Víctor Claver

Víctor Claver Arocas (born 30 August 1988) is a Spanish professional basketball player for FC Barcelona of the Liga ACB and the EuroLeague.

Professional career

Valencia
Claver began his playing career with the youth teams of his school, Sagrado Corazón HH. Maristas de Valencia, in Spain. He made his professional debut with Valencia during the 2006–07 season. He played in Europe's second-tier level competition, the EuroCup, for the first time during the 2007–08 season, and won it in the 2009–10 season.

He also played in the European top-tier level, EuroLeague, during the 2010–11 season, and averaged 10.9 points and 4.9 rebounds, over 14 games.

NBA
Claver was drafted 22nd overall by the Portland Trail Blazers in the 2009 NBA draft. On 12 July 2012, Claver signed a three-year deal with the Trail Blazers. He was assigned to the NBA D-League's Idaho Stampede on 7 December 2012, and recalled on 9 December 2012. The following day, he made his first NBA start, scoring 4 points in 25 minutes with 5 rebounds and 2 assists. Claver was reassigned to the Idaho Stampede on 14 December 2012, and recalled on 16 December 2012.  He played 80 games for the Trail Blazers over the next three years.

On 19 February 2015, Claver was traded, along with Will Barton, Thomas Robinson and a lottery-protected 2016 first-round pick, to the Denver Nuggets in exchange for Arron Afflalo and Alonzo Gee. Three days later, he was waived by the Nuggets before playing in a game for them.

Russia
On 1 March 2015, Claver signed with BC Khimki of Russia for the rest of the season. On 17 July 2015 Claver and Khimki parted ways.

Attempt to return to Spain
In August 2015, he had a deal with Baskonia, but Valencia using the right of first refusal, matched the offer, and kept his rights in Spain. Baskonia obviously was not eager to pay a buy out to Valencia, but the Spanish forward did not want to play in his old team. According to reports, he did not want to play in a team which in the next season was going to compete in the second-tier level EuroCup, which led to Claver signing with a team outside the Spanish borders.

Escape to Russia
On 2 September 2015, the Russian team Lokomotiv Kuban announced a one-year deal with Claver for the 2015–16 season.

Return to Spain
On 20 July 2016, he signed a three-year deal with FC Barcelona Lassa, after his new team reached an agreement with Valencia for acquiring his rights. On 2 July 2019, he extended his contract with FC Barcelona through 2022.

Spanish national team
As a member of the junior national teams of Spain, Claver won a bronze medal at the 2006 FIBA Europe Under-18 Championship, and a bronze medal at the 2008 FIBA Europe Under-20 Championship. He has also played with the senior national team. With Spain's senior team, he won a gold medal at the EuroBasket 2009, EuroBasket 2011, and at the EuroBasket 2015, and a bronze medal at the EuroBasket 2013.

He also played at the 2010 FIBA World Championship, the 2014 FIBA Basketball World Cup, at the 2012 Summer Olympics, where he won a silver medal, and at the 2016 Summer Olympics, where he won a bronze medal.

Career statistics

NBA

Regular season

|-
| style="text-align:left;"|
| style="text-align:left;"|Portland
| 49 || 16 || 16.6 || .392 || .287 || .467 || 2.4 || .9 || .5 || .2 || 3.8
|-
| style="text-align:left;"|
| style="text-align:left;"|Portland
| 21 || 0 || 8.8 || .405 || .167 || .909 || 1.9 || .6 || .1 || .1 || 2.2
|-
| style="text-align:left;"|
| style="text-align:left;"|Portland
| 10 || 0 || 7.6 || .450 || .545 || .000 || 2.0 || .1 || .1 || .1 || 2.4
|- class="sortbottom"
| style="text-align:center;" colspan="2"|Career
| 80 || 16 || 13.4 || .398 || .293 || .585 || 2.2 || .7 || .4 || .2 || 3.2

Playoffs

|-
| style="text-align:left;"|2014
| style="text-align:left;"|Portland
| 2 || 0 || 3.5 || .000 || .000 || .000 || 0.5 || .0 || .0 || .0 || .0
|- class="sortbottom"
| style="text-align:center;" colspan="2"|Career
| 2 || 0 || 3.5 || .000 || .000 || .000 || 0.5 || .0 || .0 || .0 || .0

EuroLeague

|-
| style="text-align:left;"|2010–11
| style="text-align:left;"|Valencia
| 14 || 14 || 27.6 || .472 || .368 || .841 || 4.9 || 1.4 || 1.4 || .4 || 10.9 || 12.1
|-
| style="text-align:left;"|2015–16
| style="text-align:left;"|Lokomotiv
| 31 || 31 || 27.0 || .500 || .400 || .780 || 5.9 || 1.6 || .8 || .5 || 9.4 || 13.5
|-
| style="text-align:left;"|2016–17
| style="text-align:left;" rowspan=3 |Barcelona
| 27 || 19 || 22.0 || .459 || .377 || .778 || 4.4 || 1.3 || .9 || .3 || 6.0 || 8.4
|-
| style="text-align:left;"|2017–18
| 24 || 6 || 16.0 || .451 || .422 || .800 || 2.9 || 1.2 || .6 || .3 || 4.6 || 5.4
|-
| style="text-align:left;"|2018–19
| 35 || 25 || 21.2 || .476 || .339 || .821 || 3.2 || 1.0 || .6 || .4 || 5.3 || 7.5
|- class="sortbottom"
| style="text-align:center;" colspan="2"|Career
| 131 || 95 || 22.5 || .477 || .382 || .806 || 4.2 || 1.3 || .8 || .4 || 6.9 || 9.2

References

External links

Víctor Claver at acb.com

Víctor Claver at fiba.com
Víctor Claver at euroleague.net

Category:1988 births
Category:Living people
Category:2010 FIBA World Championship players
Category:2014 FIBA Basketball World Cup players
Category:2019 FIBA Basketball World Cup players
Category:Basketball players at the 2012 Summer Olympics
Category:Basketball players at the 2016 Summer Olympics
Category:BC Khimki players
Category:FC Barcelona Bàsquet players
Category:Idaho Stampede players
Category:Liga ACB players
Category:Medalists at the 2012 Summer Olympics
Category:Medalists at the 2016 Summer Olympics
Category:National Basketball Association players from Spain
Category:Olympic basketball players of Spain
Category:Olympic bronze medalists for Spain
Category:Olympic medalists in basketball
Category:Olympic silver medalists for Spain
Category:PBC Lokomotiv-Kuban players
Category:Portland Trail Blazers draft picks
Category:Portland Trail Blazers players
Category:Power forwards (basketball)
Category:Small forwards
Category:Spanish expatriate basketball people in Russia
Category:Spanish expatriate basketball people in the United States
Category:Spanish men's basketball players
Category:Sportspeople from Valencia
Category:Valencia Basket players


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A bit more: established in 2000, the NNRF is an independent voluntary organisation which offers practical advice, information, support and friendship to asylum seekers and refugees who have arrived in the city. The Sycamore community centre, located just north of the city between St Anns and Mapperly, is the base for all of the NNRF’s work and aims to provide a welcoming environment for individuals to gather. A wide range of services are run out of the centre, from specialised advice and support services, to programmes for developing skills, knowledge and confidence, through to advocating in cases of injustice and hardship. With the ever-shifting political landscape, the demands on the NNRF are high and the services provided are in a constant state of change.


---
abstract: 'Different types of transformations of a dynamical system, that are compatible with the Hamiltonian structure, are discussed making use of a geometric formalism. Firstly, the case of canonoid transformations is studied with great detail and then the properties of master symmetries are also analyzed. The relations between the existence of constants of motion and the properties of canonoid symmetries is discussed making use of a family of boundary and coboundary operators.'
author:
- |
    José F. Cariñena$^{a)}$, Fernando Falceto$^{b)}$ and Manuel F. Rañada$^{c)}$\
    [*Departamento de Física Teórica, Facultad de Ciencias*]{}\
    [*Universidad de Zaragoza, 50009 Zaragoza, Spain*]{}
title: Canonoid transformations and master symmetries
---

> [*Keywords:*]{}Canonoid transformation, master symmetry, constants of motion
>
> [*Running title:*]{} Canonoid transformations and master symmetries
>
> [*MSC Classification:*]{} Primary: 70H15, 53D22; Secondary: 37J05, 37J15
>
> [*PACS numbers:*]{} 02.40.Yy, 45.20.Jj,

Introduction
============

In the search for the general solution of a dynamical equation one can use an appropriate transformation of the given equation into a simpler one, but one can also make use of reduction procedures leaving to related simpler systems. Such reduction processes are based on the determination of constants of the motion, on one side, or infinitesimal symmetries of the dynamics, on the other.

In differential geometric terms, the dynamics is described by means of a vector field and therefore the theory of transformations of such vector fields is a very important geometric ingredient. In the above mentioned reduction processes the constants of the motion give rise to invariant foliations. On the other side, infinitesimal symmetries of the dynamics allow us to introduce adapted coordinates. Then, the system of differential equations splits into a simpler one involving one less coordinate and another single equation to be solved once the other subsystem has been solved.

