# Infinite-Dimensional Hamiltonian Description of Dissipative Mechanical
Systems

Tianshu Luo<EMAIL_ADDRESS>Institute of Applied Mechanics, Department
of Applied Mechanics, Zhejiang University,
Hangzhou, Zhejiang, 310027, P.R.China
Yimu Guo<EMAIL_ADDRESS>Institute of Applied Mechanics, Department of
Applied Mechanics, Zhejiang University,
Hangzhou, Zhejiang, 310027, P.R.China

###### Abstract

In this paper an approach is proposed to define an infinite-dimensional
Hamiltonian formalism to represent dissipative mechanical systems. This
approach is based upon below viewpoints: for any non-conservative classical
mechanical system and any initial condition, there exists a conservative one;
the two systems share one and only one common phase curve; the value of the
Hamiltonian of the conservative system is equal to the sum of the total energy
of the non-conservative system on the aforementioned phase curve and a
constant depending on the initial condition. We called the conservative system
as the substituting conservative system. The infinite-dimensional Hamilton’s
description of the ideal fluid in Lagrangian and Poisson-Vlasov equation
motivate us to consider a dissipative mechanical system as a special fluid in
a domain $D$ of the phase space, viz. a collection of particles in the domain.
By comparing the description of the ideal fluid in Lagrangian coordinates, the
Hamiltonian and the Lagrangian can be thought of as the integrals of the
Hamiltonian and the Lagrangian of the substituting conservative system over
the initial value space and a new Poisson bracket is defined to represent the
Hamilton’s equation. The advantage of the approach is: the value of the
canonical momentum density $\pi$ is identical with that of the mechanical
momentum and the value of canonical coordinate $q$ is identical with that of
the coordinate of the dissipative mechanical system.

Hamiltonian formalism, dissipation, non-conservative system, damping

###### pacs:

45.20.Jj

††preprint: AIP/123-QED

##  I Introduction

Since Hamilton originated Hamilton equations of motion and Hamiltonian
formalism, it has been stated in most classical textbooks that the Hamiltonian
formalism focuses on solving conservative problems.

In 1960s, Hori and Brouwer (1961) utilized the classical Hamiltonian formalism
and a perturbation theory to solve a non-conservative problem. They did not
attempt to derive the Hamiltonian formalism for non-conservative problems.
Several authors have attempted to enlarge the scope of Hamiltonian formalism
to dissipative problems. Some significant works in this area were reported by
Vujanovic (1970, 1978) and Djukic (1973); Djukic and Vujanovic (1975); Djukic
(1975). They have proposed a technique for systems with gauge variant
Lagrangian. A.Mukherjee and A.Dasgupta (2006) considered that the technique is
rather algebraic in nature. To overcome the limitations, A.Mukherjee (1994)
proposed a modified equation with an introduction of an additional time like
variable called ’umbra time’ and extending this notion to the co-kinetic
kinetic, potential, complimentary energies as well as Lagrangian itself.
Amalendu Mukherjee (1997) introduced a procedure for getting umbra-Lagrangian
through system bond graphs and extended the basic idea of Karnopp (1977).
Mukherjee (2001) consolidated this idea and presented an important idea of
invariants of motion. A gauge variant Lagrangian implies a new definition of
canonical momentum, which might not be identical with mechanical momentum.
Some other literature of Jerrold E. Marsden (1994) and Morrison (2006, 1998),
Salmon (1988) in the geometrical mechanics field focused on the conservative
system or some special dissipative systems, e.g. an oscillator with gyroscopic
damping. Morrison (1998) had written so: ’the ideal fluid description is one
in which viscosity or other phenomenological terms are neglected. Thus, as is
the case for systems governed by Newton’s second law without dissipation, such
fluid descriptions posses Lagrangian and Hamiltonian descriptions.’
Krechetnikov and Marsden (2007) and other researchers applied the equations as
below to the problem of stability of dissipative system,

$\displaystyle\dot{p}_{i}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial q_{i}}+\bm{F}\left(\frac{\partial{r}}{\partial q_{i}}\right)$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}},$ (1)

where $\\{q,p\\}$ denote the coordinate and momentum, and the position vector
$r$ depends on the canonical variable $\\{q,p\\}$, i.e. $r(q,p)$, $H$ denotes
Hamiltonian, $\bm{F}(\partial{r}/\partial{q_{i}})$ denotes a generalized force
in direction $i$. Marsden considered that Eqs. (1) was composed of a
conservative part and a non-conservative part. Eq. (1) apparently is not a
Hamilton’s equation but only a representation of dissipative mechanical
systems in the phase space.

In this paper an $n$-dimensional dissipative mechanical system as the
following is considered:

$\ddot{\bm{q}}+\mathsfsl{c}\dot{\bm{q}}+\mathsfsl{k}\bm{q}=0,$ (2)

where $\mathsfsl{c}$ denotes the damping coefficient matrix, $\mathsfsl{k}$
denotes the stiffness coefficient matrix. In light of the proposition proposed
by Luo and Guo (2010) an attempt is made to represent the dissipative
mechanical system (2) as an infinite-dimensional Hamilton’s equation. This
proposition asserts that for any non-conservative classical mechanical system
and any initial condition, there exists a conservative one; the two systems
share one and only one common phase curve; the Hamiltonian of the conservative
system is the sum of the total energy of the non-conservative system on the
aforementioned phase curve and a constant depending on the initial condition.
In sec. II the demonstration of the proposition is first reported. Analogous
to Hamiltonian description of ideal fluid in Lagrangian variables and that of
Poisson-Vlasov equations, we attempt to define Lagrangian and Hamiltonian as
an integral over the entire initial value space. The generalized coordinates
and the canonical momentum will be thought of as the function of the initial
value and time. A new Poisson bracket will be defined to represent Eq. (2) as
an infinite-dimensional Hamilton’s Equation. This process will be in detail
presented in Sec. III.

