

in a flat FRW space-time, whose metric in Cartesian coordinates is

$ds^{2}=-dt^{2}+a^{2}\left(t\right)\delta_{ij}dx^{i}dx^{j}$

and $L_{m}$ is the Lagrangian of dust matter of density $\rho_{D}$ (for
commoving observers).

The Lagrangian of the field equations is (9.13) are the Hamiltonian (total
energy density) (9.14) and the Euler-Lagrange equations

$\frac{d}{dt}\frac{\partial
L_{NM}}{\partial\left(\dot{a},\dot{\psi}\right)}-\frac{\partial
L}{\partial\left(a,\psi\right)}=0.$

If the Hamiltonian (9.14) is $E\neq 0$ then the space admits dust which,
however does not interact with the scalar field and has energy density
$\rho_{D}=\frac{\left|E\right|}{a^{3}}.$If $E\,$ $=0$ the space does not admit
dust. In the following we determine all potentials $V\left(\psi\right)$ for
which this dynamical system admits Noether point symmetries beyond the trivial
one $\partial_{t}$. Subsequently we use the resulting Noether integrals to
find analytical solutions for the field equations for each of these
potentials.

In order to determine the Noether point symmetries of the Lagrangian (9.13) we
shall follow the results of chapter 4. That is we brake the Lagrangian in the
kinematic part which defines the kinematic metric and the remaining part which
we consider to be the potential. Then we apply theorem 4.3.2 which states that
the Noether point symmetries of the Lagrangian follow form the homothetic
algebra of the kinematic metric. The kinematic metric admits a non-trivial
homothetic (not necessarily proper homothetic) algebra if a given condition is
satisfied which involves the symmetry vector and the potential. The solution
of this relation provides all the potentials for which extra Noether
symmetries are admitted. We use the Noether integrals of these extra Noether
symmetries to find an analytic solution for each of the corresponding
potentials..

From the Lagrangian (9.13) we define the kinematic metric:

$ds_{KM}^{2}=12F\left(\psi\right)a\dot{a}^{2}+12F_{\psi}\left(\psi\right)a^{2}\dot{a}\dot{\psi}+\varepsilon
a^{3}\dot{\psi}^{2}.$ (9.23)

This is a two dimensional metric in the space $(a,\psi).$ Because the
homothetic algebra of a 2 dimensional metric is different for a flat and a
non-flat (but conformally flat because all two dimensional spaces are
conformally flat) we consider the case the metric (9.23) is maximally
symmetric222The Ricci scalar is $R=K~{}$where $K$ is a constant.

The Ricci scalar of the metric (9.23) is computed to be:

$R_{\left(KM\right)}=\frac{\varepsilon}{4a^{3}}\frac{\left(2F_{\psi\psi}F-F_{\psi}^{2}\right)}{\left(\varepsilon
F-3F_{\psi}^{2}\right)^{2}}$ (9.24)

Hence if the metric (9.23) is maximally symmetric then it follows that
$R_{\left(KM\right)}=0$, that is, the kinetic metric (9.23) must be flat.

#####  9.5.1 Case A. $R_{KM}=0$

Condition $R_{KM}=0$ and (9.24) give:

$2F_{\psi\psi}F-F_{\psi}^{2}=0$ (9.25)

provided

$\varepsilon F-3F_{\psi}^{2}\neq 0.$ (9.26)

The solution of (9.25) is

$F\left(\psi\right)=-\frac{F_{0}\varepsilon}{12}\left(\psi+\psi_{0}\right)^{2}$
(9.27)

where $\varepsilon=\pm 1$ and $F_{0}\varepsilon>0$. We note that this
$F\left(\psi\right)$ satisfies condition (9.26) therefore it is acceptable.