The existence of additional structures compatible with the dynamics provides us with additional tools. In particular, a compatible symplectic structure gives us an identification of vector fields with 1-forms, and therefore there is a distinguished class of vector fields, those associated with exact (or at least closed) 1-forms. Consequently, functions play the additional role of being generators of Hamiltonian vector fields. Noether theorem in Hamiltonian dynamics identifies constants of motion with generators of infinitesimal strictly canonical symmetries of the Hamiltonian, therefore the knowledge of a constant of the motion reduces the problem to another one involving two less degrees of freedom. Of course functionally independent constants of the motion cannot be used simultaneously in this way unless they are in involution.

The main objective of this article is to develop a deeper analysis of the theory of transformations on symplectic manifolds. The paper is organized as follows. In Section 2, we introduce the notation and give a short review of the theory of canonical transformations, non-strictly canonical transformations, and canonoid transformations using the symplectic formalism as an approach. Section 3 is devoted to the study of the one-parameter groups of master symmetries and canonoid symmetries and in Section 4 the relation with the existence of constants of motion is studied. Finally, in Section 5 we make some final comments. Appendix A summarizes some properties of two homological differential operates and Appendix B shows the possibility of choosing an appropriate 1-form used in Section 3.

Transformations in symplectic manifolds
=======================================

We first recall that a symplectic manifold is a pair $(M,\omega)$, where $M$ is a differentiable manifold endowed with a symplectic form $\omega$, which is a nondegenerate closed 2-form in $M$, $d\omega = 0$, i.e. $\omega\in Z^2(M)$ (see e.g. [@AbrMarsden]-[@MoFelV90]). There is then a one-to-one $C^\infty(M)$-linear correspondence between the $C^\infty(M)$-module of vector fields and that of 1-forms: If $X$ is in the $C^\infty(M)$-module $\mathfrak{X}(M)  $ of vector fields in $M$, the corresponding 1-form is denoted $\beta_X=i(X)\omega$, and if $\beta\in \forms^1(M)$, its associated vector field $X_\beta$ is the one such that $i(X_\beta)\omega=\beta$. Those vector fields associated to closed 1-forms are called locally-Hamiltonian vector fields and in particular vector fields associated with exact 1-forms $df$ are said to be Hamiltonian vector fields and are denoted $X_f$ instead of $X_{df}$. The set of locally-Hamiltonian and Hamiltonian vector fields are $\mathbb{R}$-linear spaces to be denoted, respectively, $\mathfrak{X}_{\rm LH}(M,\omega)$ and $\mathfrak{X}_{\rm H}(M,\omega)$.

A diffeomorphism $\Phi$ of a manifold $M$ push-forward tensorial fields in $M$. We use the notation $\Phi_*$ instead of $(\Phi^{-1})^*$ for covariant tensors. So, $\Phi_*f=(\Phi^{-1})^*f=f\circ\Phi^{-1}$, $\forall f\in C^\infty(M)$, while $\Phi_*(X)$ is obtained from $\Phi_*(X)(\Phi_*f)=\Phi_*(Xf)$, $\forall f\in C^\infty(M)$.

Those diffeomorphisms leaving invariant a particular tensor field are called symmetries of such a tensor field. Next we consider three fundamental examples:

1.  If $f$ is a function defined in $M$, $f\in C^\infty(M)$, then a symmetry of $f$ is a diffeomorphism $\Phi$ of $M$ such that $\Phi_*(f)=f$.

2.  If $X$ is a vector field on $M$, $X\in\mathfrak{X}(M)$, then a symmetry of $X$ is a diffeomorphism $\Phi$ of $M$ such that $\Phi_*(X)=X$.

3.  If $\alpha$ is a $k$-form in $M$, $\alpha\in\forms^k(M)$, then a symmetry of $\alpha$ is a diffeomorphism $\Phi$ of $M$ such that $\Phi_*(\alpha)=\alpha$.

In this section we study, by making use of a geometrical approach, three different classes of transformations related with the properties of the Hamiltonian formalism: strictly canonical transformations, non-strictly canonical transformations, and canonoid transformations.

In Classical Hamiltonian Mechanics, those transformations of the phase space that preserve the Hamiltonian form of the Hamilton equations, whatever the Hamiltonian function is, are called canonical. These transformations are characterized by the existence of a real number $\lambda$, called valence, such that the Poisson bracket of two transformed functions is $\lambda $ times the transformed of the Poisson bracket of the original functions . The set of canonical transformations is endowed with a group structure and the set of strictly canonical transformations, those corresponding to $\lambda=1$, is a normal subgroup. In differential geometric terms the phase space is a symplectic manifold $(M,\omega)$ and strictly canonical transformations are represented by diffeomorphisms $\Phi\in \Diff(M)$ that preserve the symplectic form, that is, $\Phi_*(\omega)=\omega$.

Strictly canonical transformations
-----------------------------------

In a symplectic manifold $(M,\omega)$ the symmetries of $\omega$, to be called symplectomorphisms, are diffeomorphisms of $M$ such that $\Phi_*(\omega) = \omega$, what is equivalent to $ \Phi^*(\omega) = \omega$. The remarkable point is that if $H$ is a Hamiltonian function and $\Ga_H$ is the associated vector field representing the dynamics, i.e. satisfying the equation $$i(\Ga_H)\omega = dH  \,, \label{Hdyneq}$$ then the following equation is also true $$i(\Phi_*(\Ga_H))\Phi_*(\omega) = d(\Phi_*(H))   \,.       \label{tlaw}$$ Therefore symplectomorphisms that are symmetries of $H$ are also symmetries of $\Ga_H$. However, the above equation permits the existence of symmetries of $\Ga_H$ that are not symplectomorphisms. Of course in this last case the new 2-form $\Phi_*(\omega) \ne\omega $ is admissible for $\Ga_H$ and, as pointed out in [@CaR88], the vector field $\Ga_H$ turns out to be a bi-Hamiltonian system [@Mag78] and therefore non-Noether constant of motion can be found [@CI83].

Given two functions $f$ and $g$ in a symplectic manifold $(M,\omega)$, then the symplectic product of the corresponding Hamiltonian vector fields represents the so called Poisson bracket of these two functions $$\{f,g\} = \om(X_f,X_g)=-X_fg=X_gf \,.$$

Note that for symplectomorphisms $\Phi$ of $M$, $\Phi_*X_f=X_{\Phi_*f}$, and this property leads to $$\Phi^*\{f,g\} = \{\Phi^*f,\Phi^*g\},$$ because $$\{\Phi^*f,\Phi^*g\}=X_{\Phi^{-1}_*f}\Phi^*g=\Phi^{-1}_*(X_f)(\Phi^*g)=\Phi^*(X_fg)= \Phi^*\{f,g\},$$ and therefore the symplectomorphisms preserve the Poisson brackets of any pair of functions. Consequently, they correspond to strictly canonical transformations.

At the infinitesimal level, one-parameter subgroups of symmetry transformations of tensor fields are characterized by the vanishing of the Lie derivative of the tensor field with respect to the vector field $X$ generating the one-parameter subgroup, that is, (i) $\mathcal{L}_Xf:=X(f)=0$ for functions, (ii) $\mathcal{L}_X\Ga:=[X,\Gamma]=0$ for vector fields $\Gamma$, and in general (iii) $\mathcal{L}_X\alpha :=(d\circ i(\Ga)+i(\Ga)\circ d)\alpha= 0$ for a $k$-form $\alpha$. In particular, locally-Hamiltonian vector fields in a symplectic manifold are infinitesimal symplectomorphisms.

Of course, when there exist tensorial relationships among tensorial objects their infinitesimal symmetries are also related. Next we consider two particular situations.

\(1) Let us first consider the Lie derivative with respect to a vector field $X$ of the dynamical equation (\[Hdyneq\]) and use the property $\mathcal{L}_Xi(Y)\alpha-i(Y)\mathcal{L}_X\alpha=i([X,Y])\alpha$, $\forall X,Y\in\mathfrak{X}(M)$ and $\alpha\in\forms(M)$, and we find $$\mathcal{L}_X\Bigl(i(\Ga_H)\omega-dH\Bigr) =  i(\Ga_H) \mathcal{L}_X\omega+i([X,\Ga_H])\omega-d(XH) = 0 \,. \label{Liederdyn}$$ Then, if $X$ is an infinitesimal symmetry of both $\omega$ and $H$ (that is, $\mathcal{L}_X\omega=0$ and $XH=0$) we obtain $$i([X,\Ga_H])\omega = 0\,,$$ and, as $\omega$ is nondegenerate, this means that $X$ is a symmetry of the dynamical vector field $\Ga_H$.

\(2) Let us now suppose that $X$ is such that $\mathcal{L}_X\omega=0$ and $\mathcal{L}_X\Ga_H=[X,\Ga_H]=0$.