##  II Corresponding Conservative Mechanical Systems

###  II.1 Common Phase Flow Curve

First we represented Eq. (2) as Eq. (1).Under general circumstances, the force
$\bm{F}$ is a damping force that depends on the variable set
$q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n}$. We denote by $F_{i}$ the
components of the generalized force $\bm{F}$.

$F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})=\bm{F}\left(\frac{\partial{r}}{\partial
q_{i}}\right).$ (3)

Thus we can reformulate the Eq. (2) as follows:

$\displaystyle\dot{p}_{i}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial q_{i}}+F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}}.$ (4)

Suppose the Hamiltonian quantity of a conservative system without damping is
$\hat{H}$. Thus we may write a Hamilton’s equation of the conservative system:

$\displaystyle\dot{p}_{i}$ $\displaystyle=$
$\displaystyle-\frac{\partial{\hat{H}}}{\partial q_{i}}$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$
$\displaystyle\frac{\partial\hat{H}}{\partial p_{i}}.$ (5)

We do not intend to change the definition of momentum in classical mechanics,
but we do require that a special solution of Eq. (5) is the same as that of
Eq. (4). We may therefore assume a phase curve $\gamma$ of Eq. (4) coincides
with that of Eq. (5). The phase curve $\gamma$ corresponds to an initial
condition $q_{i0},p_{i0}$. Consequently by comparing Eq. (4) and Eq. (5), we
have

$\displaystyle\left.\frac{\partial{\hat{H}}}{\partial{q_{i}}}\right|_{\gamma}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial H}{\partial
q_{i}}\right|_{\gamma}-\left.F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right|_{\gamma}$
$\displaystyle\left.\frac{\partial{\hat{H}}}{\partial{p_{i}}}\right|_{\gamma}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial H}{\partial
p_{i}}\right|_{\gamma},$ (6)

where
$\left.\frac{\partial{\hat{H}}}{\partial{q_{i}}}\right|_{\gamma},\left.\frac{\partial
H}{\partial
q_{i}}\right|_{\gamma},\left.\frac{\partial{\hat{H}}}{\partial{p_{i}}}\right|_{\gamma}$
and $\left.\frac{\partial H}{\partial p_{i}}\right|_{\gamma}$ denote the
values of these partial derivatives on the phase curve $\gamma$ and
$\left.F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right|_{\gamma}$
denotes the value of the force $F_{i}$ on the phase curve $\gamma$. In
classical mechanics the Hamiltonian $H$ of a conservative mechanical system is
mechanical energy and can be written as:

$H=\int_{\gamma}\left(\frac{\partial{H}}{\partial{q_{i}}}\right)\mathrm{d}q_{i}+\int_{\gamma}\left(\frac{\partial
H}{\partial p_{i}}\right)\mathrm{d}p_{i}+const_{1},$ (7)

where $const_{1}$ is a constant that depends on the initial condition
described above. The mechanical energy $H$ of the system (4) can be evaluated
via Eq. (7) too. If $q_{i}=0,p_{i}=0$, then $const_{1}=0$. The Einstein
summation convention has been used this section. Thus an attempt has been made
to find $\left.\hat{H}\right|_{\gamma}$ through line integral along the phase
curve $\gamma$ of the dissipative system

$\displaystyle\int_{\gamma}\left(\frac{\partial{\hat{H}}}{\partial{q_{i}}}\right)\mathrm{d}q_{i}$
$\displaystyle=$ $\displaystyle\int_{\gamma}\left[\left(\frac{\partial
H}{\partial
q_{i}}\right)-F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right]\mathrm{d}q_{i}$
$\displaystyle\int_{\gamma}\left(\frac{\partial\hat{H}}{\partial
p_{i}}\right)\mathrm{d}p_{i}$ $\displaystyle=$
$\displaystyle\int_{\gamma}\left(\frac{\partial H}{\partial
p_{i}}\right)\mathrm{d}p_{i}.$ (8)

Analogous to Eqs.(7), we have

$\hat{H}=\int_{\gamma}\left(\frac{\partial{\hat{H}}}{\partial{\hat{q}_{i}}}\right)_{\hat{q}\hat{p}}\mathrm{d}\hat{q}_{i}+\int_{\gamma}\left(\frac{\partial{\hat{H}}}{\partial\hat{p}_{i}}\right)_{\hat{q}\hat{p}}\mathrm{d}\hat{p}_{i}+const_{2},$
(9)

where $const_{2}$ is a constant which depends on the initial condition.
Substituting Eqs.(7)and Eqs.(8) into Eq. (9), we have

$\left.\hat{H}\right|_{\gamma}=H-\int_{\gamma}F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\mathrm{d}q_{i}+const.$
(10)

where $const=const_{2}-const_{1}$, and $H=\left.H\right|_{\gamma}$ because $H$
is mechanical energy of the non-conservative system (4). According to the
physical meaning of Hamiltonian, $const_{1}$, $const_{2}$ and $const$ are
added into Eq. (7)(9)(10) respectively such that the integral constant
vanishes in the Hamiltonian quantity. Arnold. (1997) had presented the Newton-
Laplace principle of determinacy as, ’This principle asserts that the state of
a mechanical system at any fixed moment of time uniquely determines all of its
(future and past) motion.’ In other words, in the phase space the position
variable and the velocity variable are determined only by the time $t$.
Therefore, we can assume that we have already a solution of Eq. (4)