In order to determine the homothetic algebra of the kinematic metric (9.23) we
write it in a more familiar form. We introduce the coordinates $A,\Psi$ by the
relations

$\displaystyle A$ $\displaystyle=$ $\displaystyle\sqrt{-2F}a$ (9.28)
$\displaystyle d\Psi$ $\displaystyle=$
$\displaystyle\sqrt{\left(\frac{3\varepsilon
F_{\psi}^{2}-F}{2F^{2}}\right)}~{}d\psi.$ (9.29)

In the new coordinates $A,\Psi$ the metric (9.23), the Lagrangian (9.13) and
the non minimal coupling function $F\left(\Psi\right)$ take the following form

For $\varepsilon=1$ and $F_{0}>0~{}$

$\displaystyle ds_{KM}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-2F\left(\Psi\right)}}\left[-3A\dot{A}^{2}+\frac{1}{2}A^{3}\dot{\Psi}^{2}\right]$
(9.30) $\displaystyle L_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-2F\left(\Psi\right)}}\left[-3A\dot{A}^{2}+\frac{1}{2}A^{3}\dot{\Psi}^{2}\right]-\frac{A^{3}}{\left(-2F\left(\Psi\right)\right)^{\frac{3}{2}}}V\left(\Psi\right)~{}.$
(9.31) $\displaystyle F\left(\Psi\right)$ $\displaystyle=$
$\displaystyle-\frac{F_{0}}{12}\exp\left(\pm\frac{\sqrt{6}}{3}\sqrt{\frac{F_{0}}{F_{0}+1}}\Psi\right)~{},~{}F_{0}>0$
(9.32)

For $\varepsilon=-1~{}\,$ and $-1<F_{0}<0~{}$

$\displaystyle ds_{KM}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-2F\left(\Psi^{\ast}\right)}}\left[-3A\dot{A}^{2}-\frac{1}{2}A^{3}\dot{\Psi}^{2}\right]$
(9.33) $\displaystyle L_{NM}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-2F\left(\Psi^{\ast}\right)}}\left[-3A\dot{A}^{2}-\frac{1}{2}A^{3}\dot{\Psi}^{2}\right]-\frac{A^{3}}{\left(-2F\left(\Psi\right)\right)^{\frac{3}{2}}}V\left(\Psi\right)$
(9.34) $\displaystyle F\left(\Psi\right)$ $\displaystyle=$
$\displaystyle-\frac{\left|F_{0}\right|}{12}\exp\left(\pm\frac{\sqrt{6}}{3}\sqrt{\frac{\left|F_{0}\right|}{1-\left|F_{0}\right|}}\Psi\right)~{},~{}-1<F_{0}<0$
(9.35)

For $\varepsilon=-1~{}$and $F_{0}<-1$ we introduce a new real field
$\Psi^{\ast}=i\Psi$ and get the real field $F\left(\Psi^{\ast}\right)$

$\displaystyle ds_{KM}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-2F\left(\Psi^{\ast}\right)}}\left[-3A\dot{A}^{2}+\frac{1}{2}A^{3}\dot{\Psi}^{\ast
2}\right]$ (9.36) $\displaystyle L_{NM}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-2F\left(\Psi^{\ast}\right)}}\left[-3A\dot{A}^{2}+\frac{1}{2}A^{3}\dot{\Psi}^{\ast
2}\right]-\frac{A^{3}}{\left(-2F\left(\Psi^{\ast}\right)\right)^{\frac{3}{2}}}V\left(\Psi^{\ast}\right)$
(9.37) $\displaystyle F\left(\Psi^{\ast}\right)$ $\displaystyle=$
$\displaystyle-\frac{\left|F_{0}\right|}{12}\exp\left(\mp\frac{\sqrt{6}}{3}\sqrt{\frac{\left|F_{0}\right|}{\left|F_{0}\right|-1}}~{}\Psi^{\ast}\right)~{},~{}F_{0}<-1.$
(9.38)

We see that in the region $-\infty<F_{0}<-1$ starting from a phantom field
$\Psi$ we end up with a real scalar field $\Psi^{\ast}.$