Then (\[Liederdyn\]) shows that $\mathcal{L}_XdH=0$, and therefore, if $M$ is connected $\mathcal{L}_XH={\rm const.}$ In particular, if a Hamiltonian vector field $X_f$ is a symmetry of $\Ga_H$ (but not of $H$) then $f$ is not necessary a constant of the motion since the vanishing of the Lie bracket $[X_f,\Ga_H]=0$ only means the vanishing of the differential of the Poisson bracket $$d \bigl(\{f,H\}\bigr) = 0,$$ and from here we can conclude when $M$ is connected that $\mathcal{L}_{\Ga_H}f=\{f, H\}$ is a (not necessarily zero) constant [@LoMR99].

Finally, let us mention that, given a Hamiltonian system $(M,\omega,H)$, one usually look for vector fields whose flows are symplectomorphisms that are also symmetries of $H$ and, therefore, symmetries of $\Ga_H$. Then for each $g\in C^\infty(M)$, the relation $$\mathcal{L}_{\Ga_H}g = \{g,H\} = -\,\mathcal{L}_{X_g}(H)$$ shows that $X_g$ is a symmetry of $H$ if and only if $g$ is a constant of motion. This is a very important property, sometimes called Noether’s theorem in Hamiltonian formalism, because it suggests us a method for finding constants of the motion which are very useful in the process of reduction of the dynamical equation.

The usefulness of non-strictly canonical infinitesimal symmetries (see e.g a generalisation of the virial theorem that can be found in the recent paper [@CFR12]) and the more general case of canonoid transformations has been less analyzed and is worthy of a deeper analysis. Several applications of canonoid transformations can be seen at [@LS07; @TALS] (see also [@DTH] for the Nambu formulation).

Non-strictly canonical transformations
--------------------------------------

As indicated above canonical transformations are those preserving the form of Hamilton equations whatever the Hamiltonian is, or in an equivalent way, preserving the Poisson bracket of any two functions up to a nonzero multiplicative constant: the valence. A transformation with valence different from one is called non-strictly canonical [@SaletanCromer], while those with valence equal to one are said to be strictly canonical. In differential geometric terms these canonical transformations in a symplectic manifold $(M,\omega)$ are represented by diffeomorphisms $\Phi$ such that $$\Phi^*(\omega)=r\,\omega, \quad r\in\mathbb{R}\,,        \label{nsct}$$ and strictly canonical ones are those with $r=1$.

Let the vector field $X$ be the generator of a one-parameter group of canonical transformations $\Phi_\epsilon^*(\omega)=r(\epsilon)\,\omega$. Then, there exists a real number $a\ne 0$ such that $$\mathcal{L}_{X}\omega=a\,\omega ,\label{infnsct}$$ with $r$ and $a$ related by $r(\epsilon)=e^{a\epsilon}$. In an equivalent way, in terms of $\beta_X=i(X)\omega$, the canonicity condition (\[infnsct\]) reads $d\beta_X=a\,\omega$. In particular, the flow of $X$ is made up of strictly canonical transformations (symplectomorphisms) when $a=0$, i.e. when $\beta_X$ is closed.

A diffeomorphism $\Phi$ on $(M,\omega)$ such that $$\Phi^*(\omega)=r\,\omega \,,{\qquad} \Phi^*(H)=r\,H \,,\qquad {\rm with}\quad r\in\mathbb{R},$$ preserves the Hamiltonian vector field $\Ga_H$, because of (\[tlaw\]). At the infinitesimal level, if $X$ is such that $ \mathcal{L}_X\omega=a\,\omega$ and $X(H)=a\, H$, then $$i([X,\Ga_H])\omega =  \mathcal{L}_X[i({\Ga_H}) \omega] -i({\Ga_H})\mathcal{L}_X\omega = \mathcal{L}_X(dH)-a\,i(\Ga_H)\omega=d(a\,H)-a(dH)=0 ,$$ and, using that $\omega$ is nondegenerate, we arrive at $$[X,\Ga_H] = 0$$ so that $X$ is a symmetry of the dynamical vector field.

On the other side, if $X$ is such that $ \mathcal{L}_X\omega=a\,\omega$, then we have $$i([X,\Ga_H])\omega  = ( \mathcal{L}_Xi(\Gamma_H)-i(\Gamma_H) \mathcal{L}_X)\omega=\mathcal{L}_X (dH) -a\,i(\Ga_H)\omega=d(\mathcal{L}_XH-aH). \label{condcansym}$$ Therefore, when $ \mathcal{L}_X\omega=a\,\omega$ and $M$ is connected, $ [X,\Ga_H]=0$ if and only if $\mathcal{L}_XH-aH$ is a numerical constant.

Let us choose a vector field $X_1$ such that $$\mathcal{L}_{X_1}\omega=-\omega.$$ This is only possible when $\omega$ is exact because $\omega=d(-i(X_1) \omega)$. For instance, when $(M,\omega=-d\theta )$ is an exact symplectic manifold the vector field $X_1$ can be chosen to be defined by (see [@NachVer86]) $$i(X_1)\omega=\theta,\qquad {\rm i.e.}\ X_1=X_\theta,$$ because then $$\mathcal{L}_{X_1}\theta=i(X_1)d\theta+d(i(X_1)\theta)=-i(X_1)\omega +d(i(X_1)\theta)=-\theta+d(i(X_1)\theta),$$ and therefore $$\mathcal{L}_{X_1}\omega=-\mathcal{L}_{X_1}(d\theta)=-d\mathcal{L}_{X_1}\theta=d\theta=-\omega.$$

Given a vector field $X$ generating a one-parameter group of non-strictly canonical transformations, we know that there exists a real number $a$ such that $\mathcal{L}_{X}\omega=a\,\omega$ and then the vector field $X+a\,X_1$ is locally-Hamiltonian, because $$\mathcal{L}_{X+a\,X_1}\omega=a\,\omega -a\, \omega =0.$$ That means that there exists a closed 1-form $\alpha$ such that $$i(X)\omega+a\, i(X_1)\omega=\alpha.$$ Conversely, given a closed 1-form $\alpha$ the preceding relation defines a vector field $X$ such that $\mathcal{L}_{X}\omega=a\,\omega$ and then $X$ generates a one-parameter (local) subgroup of non-strictly canonical transformations $\Phi_\epsilon$ with valence $e^{a\epsilon}$.

Canonoid transformations
-------------------------

As indicated above the set of canonical transformations is a catalogue of transformations preserving the Hamilton form of the dynamical equation. However, these are not the only ones that may be relevant for the study of the dynamics. Here, we shall be interested in a type of transformation that preserves the Hamiltonian character of a particular given Hamiltonian system: they are called canonoid transformations with respect to this particular Hamiltonian system [@SaletanCromer; @LeM84; @NeOT87]. They can be useful for the given specific problem, but not for other Hamiltonian systems. Of course, all the canonical transformations are canonoid but the converse is not true.

In geometric terms, given a Hamiltonian vector field $\Ga\in {\mathfrak X}_{{\rm H}}(M,\omega)$ in a symplectic manifold $(M,\omega)$, i.e. there exists a function $H\in C^\infty(M)$ such that $i(\Gamma)\omega=dH$, a transformation $\Phi\in \Diff(M)$ is said to be canonoid with respect to $\Gamma$, or with respect to its Hamiltonian $H$, if the transformed field $\Phi_*\Ga$ is also Hamiltonian, that is, $\Phi_*\Ga\in \mathfrak{X}_{\rm H}(M,\omega)$. Since $\Phi$ is a diffeomorphism, we have that the vector field $\Phi_*\Ga$ is Hamiltonian with respect to $\omega$ if and only if $\Ga$ is Hamiltonian with respect to the transformed 2-form $\Phi^*(\omega)$ [@CaR88], i.e. there exists a function $H'\in C^\infty(M)$ such that $$i(\Ga)\Phi^*(\omega) = d H' \,. \label{defcanonoid}$$ This means that if $\Phi$ is a canonoid transformation for $\Ga$ then $\Ga$ admits a new and different Hamiltonian structure. Therefore, this vector field $\Gamma$ will be a bi-Hamiltonian system, that is, it is Hamiltonian with respect to two different symplectic structures: the original symplectic form $\omega$ and the new one $\Phi^*(\omega)$.

Canonical transformations, either strictly canonical (that satisfy $\Phi^*(\omega)=\omega$) or non-strictly canonical transformations (that satisfy $\Phi^*(\omega)=r\,\omega$, with $0\ne r\in \mathbb{R}$), are canonoid with respect to any Hamiltonian in a trivial way. The converse property is also true and if a given transformation is canonoid with respect to any Hamiltonian function, it is canonical. Even it is enough when the transformation is canonoid with respect to a more reduced family of Hamiltonians (see [@CuS72], [@CaR90] and [@CGILV; @S98] and references therein).