$\displaystyle q_{i}$ $\displaystyle=$ $\displaystyle q_{i}(t)$
$\displaystyle\dot{q_{i}}$ $\displaystyle=$ $\displaystyle\dot{q_{i}}(t),$
(11)

where the solution satisfies the initial condition. We can divide the whole
time domain into a group of sufficiently small domains and in these domains
$q_{i}$ is monotone, and hence we can assume an inverse function $t=t(q_{i})$.
If $t=t(q_{i})$ is substituted into the non-conservative force
$\left.F_{i}\right|_{\gamma}$, we can assume that:

$\left.F_{i}(q_{1}(t(q_{i})),\cdots,q_{n}(t(q_{i})),\dot{q}_{1}(t(q_{i})),\cdots,\dot{q}_{n}(t(q_{i})))\right|_{\gamma}=\mathcal{F}_{i}(q_{i}),$
(12)

where $\mathcal{F}_{i}$ is a function of $q_{i}$ alone. In Eq. (12) the
function $F_{i}$ is restricted on the curve $\gamma$, such that a new function
$\mathcal{F}_{i}(q_{i})$ yields. Thus we have

$\displaystyle\int_{\gamma}F_{i}\mathrm{d}q_{i}$ $\displaystyle=$
$\displaystyle\int_{q_{i0}}^{q_{i}}\mathcal{F}_{i}(q_{i})\mathrm{d}q_{i}=W_{i}(q_{i})-W_{i}(q_{i0}).$
(13)

According to Eq. (13) the function $\mathcal{F}_{i}$ is path independent, and
therefore $\mathcal{F}_{i}$ can be regarded as a conservative force. For that
Eq. (12) represents an identity map from the non-conservative force $F$ on the
curve $\gamma$ to the conservative force $\mathcal{F}_{i}$ which is distinct
from $F_{i}$. Eq. (12) is tenable only on the phase curve $\gamma$.
Consequently the function form of $\mathcal{F}_{i}$ depends on the
aforementioned initial condition; from other initial conditions
$\mathcal{F}_{i}$ with different function forms will yield.

According to the physical meaning of Hamiltonian, $const$ is added to Eq. (10)
such that the integral constant vanishes in Hamiltonian quantity. Hence
$const=-W_{i}(q_{i0})$. Substituting Eq. (13) and $const=-W_{i}(q_{i0})$ into
Eq. (10), we have

$\left.\hat{H}\right|_{\gamma}=H-W_{i}(q_{i})$ (14)

where $-W_{i}(q_{i})$ denotes the potential of the conservative force
$\mathcal{F}_{i}$ and $W_{i}(q_{i})$ is equal to the sum of the work done by
the non-conservative force $F$ and $const$. In Eq. (14) $\hat{H}$ and $H$ are
both functions of $q_{i}$ and $W_{i}(q_{i})$ a function of $q_{i}$. Eq. (14)
and Eq. (10) can be thought of as a map from the total energy of the
dissipative system (4) to the Hamiltonian of the conservative system (5).
Indeed, $\left.\hat{H}\right|_{\gamma}$ and the total energy differ in the
constant $const=-W_{i}(q_{i0})$. When the conservative system takes a
different initial condition, if one does not change the function form of
$\left.\hat{H}\right|_{\gamma}$, one can consider
$\left.\hat{H}\right|_{\gamma}$ as a Hamiltonian quantity $\hat{H}$,

$\hat{H}=\left.\hat{H}\right|_{\gamma}=H-W_{i}(q_{i})$ (15)

and the conservative system (5) can be thought of as an entirely new
conservative system.

Based on the above, the following proposition is made:

######  Proposition II.1.

For any non-conservative classical mechanical system and any initial
condition, there exists a conservative one; the two systems share one and only
one common phase curve; the value of the Hamiltonian of the conservative
system is equal to the sum of the total energy of the non-conservative system
on the aforementioned phase curve and a constant depending on the initial
condition.

###### Proof.

First we must prove the first part of the Proposition II.1, i.e. that a
conservative system with Hamiltonian presented by Eq. (15) shares a common
phase curve with the non-conservative system represented by Eq. (4). In other
words the Hamiltonian quantity presented by Eq. (15) satisfies Eq. (6) under
the same initial condition. Substituting Eq. (15) into the left side of Eq.
(6), we have

$\displaystyle\frac{\partial{\hat{H}(q_{i},p_{i})}}{\partial{q_{i}}}$
$\displaystyle=$ $\displaystyle\frac{\partial
H(q_{i},p_{i})}{\partial{q_{i}}}-\frac{\partial{W_{j}(q_{j})}}{\partial{q_{i}}}$
$\displaystyle\frac{\partial{\hat{H}(q_{i},p_{i})}}{\partial{p_{i}}}$
$\displaystyle=$ $\displaystyle\frac{\partial
H(q_{i},p_{i})}{\partial{p_{i}}}-\frac{\partial{W_{j}(q_{j})}}{\partial{p_{i}}}.$
(16)

It must be noted that although $q_{i}$ and $p_{i}$ are considered as distinct
variables in Hamilton’s mechanics, we can consider $q_{i}$ and $\dot{q_{i}}$
as dependent variables in the process of constructing of $\hat{H}$. At the
trajectory $\gamma$ we have

$\displaystyle\frac{\partial{{W_{j}(q_{j})}}}{\partial{q_{i}}}$
$\displaystyle=$
$\displaystyle\frac{\partial{(\int_{q_{j0}}^{q_{j}}\mathcal{F}_{j}(q_{j})\mathrm{d}q_{j}+W_{i}(q_{i0}))}}{\partial{q_{i}}}=\mathcal{F}_{i}(q_{i})$
$\displaystyle\frac{\partial{{W_{j}(q_{j})}}}{\partial{p_{i}}}$
$\displaystyle=0,$ (17)

where $\mathcal{F}_{i}(q_{i})$ is equal to the damping force $F_{i}$ on the
phase curve $\gamma$. Hence under the initial condition $q_{0},p_{0}$, Eq. (6)
is satisfied. As a result, we can state that the phase curve of Eq. (5)
coincides with that of Eq. (4) under the initial condition; and $\hat{H}$
represented by Eq. (15) is the Hamiltonian of the conservative system
represented by Eq. (5).