In order to consider the three Lagrangians (9.31),(9.34),(9.37) at the same
time we consider the Lagrangian

$L=N^{2}\left(\Psi\right)\left[-3A\dot{A}^{2}+\frac{\varepsilon_{k}}{2}A^{3}\dot{\Psi}^{2}\right]-A^{3}\bar{V}\left(\Psi\right)$
(9.39)

where $N^{2}\left(\Psi\right)=\frac{1}{\sqrt{-2F\left(\Psi\right)}}~{}$and
$\bar{V}\left(\Psi\right)=\frac{V\left(\Psi\right)}{\left(-2F\left(\Psi\right)\right)^{\frac{3}{2}}}$
where $F\left(\Psi\right)<0$. The constant $\varepsilon_{k}=\pm 1$ where the
value $+1$ is for $\varepsilon=1,~{}F_{0}>0~{}$ and
$\varepsilon=-1~{},~{}F_{0}<-1$ and the value $-1$ is for
$\varepsilon=-1~{},~{}-1<F_{0}<0.$ Then the new kinematic metric is written as

$ds_{KM}^{2}=N^{2}\left(\Psi\right)\left[-3A\dot{A}^{2}+\frac{\varepsilon_{k}}{2}A^{3}\dot{\Psi}^{2}\right].$
(9.40)

We simplify this metric by introducing new coordinates $r,\theta$ defined by
the transformation:

$r=\sqrt{\frac{8}{3}}A^{\frac{3}{2}}~{},~{}\theta=\sqrt{\frac{3\varepsilon_{k}}{8}}~{}\Psi$
(9.41)

This step is necessary in order to deduce the homothetic algebra of the metric
from well known previous results. In the new coordinates the metric (9.40)
takes the simple form:

$ds_{KM}^{2}=N^{2}\left(\theta\right)\left(-dr^{2}+r^{2}d\theta^{2}\right).$
(9.42)

that is, it is directly related to the flat 2d Lorentzian space with metric

$ds^{2}=-dr^{2}+r^{2}d\theta^{2}$

with conformal factor $N^{2}\left(\theta\right).$ In the new coordinates the
curvature scalar is

$R_{KM}=-\frac{2}{r^{4}N^{3}\left(\theta\right)}\left(N_{,\theta\theta}-\frac{1}{N\left(\theta\right)}\left(N_{,\theta}\right)^{2}\right)$
(9.43)

hence the condition $R_{KM}=0$ gives the function $N\left(\theta\right)$:

$N\left(\theta\right)=N_{0}e^{k\theta}~{},~{}k\in\mathbb{C}~{},N_{0}\in\mathbb{R}$
(9.44)

where $k$ is a new constant.

Taking this into account we have that in the $r,\theta$ coordinates the
Lagrangian (9.39) becomes:

$L=N_{0}^{2}e^{2k\theta}\left(-\frac{1}{2}\dot{r}^{2}+\frac{1}{2}r^{2}\dot{\theta}^{2}\right)-r^{2}\bar{V}\left(\theta\right).$
(9.45)

The constant $k$ is related to the previous constant $F_{0}$ via the function
$N(\theta)$. Using (9.41) we express $N\left(\theta\right)$ in terms of
$\Psi$:

$N\left(\Psi\right)=N_{0}e^{k\sqrt{\frac{3\varepsilon_{k}}{8}}\Psi}.$ (9.46)

Comparing with $F\left(\Psi\right)$ and eliminating $N\left(\Psi\right)$ we
find:

$F\left(\Psi\right)=-\frac{1}{2N_{0}^{4}}e^{-4k~{}\sqrt{\frac{3\varepsilon}{8}}~{}\Psi}.$
(9.47)

This is a second expression of $F\left(\Psi\right)$ in terms of the constant
$k.$ Comparing with the previous expression (9.32),(9.35) and (9.38), which
expresses $F\left(\Psi\right)$ in terms of $F_{0}$ (and holds for all ranges
of values of $F_{0}!),$ we find:

$-\frac{F_{0}\varepsilon}{12}\exp\left(\pm\frac{\sqrt{6}}{3}\sqrt{\frac{\varepsilon
F_{0}}{F_{0}+1}}\Psi\right)=-\frac{1}{2N_{0}^{4}}e^{-4k~{}\sqrt{\frac{3\varepsilon_{k}}{8}}~{}\Psi}.$
(9.48)

This relation must hold identically which leads to the conditions

$~{}N_{0}=\left(\frac{6}{\varepsilon F_{0}}\right)^{1/4}$ (9.49)

and

$~{}\left|k\right|=\frac{1}{3}\sqrt{\frac{\varepsilon F_{0}}{F_{0}+1}}.$
(9.50)

In Appendix 9.A we give the relation between the various ranges of the
constant $F_{0}$ and the corresponding ranges of the constant
$\left|k\right|.$

###### Case $\left|k\right|\neq 1$.

For $\left|k\right|\neq 1$ the homothetic algebra consists of the gradient KVs
vectors

$\displaystyle K^{1}$ $\displaystyle=$
$\displaystyle\frac{e^{\left(1-k\right)\theta}r^{k}}{N_{0}^{2}}\left(-\partial_{r}+\frac{1}{r}\partial_{\theta}\right)$
(9.51) $\displaystyle K^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{-\left(1+k\right)\theta}r^{-k}}{N_{0}^{2}}\left(\partial_{r}+\frac{1}{r}\partial_{\theta}\right)$
(9.52)

the non gradient KV

$K^{3}=r\partial_{r}-\frac{1}{k}\partial_{\theta}$ (9.53)

and the gradient HV

$H^{i}=\frac{1}{N_{0}^{2}\left(k^{2}-1\right)}\left(-r\partial_{r}+k\partial_{\theta}\right)~{},~{}H\left(r,\theta\right)=\frac{1}{2}\frac{r^{2}e^{2k\theta}}{k^{2}-1}.$
(9.54)

Using the results of chapter 4 we find the following results

  1. 1.

The gradient KV $K^{1}$ produces Noether symmetries for the following
potentials

a) For $V\left(\theta\right)=V_{0}e^{2\theta}$ we have the Noether symmetries
$K^{1},~{}tK^{1}$ with Noether integrals

$I_{1}=\frac{d}{dt}\left(\frac{r^{1+k}e^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)~{},~{}I_{2}=t\frac{d}{dt}\left(\frac{r^{1+k}e^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)-\left(\frac{r^{1+k}e^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)$
(9.55)

b) For
$V\left(\theta\right)=V_{0}e^{2\theta}-\frac{mN_{0}^{2}}{2\left(k^{2}-1\right)}e^{2k\theta}$
we have the Noether symmetries $e^{\pm\sqrt{m}t}K^{1}$ $m=$constant, with
Noether integrals

$I_{\pm}^{\prime}=e^{\pm\sqrt{m}t}\left[\frac{d}{dt}\left(\frac{r^{1+k}e^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)\mp\sqrt{m}\left(\frac{r^{1+k}e^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)\right]$
(9.56)

From the Noether integrals we construct the time independent first integral
$I_{K^{1}}=I_{+}I_{-}.$

  2. 2.

The gradient KV $K^{2}$ produces the following Noether symmetries for the
following potentials

a) For $V\left(\theta\right)=V_{0}e^{-2\theta}$,we have the Noether symmetries
$K^{1},~{}tK^{1}$ with Noether integrals

$J_{1}=\frac{d}{dt}\left(\frac{r^{1-k}e^{-\left(1-k\right)\theta}}{k-1}\right)~{},~{}J_{2}=t\frac{d}{dt}\left(\frac{r^{1-k}e^{-\left(1-k\right)\theta}}{k-1}\right)-\frac{r^{1-k}e^{-\left(1-k\right)\theta}}{k-1}$
(9.57)

b) For
$V\left(\theta\right)=V_{0}e^{-2\theta}-\frac{mN_{0}^{2}}{2\left(k^{2}-1\right)}e^{2k\theta}$,
we have the Noether symmetries $e^{\pm\sqrt{m}t}K^{2}$ $m=$constant, with
Noether integrals