These concepts can be generalised to locally-Hamiltonian systems instead of Hamiltonian ones: If $(M,\omega,\Gamma)$ is a locally-Hamiltonian dynamical system, a diffeomorphism $\Phi:M\to M$ is a canonoid transformation with respect to $\Gamma$ when $\Gamma $ is locally-Hamiltonian with respect to $\Phi^*\omega$, i.e. if and only if $$\mathcal{L}_{\Ga}\,\Phi^*(\omega) = 0 \,, \label{definfcanonid}$$ i.e. $$d [i(\Ga)\Phi^*(\omega)]=0.\label{definfcanonoid2}$$

If we consider not just one transformation $\Phi$ but a one-parameter group of canonoid transformations $\Phi_\ep$, then this family of transformations is canonoid with respect to a Hamiltonian vector field $\Gamma_H$ if and only if its infinitesimal generator $X$ is such that there exists a function $K\in C^\infty(M)$ $$i(\Gamma_H) \mathcal{L}_{X}\omega =dK,\label{definfcanonoid}$$ as one easily sees from $i(\Ga_H)\Phi_\ep^*(\omega) = d _\ep$ when taking the derivative with respect to $\epsilon$ at $\epsilon=0$. Analogously, if $\Gamma$ is locally-Hamiltonian with respect to $\omega$, then $X$ induces a family of canonoid transformations of $\Gamma$ if and only if $$\mathcal{L}_{\Ga} \mathcal{L}_{X}\omega = 0 \,. \label{definfcanonoidloc}$$ Later we shall discuss further characterizations and properties of these transformations, but before that, we are going to introduce a generalization of symmetry and constant of motion that it happens to be closely related to canonoid transformations.

Master symmetries
=================

A function $T$ in a symplectic manifold is said to be a generator of constants of motion of degree $m$ if it is not preserved by the dynamics but it generates a constant of the motion by taking $m$ times its time derivative in an iterative way: $$\frac{d}{dt}T \ne 0\,,\dots\,,\ \frac{d^{m}}{dt^{m}}T \ne 0\,,\
 \frac{d^{m+1}}{dt^{m+1}}T = 0 \,.$$ Of course, for $m=0$ we recover the usual definition of constant of motion.

In differential geometric terms, if $m>0$ and the dynamics is given by a vector field $\Gamma$, these conditions are $$\label{masterfun}
\mathcal{L}_\Gamma T\ne 0\,,\dots,\ \mathcal{L}_\Gamma^{m}T\ne 0\,,\ \mathcal{L}_\Gamma^{m+1}T= 0\,.$$

We can introduce a time dependent observable associated to $T$ $$A=\sum_{n=0}^m(-1)^n \frac{A_n}{n!}t^n,\quad{\rm with}\ A_n=\mathcal{L}_\Ga^n T,$$ that is conserved along the motion, in the sense that $$\frac{d}{dt}A=\left(\mathcal{L}_\Ga+\frac{\partial}{\partial t}\right) A=0.$$

Similarly, as a symmetry of the dynamics $\Gamma$ is a vector field $Z$ such that $[Z,\Ga]   = 0$, a vector field $Z$ that satisfies the following two properties $$\label{masterfielduno}
 [Z,\Ga]   \ne 0  \,,\qquad
 [\,[Z \,,\Ga] \,,\Ga] = 0  \,,$$ is called a ‘master symmetry’ or a generator of symmetries of degree one for $\Ga$. If $m>1$ and $Z$ is such that $$\label{masterfieldm}
[Z\,,\Ga] \ne 0  \,,\ldots,\
[\cdots[\,[Z \,,\underbrace{\Ga] \,,\Ga],\ldots,\Ga}_m]  \ne  0\,,
\ [\cdots[\,[Z \,,\underbrace{\Ga] \,,\Ga],\ldots,\Ga}_{m+1}]  =  0$$ then it is called a ‘master symmetry’ or a generator of symmetries of degree $m$ for $\Gamma$ [@Da93; @Fe93; @Ra99; @Smir96; @FinFo02; @Cas02; @DaSo05; @Ra12]. Last condition in (\[masterfieldm\]) can also be written as $\mathcal{L}_{\Gamma}^{m+1}(Z)=0$, in complete analogy to (\[masterfun\]). As we will see below, for a Hamiltonian dynamical system this analogy goes further.

Let us now consider a locally Hamiltonian dynamical system $(M,\omega,\Gamma)$. Observe that the relation $\mathcal{L}_\Gamma (i(X)\omega)=i(\mathcal{L}_\Gamma X)\omega$ can be generalised to $$\mathcal{L}_\Gamma^k (i(X)\omega)=i(\mathcal{L}_\Gamma^k X)\omega,\quad \forall k\in \mathbb{N},\label{genleib}$$ which can easily be checked by induction on the number $k$, because it is valid for $k=1$ and if assumed true for a given index $k$, then $$\mathcal{L}_\Gamma^{k+1} (i(X)\omega)=\mathcal{L}_\Gamma(i(\mathcal{L}_\Gamma^k X)\omega)=i(\mathcal{L}_\Gamma^{k+1} X)\omega.$$ Using this property (\[genleib\]) for the Hamiltonian vector field $X_T$ associated to the function $T \in C^\infty(M)$ we obtain $$\mathcal{L}_\Gamma^k (i(X_T)\omega)=i(\mathcal{L}_\Gamma^k X_T)\omega,$$ and therefore if $T$ is the generator of constants of motion of degree $m$, then its associated Hamiltonian vector field $X_T$ is a master symmetry of degree $m$, because if $\mathcal{L}_\Gamma^{m+1}T=0$, then $d(\mathcal{L}_\Gamma^{m+1}T)=0$, and therefore from $$i(\mathcal{L}_\Gamma^{m+1}X_T)\omega=\mathcal{L}_\Gamma^{m+1}\left(i(X_T)\omega\right)=\mathcal{L}_\Gamma^{m+1}(dT)=d(\mathcal{L}_\Gamma^{m+1}T)=0,$$ we obtain that $\mathcal{L}_\Gamma^{m+1}X_T=0$. We call $T$ the generator of such Hamiltonian master symmetry. The converse is not true, in general, because in order for the Hamiltonian vector field $X_T$ to be a master symmetry of degree $m$, the preceding relation shows that it is enough to demand that $X_{\mathcal{L}_{\Gamma_H}^{m+1}T}=0$, or equivalently $$\label{hammassym}
d\,\mathcal{L}_{\Ga_H}^{m+1}\, T=0.$$

Recall that in the particular case of a Hamiltonian dynamical system $(M,\omega,H)$, $$\mathcal{L}_{\Gamma_H}^{k} T=\{\cdots\{T \,,\underbrace{H\} \,,H\},
\ldots,H\}}_k.$$ And the previous property can be rephrased by saying that a Hamiltonian vector field $X_T$ is a master symmetry of degree $M$ if and only if $$d\{\cdots\{T \,,\underbrace{H\} \,,H\},
\ldots,H\}}_{m+1}=0.$$

Next we illustrate this situation with a simple example. The Hamiltonian $H$ and the vector field $\Ga_H$ of the one dimensional free particle are given by $$H = \frac{1}{2}\,p^2 \,,{\quad} \Ga_H = p\,\pd{}{q} \,.$$ Then $X = \partial/\partial p$ satisfies $$\Bigl [p\,\pd{}{q},\pd{}{p}\Bigr]   = \pd{}{q}  {\quad}{\rm and}{\quad}
\Bigl [p\,\pd{}{q},\pd{}{q},\Bigr]   =  0\,,\label{master}$$ i.e. $\mathcal{L}_{\Gamma_H}^2(X)=0$. Therefore $X$ is a master symmetry of degree $m=1$ for $\Gamma_H$.

In this example the vector field is Hamiltonian, $X=X_G$, with $G(q,p)=-q$, and the relation (\[master\]) can be rephrased in terms of Poisson brackets $$\{G,H\}=-p\not=0\,,{\qquad}\{\{G,H\},H\}=0,$$ i.e. $$\mathcal{L}_{\Gamma_H} G=-p\ne 0, \qquad \mathcal{L}_{\Gamma_H}^2 G=0,$$ and consequently $G$ is the generator of a master symmetry of degree $m=1$.

Infinitesimal canonoid transformations and constants of the motion
==================================================================

In this section we shall describe different ways of characterizing one-parameter groups of canonoid transformations for Hamiltonian and locally-Hamiltonian systems by using their infinitesimal generators.

Let $X\in \mathfrak{X}(M)$ be a vector field in a symplectic manifold $(M,\omega)$ with a flow made of canonoid transformations with respect to a Hamiltonian $H$, then, as indicated before, $i(\Ga_H)\mathcal{L}_X\omega$ is an exact 1-form and therefore $$i(\Ga_H)\mathcal{L}_X\omega=dK\label{defK}.$$ Now a contraction with $\Ga_H$ of both sides of (\[defK\]) shows that such a function $K$ is a constant of the motion. In particular, when the flow of $X$ is made up of non-strictly canonical transformations, there exists a nonzero real number $a$ such that $\mathcal{L}_X\omega=a\,\omega$ holds and the function $K$ turns out to be $K=a\, H$.