Then we must prove the second part of Proposition II.1: the uniqueness of the
common phase curve.

We assume that Eq. (5) shares two common phase curves, $\gamma_{1}$ and
$\gamma_{2}$, with Eq. (4). Let a point of $\gamma_{1}$ at the time $t$ be
$z_{1}$, a point of $\gamma_{2}$ at the time $t$ $z_{2}$, and $g^{t}$ the
Hamiltonian phase flow of Eq. (5). Suppose a domain $\Omega$ at $t$ which
contains only points $z_{1}$ and $z_{2}$, and $\Omega$ is not only a subset of
the phase space of the non-conservative system (4) but also that of the phase
space of the conservative system (5). Hence there exists a phase flow
$\hat{g}^{t}$ composed of $\gamma_{1}$ and $\gamma_{2}$, and $\hat{g}^{t}$ is
the phase flow of Eq. (4) restricted by $\Omega$. According to the following
Louisville’s theorem in the book of Arnold. (1978):

######  Theorem II.1.

The phase flow of Hamilton’s equations preserves volume: for any region $D$ in
the phase space we have

$volume\ of\ g^{t}D=volume\ of\ D$

where $g^{t}$ is the one-parameter group of transformations of phase space

$g^{t}:(p(0),q(0))\longmapsto:(p(t),q(t))$

$g^{t}$ preserves the volume of $\Omega$. This implies that the phase flow of
Eq. (4) $\hat{g}^{t}$ preserves the volume of $\Omega$ too. But the system (4)
is not conservative, which conflicts with Louisville’s theorem; hence only a
phase curve of Eq. (5) coincides with that of Eq. (4).

∎∎

###  II.2 Obtaining the Equivalent Stiffness Matrix $\tilde{K}$

According to Proprostion II.1, an attempt is made to find a new conservative
mechanical system which is corresponding to the dissipative system (2) and an
initial condition. Under the initial condition, the dissipative system (2)
posses a phase curve $\gamma$. As in Eq. (12) we can consider that the damping
forces are equal to some conservative force on the phase curve $\gamma$

$\begin{array}[]{ccc}c_{11}\dot{q}_{1}=\varrho_{11}(q_{1})&\dots&c_{1n}\dot{q}_{n}=\varrho_{1n}(q_{1})\\\
\vdots&\ddots&\vdots\\\
c_{n1}\dot{q}_{1}=\varrho_{21}(q_{n})&\dots&c_{nn}\dot{q}_{n}=\varrho_{nn}(q_{n}).\end{array}$
(18)

For convenience, these conservative forces can be thought of as elastic
restoring forces:

$\begin{array}[]{ccc}\varrho_{11}(q_{1})=\kappa_{11}(q_{1})q_{1}&\dots&\varrho_{1n}(q_{1})=\kappa_{1n}(q_{1})q_{1}\\\
\vdots&\ddots&\vdots\\\
\varrho_{n1}(q_{1})=\kappa_{n1}(q_{n})q_{n}&\dots&\varrho_{nn}(q_{n})=\kappa_{nn}(q_{n})q_{n}.\end{array}$
(19)

An equivalent stiffness matrix $\mathsfsl{\tilde{K}}$ is obtained, which is a
diagonal matrix

$\mathsfsl{\tilde{K}}_{ii}=\sum_{l=1}^{n}\kappa_{il}(q_{l}).$ (20)

Consequently an $n$-dimensional conservative system is obtained

$\bm{\ddot{q}}+(\mathsfsl{K}+\mathsfsl{\tilde{K}})\bm{q}=0$ (21)

which shares the common phase curve $\gamma$ with the $n$-dimensional damping
system (2). In this paper, the conservative system is named as substituting
conservative system. The Lagrangian of Eqs.(21) is

$\hat{L}=\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(22)

and the Hamiltonian of Eqs.(21) is

$\hat{H}=\frac{1}{2}\bm{p}^{T}\bm{p}+\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}+\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(23)

where $\bm{0}$ is a zero vector, $\bm{p}=\dot{\bm{q}}$. $\hat{H}$ in Eq. (23)
is the mechanical energy of the conservative system (21), because
$\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}$ is a
potential function such that $\hat{H}$ doest not depend on any path of Eq.
(22).

##  III Definition of a Generalized Hamilton’s Equation

In this section Proposition II.1 would be represented as a uniform infinite-
dimensional Hamilton’s equation. IN infinite-dimensional Hamiltonian
formalism, techniques of functional derivative must be devoted. Morrison
(1998) introduced the definition the functional derivative simply. We would
report the introduction.

###  III.1 Introduction of Functional Derivative and Canonical Hamiltonian
Description of the Ideal Fluid in Lagrangian variables

Consider a functional $K[u]$. The first change in $K$ induced by $\delta u$ is
called the first variation, $\delta K$, and is given by

$\displaystyle\delta K[u;\delta u]$ $\displaystyle:=$
$\displaystyle\lim_{\varepsilon\rightarrow 0}\frac{K[u+\varepsilon
u]-K[u]}{\varepsilon}$ (24) $\displaystyle=$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\left.K[u+\varepsilon
u]\right|_{\varepsilon=0}$ $\displaystyle=:$
$\displaystyle\int_{x_{0}}^{x_{1}}\delta u\frac{\delta K}{\delta
u(x)}=:\langle\frac{\delta K}{\delta u},\delta u\rangle$

The quantity $\delta K/\delta u(x)$ of Eq. (24) is the functional derivative
of the functional $K$. Consider a now a more general functional, one of the
form