$J_{\pm}^{{}^{\prime}}=e^{\pm\sqrt{m}t}\left[\frac{d}{dt}\left(\frac{r^{1-k}e^{-\left(1-k\right)\theta}}{k-1}\right)\mp\sqrt{m}\frac{r^{1-k}e^{-\left(1-k\right)\theta}}{k-1}\right]$
(9.58)

From the Noether integrals we construct the time independent first integral
$J_{K^{2}}=J_{+}^{\prime}J_{-}^{\prime}.$

  3. 3.

The non gradient KV $K^{3}$ produces a Noether symmetry for the potential
$V\left(\theta\right)=V_{0}e^{2k\theta}$ with Noether integral

$I_{3}=\frac{re^{2k\theta}}{k}\left(k\dot{r}+r\dot{\theta}\right).$ (9.59)

  4. 4.

The gradient HV produces the following Noether symmetries for the following
potentials

a) For $V\left(\theta\right)=V_{0}e^{-2\frac{\left(k^{2}-2\right)}{k}\theta}$
, $k^{2}-2\neq 0$ we have the Noether symmetries
$2t\partial_{t}+H^{i}~{},~{}t^{2}\partial_{t}+tH^{i}$ with Noether integrals

$I_{H_{1}}=2tE-\frac{d}{dt}\left(\frac{1}{2}\frac{r^{2}e^{2k\theta}}{k^{2}-1}\right)~{},~{}I_{H_{2}}=t^{2}E-t\frac{d}{dt}\left(\frac{1}{2}\frac{r^{2}e^{2k\theta}}{k^{2}-1}\right)+\frac{1}{2}\frac{r^{2}e^{2k\theta}}{k^{2}-1}.$
(9.60)

We note that in this case the system is the Ermakov-Pinney dynamical system
(because it admits the Noether symmetry algebra the $sl(2,R),\ $hence the Lie
symmetry algebra is at least $sl(2,R))$ .

b) For
$V\left(\theta\right)=V_{0}e^{-2\frac{\left(k^{2}-2\right)}{k}\theta}-\frac{N_{0}^{2}m}{k^{2}-1}e^{2k\theta}~{}$
, $k^{2}-2\neq 0$ we have the Noether symmetries
$\frac{2}{\sqrt{m}}e^{\pm\sqrt{m}t}\partial_{t}\pm e^{\pm\sqrt{m}t}H^{i}$ ,
$m=$constant with Noether integrals

$I_{+,-}=e^{\pm
2\sqrt{m}t}\left(\frac{1}{\sqrt{m}}E\mp\frac{d}{dt}\left(\frac{1}{2}\frac{r^{2}e^{2k\theta}}{k^{2}-1}\right)+2\sqrt{m}\left(\frac{1}{2}\frac{r^{2}e^{2k\theta}}{k^{2}-1}\right)\right)$
(9.61)

For this potential the Noether symmetries form the $sl\left(2,R\right)$ Lie
algebra, i.e the dynamical system is the two dimensional Kepler-Ermakov system
Therefore it admits the Ermakov - Pinney invariant which we may construct with
the use of the Noether symmetries or with the use of the corresponding Killing
Tensor (see Proposition 5.6.2)

  5. 5.

The case $V\left(\theta\right)=0$ corresponds the free particle (see chapter
2).