An equivalent way of characterizing canonoid transformations for a (locally-) Hamiltonian vector field in a symplectic manifold $(M,\omega)$ is the following one:

a\) The vector field $X\in \mathfrak{X}(M)$ is the infinitesimal generator of a group of canonoid transformations for a locally-Hamiltonian vector field $\Gamma\in  \mathfrak{X}_{\rm LH}(M,\omega)$ if and only if $[X,\Gamma]$ is locally-Hamiltonian, $[X,\Gamma]\in \mathfrak{X}_{\rm LH}(M,\omega)$.b) Analogously $X$ is the infinitesimal generator of a group of canonoid transformations for a Hamiltonian $H$ if and only if $[X,\Gamma_H]$ is a Hamiltonian vector field, $[X,\Gamma_H]\in  \mathfrak{X}_{\rm H}(M,\omega)$. In this case its Hamiltonian function is $\mathcal{L}_XH-K$, where $K$ is like in (\[defK\]).

Proof.-   a) We first compute $i([X,\Gamma])\omega$ and show that it is a closed form if and only if $X$ generates canonoid transformations. In fact, if $\Gamma\in  \mathfrak{X}_{\rm LH}(M,\omega)$, then $i(\Gamma)\omega$ is a closed form and $$i([X,\Gamma])\omega= \mathcal{L}_X[i(\Gamma)\omega]-i(\Gamma)\mathcal{L}_X\omega=d\left(\omega(\Gamma,X)\right)-
i(\Gamma)\mathcal{L}_X\omega.\label{iXG}$$ Therefore, $[X,\Gamma]$ is a locally-Hamiltonian vector field if and only if $i(\Gamma)\mathcal{L}_X\omega$ is a closed 1-form, or, equivalently, if and only if $X$ is an infinitesimal canonoid transformation for $\Gamma$.

b\) Computing now $i([X,\Gamma_H])\omega$ as before we can conclude that $i([X,\Gamma_H])\omega$ is exact if and only if $i(\Gamma_H)\mathcal{L}_X\omega$ is exact. Moreover, in this case assuming that $i(\Gamma_H)\mathcal{L}_X\omega=d K$, as $\omega(\Gamma,X)=dH(X)=XH$, (\[iXG\]) for $\Gamma=\Gamma_H$ reduces to $$i([X,\Gamma_H])\omega=d(\mathcal{L}_XH)-dK,$$ which proves that $[X,\Gamma_H]\in  \mathfrak{X}_{\rm H}(M,\omega)$, and its Hamiltonian function is $\mathcal{L}_XH-K$. [$\square$]{}\
qed

As an immediate consequence of the previous proposition we see that any infinitesimal symmetry of the dynamics is the infinitesimal generator of a family of canonoid transformations.

Cohomological techniques have been used [@CI88] for studying non-canonical groups of transformations. In this paper, in order to further study infinitesimal canonoid transformations of a locally-Hamiltonian dynamical system $(M, \omega, \Gamma)$ and their relations with symmetries, we find useful to consider the differential operator of degree $-1$ on $\forms^\bullet(M)$ given by the contraction with the dynamical vector field, i.e. $$i(\Gamma):\forms^{k+1}(M)\rightarrow\forms^k(M),$$ which obviously satisfies $i(\Gamma)\circ i(\Gamma)=0$. This, together with the de Rahm differential, allows us to define new twisted boundary and coboundary operators, namely: $$\partial_\Gamma:=i(\Gamma)\circ d\circ i(\Gamma)\,,
\qquad
d_\Gamma:=d\circ i(\Gamma)\circ d\,.$$ Clearly both satisfy $\partial_\Gamma\circ\partial_\Gamma=0$, $d_\Gamma\circ d_\Gamma=0$ and the following relations $$i(\Gamma)\circ\partial_\Gamma=\partial_\Gamma\circ i(\Gamma)=0,\quad
d_\Gamma\circ d=d\circ d_\Gamma=0, \quad \mathcal{L}_\Gamma \circ d_\Gamma= d_\Gamma  \circ   \mathcal{L}_\Gamma, \quad \mathcal{L}_\Gamma \circ
\partial _\Gamma= \partial_\Gamma  \circ   \mathcal{L}_\Gamma.$$ Further properties of these operators are discussed in the Appendix A.

The closed and exact forms in $d_\Gamma$ cohomology have interesting dynamical properties. An example of this is contained in the following proposition:

Given a locally-Hamiltonian dynamical system $(M,\omega, \Gamma)$, the zero cohomology group of $d_\Gamma$, $H^0_\Gamma(M)=Z^0_\Gamma(M)=\{f\in C^\infty(M)\mid d_\Gamma f=0\}$ coincides with the set of generators of Hamiltonian dynamical symmetries.

Proof.-   Let be $f\in C^\infty(M)$ and $X_f$ its Hamiltonian vector field, i.e. $$\label{hvf}
  df=i(X_f)\omega.$$ Then, taking the Lie derivative in both sides of (\[hvf\]), we have $$\mathcal{L}_\Gamma\,df=i([\Gamma,X_f])\omega + i(X_f)\,\mathcal{L}_\Gamma\omega \,,  \label{lgdf}$$ from where using that $\Gamma$ is a locally Hamiltonian vector field, and therefore it satisfies $\mathcal{L}_\Gamma\omega=0$, and the fact that $\omega$ is nondegenerate, we arrive to $[\Gamma,X_f]=0$. That is, $X_f$ is a dynamical symmetry of $\Gamma$ if and only if $\mathcal{L}_\Gamma df= d_\Gamma f=0$, i.e. $f\in Z_\Gamma^0(M)$.\
qed

The generators of master symmetries can be characterized similarly. The result is that $G$ is the generator of a master symmetry of degree $m$ of $\Gamma$ if and only if $\mathcal{L}_\Gamma^m G\in Z^0_\Gamma(H)$, because $$d\mathcal{L}_\Gamma^{m+1}G=d\mathcal{L}_\Gamma(\mathcal{L}_\Gamma^{m}G)=(d\circ i(\Gamma)\circ d)(\mathcal{L}_\Gamma^{m}G)=d_\Gamma(\mathcal{L}_\Gamma^{m}G).$$

The space of exact 1-forms with respect to $d_\Ga$, $B_\Ga^1(M)=\{d_\Ga f \mid  f\in C^\infty(M)\}$, has also its dynamical interpretation.

For a given locally-Hamiltonian dynamical system, $(M,\omega,\Gamma)$, the vector field associated to the $d_\Gamma$-exact 1-form $\beta=d_\Gamma f$ is $X_\beta=[\Gamma, X_f]$, where $X_f$ is the Hamiltonian vector field of $f$.

Proof.-  As $\Gamma$ is a locally-Hamiltonian vector field, relation (\[lgdf\]) reduces to $$i([\Gamma,X_f])\,\omega=\mathcal{L}_\Gamma d f=d_\Gamma f.$$\
qed

Finally the space of closed 1-forms with respect to $d_\Ga$ is related to infinitesimal canonoid transformations as it is shown in the following proposition:

Let $(M,\omega,\Gamma)$ be a locally-Hamiltonian dynamical system and consider $\beta\in\forms^1(M)$. Then $X_\beta\in\vfield(M)$ such that $i(X_\beta)\omega=\beta$ is the infinitesimal generator of a canonoid transformation if, and only if, $\beta$ is $d_\Gamma$ closed, i.e. $\beta\in Z^1_\Gamma(M)=\{\beta\in\forms^1(M)\mid  d_\Gamma\beta=0\}$.

Proof.-   Recall that $X_\beta$ induces a family of canonoid transformations if, and only if, $\mathcal{L}_\Gamma(\mathcal{L}_{X_\beta}\omega)=0$, or equivalently, $(d\circ i(\Gamma)\circ d\circ i(X_\beta) )\omega=0$, where $d\omega=0$ has been used. In terms of the twisted differential $d_\Gamma$ and $\beta$ the previous relation reduces to $d_\Gamma \beta=0$.\
qed

Note that this proposition is the translation to canonoid transformations and twisted cohomology of the well known result about canonical transformations that are generated by closed forms in the de Rahm cohomology. The 1-form $\beta$ is called the generator of the transformation.

Next we shall consider infinitesimal canonoid transformations which are master symmetries of degree $m$ of the dynamics. We will show that for every such a transformation we can associate the generator of a Hamiltonian master symmetry of degree $m-1$.

Let $(M,\omega,\Gamma )$ be a locally-Hamiltonian dynamical system and assume that $\beta\in \forms^1(M)$ generates an infinitesimal canonoid transformation for $\Gamma$. Then $X_\beta$ is a master symmetry of degree $m\ge 1$ for $\Gamma$ if and only if $i(\Gamma) \beta$ is the generator of a Hamiltonian master symmetry of degree $m-1$.