$\hat{F}[u]=\int_{x_{0}}^{x_{1}}\mathcal{\hat{F}}(x,u,u_{x},u_{xx},\dots)\mathrm{d}x$
(25)

where $\mathcal{\hat{F}}$ is an ordinary, sufficiently differentiable,
function of its arguments. Note $u_{x}=\mathrm{d}u/\mathrm{d}x$, etc. This
first variation of Eq. (25) yields

$\delta\hat{F}[u,\delta
u]=\int_{x_{0}}^{x_{1}}\left(\frac{\partial\mathcal{\hat{F}}}{\partial
u}\delta u+\frac{\partial\mathcal{\hat{F}}}{\partial u_{x}}\delta
u_{x}+\frac{\partial\mathcal{\hat{F}}}{\partial u_{xx}}\delta
u_{xx}+\cdots\right)\mathrm{d}x,$ (26)

which upon integration by parts becomes

$\displaystyle\delta\hat{F}[u,\delta u]$ $\displaystyle=$
$\displaystyle\int_{x_{0}}^{x_{1}}\delta
u\left(\frac{\partial\mathcal{\hat{F}}}{\partial
u}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial\mathcal{\hat{F}}}{\partial
u_{x}}+\frac{\mathrm{d}^{2}}{\mathrm{d}^{2}x}\frac{\partial\mathcal{\hat{F}}}{\partial
u_{xx}}\delta u_{xx}-\cdots\right)\mathrm{d}x$ (27)
$\displaystyle+\left.\left(\frac{\partial\mathcal{\hat{F}}}{\partial
u_{x}}\delta u+\cdots\right)\right|_{x_{0}}^{x_{1}}.$

Usually the variations $\delta u$ are chosen so that the last term, the
boundary term, vanishes; e.g., $\delta u(x_{0})=\delta u(x_{1})=0,\ \ \delta
u_{x}(x_{0})=\delta u_{x}(x_{1})=0$, etc. Sometimes the boundary term vanishes
without a condition on $\delta u$ because of the form of $\mathcal{\hat{F}}$.
When this happens the boundary conditions are called natural. Assuming, for
one reason or the other, that the boundary term vanishes, Eq. (27) becomes

$\delta\hat{F}[u;\delta u]=\langle\frac{\delta\hat{F}}{\delta u},\delta
u\rangle,$ (28)

where the functional derivative

$\frac{\delta F}{\delta u}=\frac{\partial\mathcal{\hat{F}}}{\partial
u}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial\mathcal{\hat{F}}}{\partial
u_{x}}+\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\frac{\partial\mathcal{\hat{F}}}{\partial
u_{xx}}-\dots.$ (29)

The main objective of the calculus of variations is the extremization of
functionals. A common terminology is to call a function $\hat{u}$, which is a
point in the domain, an extremal point if $\delta\hat{F}[u]/\delta
u|_{u=\hat{u}}=0$. It could be a maxi- mum, a minimum, or an inflection point.
If the extremal point $\hat{u}$ is a minimum or maximum, then such a point is
called an extremum.

An example is the functional defined by evaluating the function $u$ at the
point $x$ . This can be written as

$u(x^{\prime})=\int_{x_{0}}^{x_{1}}\delta(x-x^{\prime})u(x)\mathrm{d}x,$ (30)

where $\delta(x-x^{\prime})$ is the Dirac delta function and where we have
departed from the $[]$ notion. Applying the definition of Eq. (24) yields

$\frac{\delta u(x^{\prime})}{\delta u(x)}=\delta(x-x^{\prime}).$ (31)

This is the infinite-dimensional or continuum analog of $\partial
x_{i}/\partial x_{j}=\delta_{ij}$, where $\delta_{ij}$ is is the Kronecker
delta function. Eq. (30) shows why it is sometimes useful to display the
argument of the function in the functional derivative.

The generalizations of the above ideas to functionals of more than one
function and to more than a single spatial variable are straightforward. An
example is given by the kinetic energy of a three-dimensional compress- ideal
fluid,

$T(\rho,\bm{v})=\int_{D}\frac{1}{2}\rho\bm{v}^{2}\mathrm{d}^{3}\bm{x}$ (32)

where the velocity has three rectangular components
$\bm{v}=\\{v_{1},v_{2},v_{3}\\}$ that depend upon
$\bm{x}=\\{x_{1},x_{2},x_{3}\\}\in D$ and
$\bm{v}^{2}=\bm{x}\cdot\bm{x}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$. The functional
derivatives are

$\frac{\delta T}{\delta v_{i}}=\rho v_{i},\ \ \frac{\delta
T}{\delta\rho}=\frac{\bm{v}^{2}}{2}.$ (33)

For a more general functional $\hat{F}[\bm{\psi}]$, where
$\bm{\psi}(\bm{x})=(\psi_{1},\psi_{2},\cdots,\psi_{n})$ and
$\bm{x}=(x_{1},x_{2},\cdots,x_{n})$, the analog of Eq. (24) is

$\delta\hat{F}[\bm{\psi};\delta\bm{\psi}]=\int_{D}\delta\psi_{i}\frac{\delta\hat{F}}{\delta\psi_{i}(\bm{x})}\mathrm{d}^{n}\bm{x}=:\langle\frac{\delta\hat{F}}{\delta\bm{\psi}},\delta\bm{\psi}\rangle.$
(34)

Salmon (1988) and Morrison (1998) described the Hamiltonian formalism of ideal
fluid in Lagrangian variables in detail. In order to state the infinite-
dimensional formalism for dissipative mechanical system, we repeat the
representation of Salmon (1988) and Morrison (1998).