###### Case $\left|k\right|=1$

We have to consider two cases i.e. $k=1$ and $k=-1.$

###### Case $k=1$

The KVs of the kinematic metric are the $K_{k=1}^{1,2}$ of (9.51,9.52) and the
vector

$K_{k=1}^{3}=-r\left(\ln\left(re^{-\theta}\right)-1\right)\partial_{r}+\ln\left(re^{-\theta}\right)\partial_{\theta}.$
(9.62)

The vectors $K_{k=1}^{1,2}$ are gradient and $K^{3}$ is non-gradient. The HV
is gradient and it is given by

$H^{i}=\frac{1}{4}r\left(2\ln\left(re^{-\theta}\right)+3\right)\partial_{r}-\frac{1}{2}\left(\ln\left(re^{-\theta}\right)+\frac{1}{2}\right)\partial_{\theta}$
(9.63)

###### Case $k=-1$

The KVs of the kinematic metric are $K_{k=-1}^{1,2}$ of (9.51,9.52) and the
vector

$\bar{K}^{3}=r\left(\ln\left(re^{\theta}\right)-1\right)\partial_{r}+\ln\left(re^{\theta}\right)\partial_{\theta}.$
(9.64)

The vectors $\bar{K}^{1,2}$ are gradient and $\bar{K}^{3}$ is non-gradient.
The gradient HV is given by

$\bar{H}^{i}=\frac{1}{4}r\left(2\ln\left(re^{\theta}\right)+3\right)\partial_{r}+\frac{1}{2}\left(\ln\left(re^{\theta}\right)+\frac{1}{2}\right)\partial_{\theta}$
(9.65)

In the following we consider only the case $k=1$. The results for the case
$k=-1$ are found if we make the change
$\theta_{\left(k=-1\right)}=-\bar{\theta}$.

Using theorem 4.3.2 and making simple calculations we find the following
results

  1. 1.

Noether symmetries generated by the KV $K^{1}$.

a) If $V\left(\theta\right)=V_{0}e^{2\theta}~{}$then we have the extra Noether
symmetries $K^{1}~{},~{}tK^{1}$ with Noether integrals the (9.55) with $k=1$.

b) If $V\left(\theta\right)=\left(V_{0}e^{2\theta}-\frac{m}{2}\theta
e^{2\theta}\right)$, then we have the Noether symmetries
$e^{\pm\sqrt{m}t}K^{1}$ with Noether integrals (9.56) with $k=1$.

  2. 2.

Noether symmetries generated by the KV $K^{2}$.

a) If $V\left(\theta\right)=V_{0}e^{-2\theta}$ then we have the Noether
symmetries $K^{2}~{},~{}tK^{2}$ with Noether integrals

$I_{2}^{\prime}=\frac{d}{dt}\left(\theta-\ln
r\right)~{},~{}I_{2}^{\prime}=t\left[\frac{d}{dt}\left(\theta-\ln
r\right)\right]-\left(\theta-\ln r\right)$ (9.66)

b) If $V\left(\theta\right)=V_{0}e^{-2\theta}-\frac{1}{4}pe^{2\theta}\,$ then
we have the Noether symmetries $K^{2}~{},~{}tK^{2}$ with Noether integrals

$I_{1}^{\prime}=\frac{d}{dt}\left(\theta-\ln
r\right)-pt~{},~{}I_{2}^{\prime}=t\left[\frac{d}{dt}\left(\theta-\ln
r\right)\right]-\left(\theta-\ln r\right)-\frac{1}{2}pt^{2}$ (9.67)

  3. 3.

If $V\left(\theta\right)=0$ then the system becomes the free particle and
admits seven extra Noether symmetries.

As we have remarked the results for $k=-1$ are obtained directly from those
for $k=1$ if we make the substitution
$\theta_{\left(k=-1\right)}=-\bar{\theta}$. Therefore there is no need to
state them explicitly.

In the next section using the Noether symmetries for the potentials we have
found, we determine the analytic solution for each case.