Proof.-   $X_\beta$ is a master symmetry of degree $m$ if and only if $$\label{masterprop}
\mathcal{L}_\Ga^{m+1 }X_\beta=0,
\qquad
{\rm with}
\quad\
\mathcal{L}_\Ga^{m}X_\beta\ne0,$$ and using that $\Gamma$ is Locally-Hamiltonian, $\mathcal{L}_{\Ga}\omega=0$, and the above mentioned property (\[genleib\]) for $k=m+1$ and $X=X_\beta$, i.e. $$\mathcal{L}_\Ga^{m+1}\left( i(X_\beta)\omega\right)=i(\mathcal{L}_\Ga^{m+1}X_\beta)\omega,$$ together with the definition of $X_\beta$, we see that (\[masterprop\]) can be equivalently written as $$\mathcal{L}_\Ga^{m+1 }\beta=0
\qquad
{\rm and}
\quad
\mathcal{L}_\Ga^{m}\beta\ne 0.$$ Now, if $X_\beta$ is an infinitesimal canonoid transformation $d_\Gamma\beta=0$, and we have, $$0=\mathcal{L}_\Ga^{m+1}\beta=\mathcal{L}_\Ga^{m}( (i(\Gamma)\circ d+d\circ i(\Gamma))\beta) =(d\circ\mathcal{L}_\Ga^{m})( i(\Ga)\beta),
\qquad {\rm for}\quad m\geq 1,$$ which, according to eq. (\[hammassym\]), is equivalent to say that $i(\Gamma)\beta$ is the generator of a Hamiltonian master symmetry of degree $m-1$.\
qed

So far we have put into relation canonoid master symmetries with Hamiltonian master symmetries of lower degree. In the paragraphs below we shall go in the opposite direction, namely we shall relate canonoid transformations which are symmetries of the dynamics (recall that every symmetry of the dynamics is a canonoid transformation) with generators of constants of motion of degree one.

With this aim we take, for a locally-Hamiltonian dynamical system $(M, \omega, \Gamma)$, a 1-form $\beta\in Z^1_\Gamma(M)$, i.e. $d_\Gamma\beta=0$, and therefore, $X_\beta$ is the infinitesimal generator of a family of canonoid transformations. Assume that, at least locally, $\beta$ can be written $$\label{definitionofalpha}
\beta=\alpha+d G,$$ where $\partial_\Gamma\alpha=0$ (it is easy to show that this can always be achieved for the generators of symmetries or around points where $\Gamma$ does not vanish, see Appendix B).

Note that $\partial_\Gamma\alpha =0$ implies $i(\Gamma)(d(i(\Gamma)\alpha))=0$ and therefore the function $i(\Gamma)\alpha$ is a constant of motion. Moreover, (\[definitionofalpha\]) shows that $d_\Gamma\alpha=0$.

On the other hand, $\mathcal{L}_\Gamma \alpha$ is a closed 1-form because $d\mathcal{L}_\Gamma \alpha=d_\Gamma \alpha=d_\Gamma \beta=0$.

The connection between canonoid symmetries and master symmetries is expressed in the following proposition:

If $X_\beta$ is the infinitesimal generator of a group of dynamical symmetries of $\Gamma\in\mathfrak{X}_{\rm LH}(M,\omega)$, with $\beta=\alpha+d G$ and $\partial_\Gamma\alpha =0$, then $\mathcal{L}_\Gamma\alpha$ is exact, i.e. there exists a function $F$, uniquely defined up to addition of a constant, such that $$\label{definitionofF}
\mathcal{L}_\Gamma\alpha=dF,$$ the function $G$ is the generator of a constant of motion of degree one and the above function $F$ can be chosen such that $$\mathcal{L}_\Gamma G+F=0,\quad \mathcal{L}_\Gamma^2G=0.\label{GGF}$$

Proof.-   Consider the following equalities $$i([X_\beta,\Gamma])\omega= -\mathcal{L}_{\Gamma} (i(X_\beta)\omega)             +
i(X_\beta)\mathcal{L}_{\Gamma}\omega=
-\mathcal{L}_{\Gamma} (\alpha+dG),$$ we see that if $i([X_\beta,\Gamma])=0$ then $\mathcal{L}_{\Gamma} \alpha$ is exact and using the defining property for $F$ (\[definitionofF\]), the function $F+\mathcal{L}_\Gamma G$ is constant in every connected component of $M$. With an adequate choice of the function $F$ in (\[definitionofF\]), $F+\mathcal{L}_\Gamma G$ can be set to zero. Moreover, the function $F$ is a constant of motion, because $$\begin{aligned}
\mathcal{L}_\Gamma F=  i(\Gamma)\ dF=i(\Gamma)\mathcal{L}_\Gamma\alpha=
 \partial_\Gamma\alpha=0,\end{aligned}$$ and therefore $\mathcal{L}_\Gamma^2G=-\mathcal{L}_\Gamma F=0$, i.e. $G$ is the generator of a constant of the motion of degree one.\
qed

This result can also be stated by saying that if the function $G$ and the 1-form $\beta$ are related by (\[definitionofalpha\]) and $X_\beta$ is a dynamical symmetry of $\Gamma$, then $X_G$ is a master symmetry of degree one. If the dynamical system is Hamiltonian with $\Gamma=\Gamma_H$ then the relations (\[GGF\]) are $$\{G,H\}+F=0,\quad {\rm and} \quad  \{F,H\}=0.\label{GGF2}$$

There is a kind of converse property. If $G$ is the generator of a constant of the motion of degree one, then the function $F=-\mathcal{L}_\Gamma G$ is a constant of the motion. Now, for each 1-form $\alpha$ such that (\[definitionofF\]) is satisfied we obtain that $\partial_\Gamma\alpha=i(\Gamma)\mathcal{L}_\Gamma \alpha=0$ and $d_\Gamma\alpha=d(\mathcal{L}_\Gamma \alpha)=0$. Therefore, the 1-form $\beta $ given by (\[definitionofalpha\]) satisfies $d_\Gamma\beta=0$, and consequently, $X_\beta$ generates a one-parameter group of canonoid transformations. Moreover, using the relations $$0=d(F+\mathcal{L}_\Gamma G)=\mathcal{L}_{\Gamma} (\alpha+dG)=
\mathcal{L}_{\Gamma} (i(X_\beta)\omega)    ,$$ as $\Gamma$ is locally-Hamiltonian, the preceding expression becomes $$0=\mathcal{L}_{\Gamma} (i(X_\beta)\omega   -
i(X_\beta)\mathcal{L}_{\Gamma}\omega=-i([X_\beta,\Gamma])\omega,$$ hence, $[X_\beta,\Gamma]=0$, and $X_\beta$ is a canonoid dynamical symmetry.

Note that the connection between (canonoid) dynamical symmetries and generators of constants of motion of degree one is a generalisation of the strictly canonical case. In the latter $\alpha=0$, $F$ vanishes and the master symmetry of degree one is actually of degree zero, i.e. $G$ is a constant of motion.

To illustrate the previous results we can consider the free particle in $\mathbb{R}$. The phase space $T^* \mathbb{R}$ is endowed with the canonical symplectic structure and the dynamics is $\Gamma=p\,\partial/\partial q$, with $H=p^2/2$. Let now $G$ be given by $$G=q\,p\,f(p),$$ and then in order for $F$ to satisfy the first relation in (\[GGF2\]), $\{G,H\}+F=0$, we must choose $F=-p^2\, f(p)$. If the 1-form $\alpha$ is given by $\alpha=-(2f(p)+pf'(p))q\ dp$, that satisfy $i(\Gamma)\alpha=0$, and therefore $\partial_\Gamma\alpha=0$, then $\beta=\alpha+dG$ generates a canonoid transformation $$X_\beta=f(p)\left(p\pd{}p+q\pd{}q\right),$$ that clearly does not preserve (even up to a factor) the symplectic form. Also notice that the previous expression (upon the addition of a Hamiltonian vector field $g(p)\,\partial/\partial q$) is the most general form for a dynamical symmetry of the one dimensional free particle. We should remark that the correspondence between (Hamiltonian) master symmetries and dynamical symmetries is not one to one. In fact, for the same master symmetry as before we can choose a different one form $\alpha$ and obtain a completely different symmetry. For instance, if we take $\alpha=-d(p\,q\, f(p))$ that satisfies all the required properties and the same function $G$ as before, we obtain $X=0$.

Also the other way around: for any canonoid dynamical symmetry $X_\beta$, as shown in the appendix, $\partial_\Gamma\beta=0$, which means that we can always take $G=0$ and the Hamiltonian master symmetry is trivial.

Final comments
==============

In this paper we have studied, from a geometric perspective, the different transformations of a dynamical system that preserve the Hamiltonian character of the equations of motion. We emphasize their similarities and discuss in depth the case of canonoid transformations that are characterized by preserving the structure of the equations for a particular Hamiltonian. This type of transformations include, in particular, all the dynamical symmetries of the system.

We present different intrinsic characterizations of the infinitesimal generators for one-parameter groups of canonoid transformations and how they are related to canonical transformations. A useful tool for achieving this goal are certain [*twisted*]{} homological and cohomological operators that are discussed in the paper.

On the other hand we introduce a generalization of symmetries and constants of motion: the so called master symmetries and generators of constants of motion. The latter, actually, can be identified with a conserved quantity that is polynomial in time. We establish two types of relations between master symmetries and canonoid transformations that are symmetries of the dynamics.

An interesting point that could be worth studying is to try to extend the relation between dynamical symmetries and canonoid transformations to the case of master symmetries. Exactly as all dynamical symmetries are canonoid transformations one could enlarge the class of transformations so that they include all master symmetries of the system.