In the the Hamiltonian description, a fluid is described as a collection of
fluid particles or elements. Suppose the coordinate of a fluid particle at
time $t$

$\bm{q}=\bm{q}(\bm{a},t),$ (35)

where $\bm{q}=\\{q_{1},q_{2},q_{3}\\}$,$\bm{a}=\\{a_{1},a_{2},a_{3}\\}$ is the
coordinate of the particle at the initial time $t=t_{0}$. We assume that
$\bm{a}$ varies over a fixed domain $D$, which is completely filled with
fluid, and that the functions $q$ map $D$ onto itself.

In Lagrangian variables $\bm{a}$ the Lagrangian quantity of the fluid particle
is considered as Lagrangian density

$\mathcal{L}_{f}(\bm{q},\dot{\bm{q}},\partial\bm{q}/\partial\bm{a},t)=\frac{1}{2}\rho_{0}\dot{\bm{q}}^{2}-\rho_{0}E(s_{0},\rho_{0}/\mathsfsl{\mathcal{J}})-\phi,$
(36)

where $\rho_{0}=\rho_{0}(\bm{a})$ is a given initial density distribution,
$\dot{\bm{q}}$ is the velocity of the fluid particle, a shorthand
$\dot{\bm{q}}^{2}=\delta_{ij}q_{i}q_{j}$ is used, $E$ is the energy per unit
mass, $s_{0}$ is the entropy per unit mass at the time $t_{0}$,
$\mathsfsl{\mathcal{J}}=\det(\partial{q}^{i}/\partial{a}^{j})$, $\phi$ is a
potential function for external conservative forces. The intensive quantities,
pressure and temperature, are obtained as follows:

$T=\frac{\partial U}{\partial s}(s,\rho),\ \ p=\rho^{2}\frac{\partial
U}{\partial\rho}(s,\rho)$ (37)

Therefore, we have the Lagrangian functional of the fluid particles of the
domain $D$:

$L_{f}[\bm{q},\dot{\bm{q}}]=\int_{D}\mathcal{L}_{f}\mathrm{d}^{3}\bm{a}=\int_{D}\left[\frac{1}{2}\rho_{0}\dot{\bm{q}}^{2}-\rho_{0}E(s_{0},\rho_{0}/\mathsfsl{\mathcal{J}})-\phi\right]\mathrm{d}^{3}\bm{a},$
(38)

where $\mathrm{d}^{3}\bm{a}=\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}a_{3}$.
Thus the action functional is given by

$S_{f}[\bm{q}]=\int_{t_{0}}^{t^{1}}\mathrm{d}t\int_{D}L_{f}[\bm{q},\dot{\bm{q}}]\mathrm{d}^{3}\bm{a}=\int_{t_{0}}^{t^{1}}\mathrm{d}t\int_{D}\left[\frac{1}{2}\rho_{0}\dot{\bm{q}}^{2}-\rho_{0}E-\phi\right]\mathrm{d}^{3}\bm{a}$
(39)

Observe that this action functional is like that for finite-degree-of-freedom
systems, as treated above, except that the sum over particles is replaced by
integration over $D$, i.e.,

$\int_{D}\mathrm{d}^{3}\bm{a}\leftrightarrow\sum_{i}$ (40)

By a Legendre transform, we have a canonical momentum density

$\bm{\varpi}_{i}(\bm{a},t)=\frac{\delta
L_{f}}{\delta\dot{\bm{q}}_{i}(\bm{a},t)}=\rho_{0}\dot{\bm{q}}_{i},$ (41)

and a generalized Hamiltonian quantity

$H_{f}[\bm{q},\bm{\varpi}]=\int_{D}\left[\bm{\varpi}\dot{\bm{q}}-\mathcal{L}_{f}\right]\mathrm{d}^{3}{\bm{a}}=\int_{D}\left[\frac{\dot{\bm{\varpi}^{2}}}{2\rho_{0}}+E+\phi\right]\mathrm{d}^{3}{\bm{a}},$
(42)

where $\rho_{0}\dot{\bm{q}}^{2}/2+E+\phi=\mathcal{H}_{f}$ can be consider as a
Hamiltonian density. A generalized Hamilton’s equation is

$\dot{\varpi}_{i}=-\frac{\delta H_{f}}{\delta q_{i}},\ \ \ \
\dot{q}_{i}=\frac{\delta H_{f}}{\delta\bm{\varpi}_{i}}.$ (43)

These equations can also be written in terms of the Poisson bracket (see
Morrison (1998)),

$\\{F,G\\}=\int_{D}\left[\frac{\delta F}{\delta\bm{q}}\cdot\frac{\delta
G}{\delta\bm{\varpi}}-\frac{\delta G}{\delta q}\cdot\frac{\delta
F}{\delta\bm{\varpi}}\right]\mathrm{d}^{3}\bm{a}$ (44)

viz.,

$\dot{\bm{\varpi}}_{i}=\\{\bm{\varpi}_{i},H_{f}\\},\ \ \ \
\dot{\bm{q}}_{i}=\\{\bm{q}_{i},H_{f}\\}$ (45)

Here $\delta q_{i}(\bm{a}/)\delta
q_{j}(\bm{a}^{\prime})=\delta_{ij}\delta(\bm{a}-\bm{a}^{\prime})$ has been
used, where $\delta(\bm{a}-\bm{a}^{\prime})$ is a three-dimensional Dirac
delta function(recall Eq. (31).