#####  9.5.2 Case A: Analytic solutions for $k=1$

We introduce new coordinates $u,v$ by the relations

$u=\left(\theta-\ln r\right)~{},~{}v=\frac{1}{2}e^{2\theta}r^{2}$ (9.68)

with inverse relations:

$r=\left(2ve^{-2u}\right)^{\frac{1}{4}}~{},~{}~{}\theta=\frac{1}{4}\ln\left(2ve^{2u}\right).$
(9.69)

In the new coordinates $u,v$ the Lagrangian (9.45) of the field equations
becomes

$L\left(u,v,\dot{u},\dot{v}\right)=\frac{N_{0}^{2}}{2}\left(\dot{u}\dot{v}\right)-U\left(u,v\right)$
(9.70)

and the field equations are

$E=\frac{N_{0}^{2}}{2}\left(\dot{u}\dot{v}\right)+U\left(u,v\right)$
$\displaystyle\frac{N_{0}^{2}}{2}\ddot{u}+U_{,v}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle\frac{N_{0}^{2}}{2}\ddot{v}+U_{,u}$
$\displaystyle=$ $\displaystyle 0$

where the potential $U\left(u,v\right)$ is one of the potentials we have found
in the last section. In the new coordinates we have ($p,m$ are constants;
recall that in general $U(r,\theta)=r^{2}V\left(\theta\right)$):

  * •

$U(r,\theta)=V_{0}r^{2}e^{-2\theta}\rightarrow U\left(u,v\right)=V_{0}e^{-2u}$
(9.71)

  * •

$U(r,\theta)=V_{0}r^{2}e^{2\theta}\rightarrow U\left(u,v\right)=2V_{0}v$
(9.72)

  * •

$U(r,\theta)=r^{2}\left(V_{0}e^{-2\theta}-\frac{1}{4}pe^{2\theta}\right)\rightarrow
U\left(u,v\right)=V_{0}e^{-2u}-\frac{p}{2}v$ (9.73)

  * •

$U(r,\theta)=r^{2}\left(V_{0}e^{2\theta}-\frac{m}{2}\theta
e^{2\theta}\right)\rightarrow U\left(u,v\right)=2\left(V_{0}-m\frac{\ln
2}{8}\right)v-\frac{m}{4}v\ln v-\frac{m}{2}uv$ (9.74)

  * •

$U(r,\theta)=0\rightarrow U\left(u,v\right)=0$ (9.75)

The Hamiltonian equals

$E=\frac{N_{0}^{2}}{2}\left(\dot{u}\dot{v}\right)+U\left(u,v\right).$ (9.76)

We write Lagrange equations for each potential and solve them taking into
consideration the first integrals for each Noether symmetry we have found and
the constraint imposed by the Hamiltonian. For each of the potentials above we
find the corresponding analytic solution given below.

  * •

$U\left(u,v\right)=V_{0}e^{-2u}$

$u\left(t\right)=u_{1}t+u_{2},\text{
}v\left(t\right)=\frac{e^{-2u_{2}}}{u_{1}^{2}}\frac{V_{0}}{N_{0}^{2}}e^{-2u_{1}t}+v_{1}t+v_{2}$
(9.77)

with Hamiltonian constraint

$E=\frac{u_{1}v_{1}}{2N_{0}^{2}}.$ (9.78)

  * •

$U\left(u,v\right)=2V_{0}v$

$u\left(t\right)=-\frac{2V_{0}}{N_{0}^{2}}t^{2}+u_{1}t+u_{2},v\left(t\right)=v_{1}t+v_{2}$
(9.79)

with Hamiltonian constrain

$E=2V_{0}v_{2}+N_{0}^{2}\frac{u_{1}v_{1}}{2}.$ (9.80)

  * •

$U\left(u,v\right)=V_{0}e^{-2u}-\frac{p}{2}v$

where

$v_{1}=\frac{2V_{0}\sqrt{\pi}}{p^{\frac{3}{2}}N_{0}}\exp\left(\frac{u_{1}^{2}N_{0}^{2}}{p}-2u_{2}\right)~{},~{}v_{2}=\frac{2V_{0}}{p}e^{-2u_{2}}$
(9.81)

with Hamiltonian constraint

$E=N_{0}^{2}\frac{v_{3}u_{1}}{2}-\frac{v_{4}p}{2}.$ (9.82)