Appendix A {#appendix-a .unnumbered}
==========

Let $X\in\mathfrak{X}(M)$ be a vector field in a connected differentiable manifold. We define a map $d_X:\forms(M) \to\forms(M)$ as follows: $$d_X:=d\circ i(X)\circ d= d\circ\mathcal{L}_X=\mathcal{L}_X\circ d.$$

It is a degree one $\mathbb{R}$-linear map $d_X:\forms^\bullet(M)\to \forms^{\bullet+1}(M)$ such that:

-   $d_X\circ d_X=0$ and $d\circ d_X=d_X\circ d=0$.

-   $d_X\circ\mathcal{L}_X=\mathcal{L}_X\circ d_X$.

-   $d_X\circ i(X)+i(X)\circ d_X=\mathcal{L}_X^2$.

-   It is not a derivation but, for $\alpha, \beta\in \forms^r(M)$, it satisfies $$d_X(\alpha\wedge \beta)= d_X\alpha\wedge \beta+ (-1)^r \alpha\wedge d_X\beta+d\alpha\wedge \mathcal{L}_X\beta+(-1)^r \mathcal{L}_X\alpha\wedge d\beta  \,.$$

As $d_X\circ d_X=0$ we can define an associated cohomology where $B^0_X(M)$ is defined as $B^0_X(M)=\{0\}$ and $Z^r_X(M)$ and $B^r_X(M)$, $r\in \mathbb{N}$, are given by $$Z^r_X(M)=\{\alpha\in \forms\,\!\!^r(M)\mid d_X\alpha=0\},$$ and $$B^r_X(M)=\{\alpha\in \forms\,\!\!^r(M)\mid \exists \beta\in\forms\,\!\!^{r-1}(M) ,\
\alpha=d_X\beta\} \,, \quad r\geq 1.$$

We remark that a consequence of the definition of $d_X$ is the following chain of inclusions $$B_X^r(M)\subset B^r(M)\subset Z^r(M)\subset Z^r_X(M).$$

We can characterize differently the space of $d_X$-closed and exact forms $$\begin{array}{rcl}
B_X^r(M)&=&\{\alpha\in \forms\,\!\!^r(M)\mid  \exists \beta\in B^{r}(M),\ \alpha ={\mathcal{L}}_X\beta \}  \\
Z_X^r(M)&=&\{\alpha\in \forms\,\!\!^r(M)\mid  \mathcal{L}_X\alpha\in Z^r(M)\}
\end{array}$$ i.e. we can say that $B_X^r(M)$ is the image of $B^r(M)$ under $\mathcal{L}_X$ while $Z_X^r(M)$ is the preimage of $Z^r(M)$.

We have also introduced a degree -1 operator $\partial_X:\forms^r(M)\rightarrow\forms^{r-1}$ of the following form: $$\partial_X:=i(X)\circ d\circ i(X)
=i(X)\circ\mathcal{L}_X
=\mathcal{L}_X\circ i(X).$$ It satisfies

-   $\partial_X\circ \partial_X=0$ and $i(X)\circ \partial_X=\partial_X\circ i(X)=0$.

-   $\partial_X\circ\mathcal{L}_X=\mathcal{L}_X\circ \partial_X$.

-   $\partial_X\circ d=i(X)\circ d_X$ and $d\circ \partial_X=d_X\circ i(X)$.

-   $\partial_X\circ d+d\circ \partial_X=\mathcal{L}_X^2$.

-   $\partial_X\circ d_X+d_X\circ \partial_X=\mathcal{L}_X^3$.

-   For $\alpha,\beta\in \forms^r(M)$, we have $$\partial_X(\alpha\wedge \beta)= \partial_X\alpha\wedge \beta+ (-1)^r \alpha\wedge \partial_X\beta+i(X)\alpha \wedge \mathcal{L}_X\beta+(-1)^r \mathcal{L}_X\alpha\wedge i(X)\beta  \,.$$

Appendix B {#appendix-b .unnumbered}
==========

Now we address the problem of existence of the [*gauge fixing*]{} 1-form i.e. given $\alpha$ such that $d_\Ga\alpha=0$ does there exist a function $f$ such that $\partial_\Gamma(\alpha+df)=i(\Gamma)di(\Gamma)(\alpha+df)=0$? We have two partial positive answers to that question: a global one when $X_\alpha$ is a symmetry of the dynamics and a local one around a point in which $\Ga$ does not vanish.

The first result is contained in the following proposition.

Given a locally-Hamiltonian dynamical system $(M,\omega,\Gamma)$, if the 1-form $\alpha$ is such that $[X_\alpha,\Ga]=0$, then $d_\Gamma\alpha=0$ and $\partial_\Gamma\alpha=0$.

Proof.-  The relation $$\mathcal{L}_\Gamma\alpha=\mathcal{L}_\Gamma(i(X_\alpha)\omega)=i([\Gamma,X_\alpha])\omega$$ shows that if $X_\alpha$ is a symmetry we have $\mathcal{L}_\Ga\alpha=0$ and applying to this identity the operator $d$ or $i(\Gamma )$ we obtain both results, because $$d(\mathcal{L}_\Gamma\alpha)=d_\Gamma\alpha=0,\quad i(\Gamma)\mathcal{L}_\Gamma\alpha=\partial_\Gamma\alpha=0.$$\
qed

The local result is made more precise in the following proposition.

Let $p$ be a point in a Hamiltonian dynamical system $(M,\omega,H)$ such that $(\Gamma_H)_p\not=0$ and $\alpha$ any 1-form in $M$. Then, there exists a function $f$, locally defined around $p$, such that $\partial_{\Gamma_H}(\alpha+df)=i(\Gamma_H)di(\Gamma_H)(\alpha+df)=0$.

Proof.-   Note first that if the function $f$ is such that $$\label{gaugefixing}
 \mathcal{L}_{\Gamma_H} f= -i(\Gamma_H)\alpha$$ applying $i_(\Gamma_H)\circ d$ to both sides we obtain $\partial_{\Gamma_H}(\alpha+df)=0$.

But using the straightening out theorem (see e.g. [@CrampPirani],[@AbrMarsdenRatiu]), if $\Gamma _H$ is different from zero at the point $p$, (\[gaugefixing\]) can be transformed into an explicit first-order ordinary differential equation around $p$, whose solution always exists locally.\
qed

As the previous proposition shows the difficulties for finding a locally defined 1-form in the family satisfying locally condition $i(\Gamma_H)\,d\,i(\Gamma_H)\alpha=0$ arise when the dynamical vector field vanishes at one point. In this case we can exhibit en example in which the equation (\[gaugefixing\]) cannot be solved.

Consider the Harmonic oscillator in one dimension with Hamiltonian given by the function in the phase space $T^*\mathbb{R}$, endowed with its canonical symplectic structure $\omega_0$, $$H=\frac12(p^2+q^2),$$ and therefore $$\Gamma_H=p\pd{}q-q\pd{}p.$$ Note that $dH$, and therefore also $\Gamma_H$, vanish at the point $(0,0)$.

Take the canonical 1-form $\alpha=\theta_0=p\, dq$. One easily sees that $i(\Gamma_H)\alpha= p^2$, $i(\Gamma_H)d\alpha=-dH$, and it shows that $d i(\Gamma_H)d\alpha=0$ but the equation (\[gaugefixing\]) in this case reads $$q\pd fp-p\pd fq=p^2,$$ and the smooth solution should satisfy $$\begin{aligned}
\pd fq(q, p)&=&q\, g(q,p)-p,\cr
\pd fp(q,p)&=&p\, g(q,p).\end{aligned}$$ for some smooth function $g$. This pair of equations cannot be solved around $p=q=0$ because from them we get $$0=\pd {}q\left(\pd fp\right)-\pd{}p\left(\pd fq\right)(q,p)=
1-q\pd gp(q,p)+p\pd{}gq(q,p)$$ and the right hand side does not vanish at $p=q=0$.

In this situation, however, instead of the stronger condition (\[gaugefixing\]) we can satisfy the weaker condition $i(\Gamma_H)\,d\,i(\Gamma_H)\alpha=0$, i.e. $$\Gamma_H(\Gamma_H f)=-i(\Gamma_H)di(\Gamma_H)\alpha,$$ or in other words $$\left(p\pd{}q-q\pd{}p\right)^2 f= 2\,q\,p$$ which can be solved by $f(q,p)=-\frac12q\,p$. The new equivalent 1.form $$\alpha'=\alpha+df=\frac12(p\,dq-q\,dp),$$ satisfies $\partial_\Gamma \alpha'=0$.

Acknowledgments {#acknowledgments .unnumbered}
===============

We acknowledge the support from research projects MTM–2009–11154, MTM–2010-12116-E, FPA–2009-09638 (MEC, Madrid) and DGA-E24/1, DGA-E24/2 (DGA, Zaragoza).