###  III.2 Derivation of Hamiltonian Description of Dissipative Mechanical
Systems

The Hamiltonian description of the ideal fluid is infinite-dimensional, and
the Hamiltonian quantity and Lagranian is the integrals over the domain $D$ in
the initial configuration space. In addition, Morrison (1980) proposed the
Hamiltonian description of Poisson-Vlasov equations with Hamiltonian quantity,
which is an integral over the phase space. These ideas of Salmon (1988),
Morrison (1998) and Morrison (1980) motivate us to consider the mechanical
system (2) as a special fluid which is a collection of fluid particles in the
phase space. In general case Hamilton’s quantity is an energy function.
Although the total energy of the oscillator with damping is conservative, the
total energy depends on the initial condition. Consequently there is a path-
dependency problem. It is well known that the energy per unit mass $E$ is the
origin of the pressure in the fluid. The mechanical system (2) describes that
a particle moves in the configuration space. One can also consider that
individual particles of the special fluid moves without interaction.
Therefore, one can assume that no internal energy function $E$. exists in the
Lagrangian density of the system (2); the Lagrangian variable of the special
fluid in a fixed domain $D$ is

$\bm{a}=(\bm{q}_{0},\dot{\bm{q}}_{0})=(q_{0}^{1},\dots,q_{0}^{n},\dot{q}_{0}^{1},\dots,\dot{q}_{0}^{n})$
(46)

; the coordinate of a particle in the configuration space is

$\bm{q}=\bm{q}(\bm{a},t)=(q_{1}(\bm{a},t),\dots,q_{n}(\bm{a},t);$ (47)

$\rho_{o}=1$. By comparing the generalized Hamilton’s equation(43) and
Hamilton’s equation in odd dimensional phase space, one can find that the
Hamiltonian density does not need to satisfy the path independency requirement
fully, according to to Eq. (29) we have

$\displaystyle\dot{\varpi}_{i}(\bm{a})=-\frac{\delta H_{f}}{\delta
q_{i}(\bm{a})}=-\frac{\partial\mathcal{H}_{f}}{\partial q_{i}(\bm{a})},$

where $q_{i}(\bm{a})$ is the value of $q_{i}$ on the path of the particle
$\bm{a}$ in the configuration space. Therefore, analogous to Eq. (36), one can
consider $\hat{L}$ in Eq. (22) as a Lagrangian density of the system (2)

$\mathcal{L}=\hat{L}=\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(48)

and consider $\hat{H}$ in Eq. (23) as a Hamiltonian density of the system (2)

$\mathcal{H}=\hat{H}=\frac{1}{2}\bm{p}^{T}\bm{p}+\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}+\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(49)

where $q_{i}(\bm{a})$ is the value of $q_{i}$ on the path of the particle
$\bm{a}$ in the phase space, suth that one can avoid the afore-mentioned path-
dependency problem. Thus the Lagrangian functional of Eq. (2) can be presented
as following:

$L[q,\dot{q}]=\int_{D}\mathcal{L}\mathrm{d}^{2n}\bm{a}=\int_{D}\left[\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}\right]\mathrm{d}^{2n}\bm{a},$
(50)

where
$\mathrm{d}^{2n}=\mathrm{d}^{n}\bm{q}_{0}\mathrm{d}^{n}\dot{\bm{q}}_{0}=\mathrm{d}q^{1}_{0}\dots\mathrm{d}q^{n}_{0}\mathrm{d}\dot{q}^{1}_{0}\dots\mathrm{d}\dot{q}^{n}_{0}$.
The Lagrangian functional Thus the action functional can be presented as
following:

$S[q]=\int^{t1}_{t0}L[q,\dot{q}]\mathrm{d}t=\int^{t1}_{t0}\mathrm{d}t\int_{D}\left[\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}\right]\mathrm{d}^{2n}\bm{a}$
(51)

According to Hamiltonian theorem, we have the functional derivative $\delta
S/\delta\bm{q}(a,t)=0$, according to the generalization Eq. (29):

$\displaystyle\frac{\delta S}{\delta\bm{q}(\bm{a},t)}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\bm{q}(\bm{a},t)}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\mathcal{L}}{\partial\dot{\bm{q}}(\bm{a},t)}$
(52) $\displaystyle=$
$\displaystyle-\ddot{\bm{q}}(\bm{a},t)-\mathsfsl{k}\bm{q}-\tilde{\mathsfsl{K}}\bm{q}=0$

The equation above implies that under the initial condition $\bm{a}$ a
conservative system exists, the control equation of which is Eq. (21), the
phase curve of which coincides with that of the oscillator with damping.
Define a canonical momentum density for the dissipative system (2) is

$\pi_{i}(\bm{a},t)=\frac{\delta
L}{\delta\dot{q}_{i}(\bm{a})}=\dot{\bm{q}_{i}},$ (53)

which is a functional derivative, while classical canonical momentum is
defined as a partial derivative. By a Legendre transform, we have the
generalized Hamiltonian $\hat{K}$ is

$\hat{K}[\bm{\pi},\bm{q}]=\int_{D}\mathrm{d}^{2n}\bm{a}\left[\bm{\pi}\cdot\dot{\bm{q}}-\mathcal{L}\right]=\int_{D}\mathrm{d}^{2n}\bm{a}\left[\frac{1}{2}\bm{p}^{T}\bm{p}+\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}+\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}\right],$
(54)

where $\bm{q}=\bm{q}(\bm{a},t)$. Thus the generalized Hamilton’s equations of
the dissipative system (2) are

$\dot{\pi}_{i}=-\frac{\delta\hat{K}}{\delta q_{i}},\ \
\dot{q}_{i}=\frac{\delta\hat{K}}{\delta\pi_{i}}.$ (55)

######  Definition III.1.

For two functionals $F[\bm{\pi}(\bm{a}),\bm{q}(a)]$ and
$G[\bm{\pi}(a),\bm{q}(a)]$ in a domain $D$ of the phase space exists a
functional