  * •

$U\left(u,v\right)=2\left(V_{0}-m\frac{\ln 2}{8}\right)v-\frac{m}{4}v\ln
v-\frac{m}{2}uv$

$\displaystyle u\left(t\right)$ $\displaystyle=$ $\displaystyle
u_{1}e^{\frac{1}{N_{0}}\sqrt{m}t}+u_{2}e^{-\frac{1}{N_{0}}\sqrt{m}t}-\frac{\sqrt{m}}{2N_{0}}t+\frac{4V_{0}}{m}-\ln\left(\sqrt{2}\right)-\frac{1}{2}~{},~{}$
(9.83) $\displaystyle v\left(t\right)$ $\displaystyle=$ $\displaystyle
e^{\frac{1}{N_{0}}\sqrt{m}t}~{}~{},~{}E=-u_{2}m$ (9.84) $\displaystyle
u\left(t\right)$ $\displaystyle=$ $\displaystyle
u_{1}e^{\frac{1}{N_{0}}\sqrt{m}t}+u_{2}e^{-\frac{1}{N_{0}}\sqrt{m}t}+\frac{\sqrt{m}}{2N_{0}}t+\frac{4V_{0}}{m}-\ln\left(\sqrt{2}\right)-\frac{1}{2}~{},~{}$
(9.85) $\displaystyle v\left(t\right)$ $\displaystyle=$ $\displaystyle
e^{-\frac{1}{N_{0}}\sqrt{m}t}~{},~{}E=-u_{1}m.$ (9.86)

  * •

$U\left(u,v\right)=0$ (the free particle)

$u\left(t\right)=u_{1}t+u_{2}~{},~{}v\left(t\right)=v_{1}t+v_{2}$ (9.87)

with Hamiltonian constraint

$E=\frac{N_{0}^{2}}{2}u_{1}v_{1}.$ (9.88)

#####  9.5.3 Case A: Analytic solutions for $\left|k\right|\neq 1$

When $\left|k\right|\neq 1$ we have to consider two cases $k^{2}>1$ and
$k^{2}<1.$ Both cases it is convenient to be discussed if we use as variables
the functions $S_{1}\left(r,\theta\right),$ $S_{2}\left(r,\theta\right)$ which
generate the Killing vectors (i.e. $K^{I,i}=S_{I}^{,i}$ $I=1,2).$ A standard
calculation gives (see appendix B for details) :

$\displaystyle K^{1}$ $\displaystyle=$
$\displaystyle\frac{e^{\left(1-k\right)\theta}r^{k}}{N_{0}^{2}}\left(-\partial_{r}+\frac{1}{r}\partial_{\theta}\right)~{},~{}S_{1}\left(r,\theta\right)=\frac{r^{1+k}e^{\left(1+k\right)\theta}}{\left(k+1\right)}$
(9.89) $\displaystyle K^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{-\left(1+k\right)\theta}r^{-k}}{N_{0}^{2}}\left(\partial_{r}+\frac{1}{r}\partial_{\theta}\right),~{}S_{2}\left(r,\theta\right)=\frac{r^{1-k}e^{-\left(1-k\right)\theta}}{k-1}.$
(9.90)

Because the metric is flat the new variables $S_{1}\left(r,\theta\right),$
$S_{2}\left(r,\theta\right)$ are Cartesian and will be denoted with $x,y.$ So
we write:

$x=\frac{r^{1+k}e^{-\left(1+k\right)\theta}}{\left(k+1\right)}~{},~{}y=\frac{r^{1-k}e^{\left(-1+k\right)\theta}}{k-1}.$
(9.91)

The inverse transformation is:

for $k^{2}>1$

$\displaystyle\theta$ $\displaystyle=$