$\\{F,G\\}[\bm{\pi}(\bm{a}),\bm{q}(\bm{a})]=\int_{D}\left[\frac{\delta
F}{\delta\bm{q}(\bm{a}^{\prime})}\cdot\frac{\delta
G}{\delta\bm{\pi}(\bm{a}^{\prime})}-\frac{\delta
G}{\delta\bm{q}(\bm{a}^{\prime})}\cdot\frac{\delta
F}{\delta\bm{\pi}(\bm{a}^{\prime})}\right]\mathrm{d}^{2n}\bm{a},$ (56)

where the functional derivative $\delta F/\delta\bm{q}(\bm{a}^{\prime})$ is
defined analogues to Eq. (24) and Eq. (34) as:

$\left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\right|_{\varepsilon=0}[\bm{q}(\bm{a}^{\prime})+\varepsilon\delta\bm{q}(\bm{a}^{\prime}),\bm{\pi}(\bm{a}^{\prime})]=\int_{D}\frac{\delta
F}{\delta\bm{q}(\bm{a}^{\prime})}\mathrm{d}^{2}n\bm{a}$

The Hamilton’s equations can also be represented in terms of the Poisson
bracket (56) viz.,

$\dot{\pi}_{i}=\\{\pi_{i},\hat{K}\\},\dot{q}_{i}=\\{q_{i},\hat{K}\\}.$ (57)

Expand $\\{\pi_{i},\hat{K}\\}$, we have

$\displaystyle\\{\pi_{i}(\bm{a}),\hat{K}\\}$ $\displaystyle=$
$\displaystyle\frac{\delta\pi_{i}(\bm{a})}{\delta
q_{j}(\bm{a}^{\prime})}\frac{\delta\hat{K}}{\delta\pi_{j}(\bm{a}^{\prime})}-\frac{\delta\pi_{i}(\bm{a})}{\delta\pi_{j}(\bm{a}^{\prime})}\frac{\delta\hat{K}}{\delta
q_{j}(\bm{a}^{\prime})}$ (58) $\displaystyle=$
$\displaystyle-\delta_{ij}\delta(\bm{a}-\bm{a}^{\prime})\frac{\delta\hat{K}}{\delta
q_{j}(\bm{a}^{\prime})}$ $\displaystyle=$
$\displaystyle-\frac{\delta\hat{K}}{\delta q_{i}(\bm{a})},$

Here $\delta q_{i}(\bm{a}/)\delta
q_{j}(\bm{a}^{\prime})=\delta_{ij}\delta(\bm{a}-\bm{a}^{\prime})$ has been
used, where $\delta(\bm{a}-\bm{a}^{\prime})$ is a three-dimensional Dirac
delta function(recall Eq. (31). Analogous to Eq. (40), we have

$\int_{D}\mathrm{d}^{2n}\bm{a}\leftrightarrow\sum_{i},\ \
\hat{K}=\sum_{i}\mathcal{H}(\bm{a})$ (59)

According to Eq. (59), from Eq. (58) we can derive

$\dot{\pi}_{i}(\bm{a})=\\{\pi_{i}(\bm{a}),\hat{K}\\}=-\frac{\delta\hat{K}}{\delta
q_{i}(\bm{a})}=-\frac{\partial\mathcal{H}(\bm{a})}{\partial q_{i}(\bm{a})}$
(60)

In the similiar way

$\dot{q}_{i}(\bm{a})=\\{q_{i}(\bm{a}),\hat{K}\\}=\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}=\frac{\partial\mathcal{H}(\bm{a})}{\partial\pi_{i}(\bm{a})}$
(61)

Therefore, we can assert that Eq. (60) and Eq. (61) describes a phase curve
which is a common phase curve of the dissipative system and a conservative
system under the initial condition $\bm{a}$.

From the Hamilton’s equation(55), we can derive the total energy conservative
principle

$\displaystyle\delta\hat{K}$ $\displaystyle=$
$\displaystyle\int_{D}\left[\frac{\delta\hat{K}}{\delta q_{i}(\bm{a})}\delta
q_{i}(\bm{a})+\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}\delta\pi_{i}(\bm{a})\right]\mathrm{d}^{2n}\bm{a}$
$\displaystyle=$ $\displaystyle\int_{D}\left[\frac{\delta\hat{K}}{\delta
q_{i}(\bm{a})}\frac{\mathrm{d}q_{i}(\bm{a})}{\mathrm{d}t}\mathrm{d}t+\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}\frac{\mathrm{d}\pi_{i}(\bm{a})}{\mathrm{d}t}\mathrm{d}t\right]\mathrm{d}^{2n}\bm{a}$
$\displaystyle=$ $\displaystyle\int_{D}\left[\frac{\delta\hat{K}}{\delta
q_{i}(\bm{a})}\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}\mathrm{d}t-\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}\frac{\delta\hat{K}}{\delta
q_{i}(\bm{a})}\mathrm{d}t\right]\mathrm{d}^{2n}\bm{a}$ $\displaystyle=$
$\displaystyle 0$

##  IV Conclusion

The following conclusions can be drawn. The infinite-dimensional
description(53),(54), (55),(56,(57) can describe a dissipative mechanical
system based on the propositon II.1: For any non-conservative classical
mechanical system and any initial condition, there exists a conservative one;
the two systems share one and only one common phase curve; the value of the
Hamiltonian of the conservative system is equal to the sum of the total energy
of the non-conservative system on the aforementioned phase curve and a
constant depending on the initial condition. In fact, if the generalized
Hamilton’s equation (55) and (57) is constrained at a initial condition
$\bm{a}$, the generalized Hamilton’s equation is a phase curve of the afore-
mentioned conservative system (21). As the classical Hamilton’s equation
represents the conservation of mechanical energy principle, the generalized
Hamilton’s equation(55,57) describes the conservation of total energy
principle. One can assert that the generalized Hamilton’s equation(55,57) are
the generalization of the classic Hamilton’s equations.

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