[99]{}

R. Abraham and J.E. Marsden, Benjamin/Cummings Pub., 2nd ed., 1978. G. Marmo, E.J. Saletan, A. Simoni, and B. Vitale, , J. Wiley, Chichester, 1985. M. Crampin and F.A.E. Pirani, Cambridge Univ. Press, Cambridge, 1986. M. de León and P.R. Rodrigues, North-Holland, Amsterdam, 1989. G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo, and C. Rubano, *The inverse problem in the calculus of variations and the geometry of the tangent bundle*, Phys. Rep. **188** (1990), 147–284. E.J. Saletan and A.H. Cromer, J. Wiley, 1971. E.J. Saletan and J.V. José, Cambridge Univ. Press, Cambridge, 1995. J.F. Cariñena and M.F. Rañada, *Canonoid transformations from a geometric perspective*, J. Math. Phys. **29** (1988), 2181–2186. F. Magri, *A simple model of the integrable Hamiltonian equation*, J. Math. Phys. **19** (1978), 1156–1162. J.F. Cariñena and L.A. Ibort, *Non-Noether constants of motion*, J. Phys. A: Math. Gen. **16** (1983), 1–7. C. López, E. Martínez, and M.F. Rañada, *Dynamical symmetries, non–Cartan symmetries, and superintegrability of the $n$–dimensional harmonic oscillator*, J. Phys. A: Math. Gen. **32** (1999), 1241–1249. J.F. Cari\~ nena, F. Falceto, and M.F. Rañada, *A geometric approach to a generalised Virial theorem*, J. Phys. A: Math. Theor. **45** (2012), 395210. G. Landolfi and G. Soliani, *On certain canonoid transformations and invariants for the parametric oscillator*, J. Phys. A: Math. Theor. **40** (2007), 3413–3423 P. Tempesta, E. Alfinito, R. A. Leo, and G. Soliani, *Quantum models related to fouled Hamiltonians of the harmonic oscillator*, J. Math. Phys. **43** (2002), 3538–3553. T. Dereli, A. Tegmen and T. Hakioglu, *Canonical Transformations in Three-Dimensional Phase-Space*, Int. J. Modern Phys. A **24** (2009), 4769–4788. B Nachtergaele and A. Verbeure, *Groups of canonical transformatioins and the virial Noether theorem*, J. Geom. Phys. **3** (1986), 315–325. C. Leubner and M. Marte, *Generalized canonical transformations and constants of the motion*, Phys. Lett. A **101** (1984), 179–184. L. Negri, L.C. Oliveira, and J.M. Teixeira, *Canonoid transformations and constants of motion*, J. Math. Phys. **28** (1987), 2369–2372. D.G. Currie and E.J. Saletan, *Canonical transformations and quadratic hamiltonians*, Nuovo Cimento B **9** (1972), 143–153. J.F. Cariñena and M.F. Rañada, *Generating functions, bi-Hamiltonian systems and the quadratic-Hamiltonian theorem*, J. Math. Phys. **31** (1990), 801–807. J.F. Cariñena, J.M. Gracia-Bondía, L.A. Ibort, C. López, and J.C. Várilly, ‘ *Distinguished Hamiltonian theorem for homogeneous symplectic manifolds*, Lett. Math. Phys. **23** (1991), 35–43. R. Schmid, *The quadratic-Hamiltonian theorem in infinite dimensions*, J. Math. Phys. **29** (1988), 2010–2011. , *Symmetries of Toda equations*, J. Phys. A: Math. Gen. **26** (1993), 3791–3796. , *On the master symmetries and bi-Hamiltonian structure of the Toda Lattice*, J. Phys. A: Math. Gen. **26** (1993), 3797–3803. , *Superintegrability of the Calogero-Moser system: constants of motion, master symmetries, and time-dependent symmetries*, J. Math. Phys. **40** (1999), 236–247. R.G. Smirnov, *On the master symmetries related to certain classes of integrable Hamiltonian systems*, J. Phys. A: Math. Gen. **29** (1996), 8133–8138. F. Finkel and A.S. Fokas, *On the construction of evolution equations admitting a master symmetry"*, Phys. Lett. A **293** (2002), 36–44. R. Caseiro, *Master integrals, superintegrability and quadratic algebras"*, Bull. Sci. Math. **126** (2002), 617–630. P.A. Damianou and Ch. Sophocleous, *Noether and master symmetries for the Toda lattice"*, Appl. Math. Lett. **18** (2005), 163–170. M.F. Rañada, *Master symmetries, non-Hamiltonian symmetries, and superintegrability of the generalized Smoridinsky-Winternitz system"*, J. Phys. A: Math. Theor. **45** (2012), 145204. J.F. Cariñena and L.A. Ibort, *Noncanonical groups of transformations, anomalies and cohomology"*, J. Math. Phys. **29** (1988), 541–545. R. Abraham, J.E. Marsden, and T. Ratiu, Applied Mathematical Sciences vo. 75, Springer, 2nd ed., 1988.


5 Women Share Exactly What It Feels Like To Have A Heart Attack

You’d totally know if you were having a heart attack, right? You’d clutch your left arm in pain, immediately fall down to the floor, and head to the hospital immediately.

Eh, not so much. Heart attacks look way different in women. While, yes, many women experience chest and arm pain, it’s entirely possible to have a heart attack without these symptoms, says Dr. Jennifer Haythe, a cardiologist and co-director of the Women’s Center for Cardiovascular Health at Columbia University.

Other sneaky heart attack symptoms you need to know include jaw pain, shortness of breath, extreme fatigue, dizziness, back pain, nausea, hot flashes, indigestion, and even something that feels like an anxiety attack.

“Too many women don’t realise that heart disease is the greatest risk to their health,” says Dr. Haythe. “Women need to be even more educated because they’re more likely to have atypical symptoms.”

If that weren’t tricky enough, each individual may experience different symptoms. Here, five heart attack survivors reveal exactly what their heart attack felt like.

“I felt like I had a large pill stuck in my throat”

“I actually had three heart attacks in three days. I experienced my first heart attack on April 10th, 2014, a second on April 11th, and a third on the 12th, just hours after being released from the hospital. The last one was so severe I was considered dead for a few minutes and it’s miracle the doctors were able to bring me back.

“Looking back, I realised I’d been having heart symptoms, like numbness in my left arm and stiffness in my neck, since November of 2013. During my actual heart attacks, my chest felt as if something was stuck, like I swallowed a large pill and it wouldn’t pass. I also experienced nausea, hot flashes, sweating, clamminess, and extreme fatigue. But if I had to sum it up, I would say my body felt like a lot of circuits misfiring at one time.

“After surgery to put a stent in, my cardiologist told me I needed to rethink my entire lifestyle, mentally and physically. I took his advice and today I’m happy to say that I exercise five days a week with a combination of yoga, cardio, and weight lifting. I admit that I have a slight advantage over most as my husband is a fitness coach; he helps me stay the course! We all need support and encouragement for this journey.”

“I felt like I had awful heartburn.”

“I had just finished teaching three high-intensity classes, and I felt a pain in my chest, but I brushed it off as heartburn. Then, as I was walking up the stairs at my gym for a meeting, I suddenly felt extremely fatigued—so much so that I didn’t think I’d be able to make it the rest of the way.

“Seconds later, I felt hot and sweat started rolling off of me, like I had just gotten out of a sauna. I also suddenly wasn’t able to catch my breath, I felt nauseous, I had cramping in my hands, and my chest pain was getting worse and worse.

“I kept saying I was fine, but my coworker knew better, so she called 911. Paramedics did an EKG and immediately threw me in the ambulance and rushed me to the hospital.

“Within an hour, I had an angiogram to remove a clot that had blocked my left anterior descending artery 100 percent. The second it was out, I felt immediate relief. My cardiologist’s best guess was that stress or exercise, or a combination of the two, caused a tear in my artery, which forced the blood to clot. I found out later that the majority of people with this same blockage die.

“I feel so fortunate to be able to lead my healthy lifestyle today, but it’s always in the back of my mind. Now I have regular checkups for my arteries and am careful not to overdo it with intense exercise. The experience has just made me so thankful for every day I have!”

–Shannon Schleicher, 39, a personal trainer and fitness coach for Life Time Athletic

“I felt like I was hiking Mount Everest—on flat ground.”

“I was on what was supposed to be an easy hike with my friend, when all of a sudden I felt super breathless (we weren’t even out of sight of the parking lot yet), and I had pain shooting down my left arm.

“I sat down on a rock and told my friend I felt fine—the hike should’ve been easy, I had just completed a half marathon and was recently checked out by my doctor for mild chest pain—but he didn’t believe me and rushed me to the doctor.

“When I got there, the doctors found that the left main artery of my heart was 98 percent blocked and the blockage was very close to my heart—a condition referred to as “the widow maker” because it’s so often lethal. I was just 33 years old and in great shape, but it turns out I carry a gene that greatly increases my risk of heart disease. (My dad had his first heart attack at 32!)

“Since that fateful day, I had three surgeries and five stents put in. I also need to follow a strict vegan, low-cholesterol diet. But today I’m doing great and back to doing the activities I love, including starting my own fitness company.”

–Mika Leah, 41, founder of Goomi and volunteer for the American Heart Association’s Go Red For Women “Real Women” campaign

“I thought I had breast cancer.”

“In early 2008 I started having some strange symptoms—my left shoulder hurt for no reason, I was having a hard time answering questions because my brain felt foggy, and I had what felt like electric shocks all down my left side. But none of them felt severe enough to get checked out, so I brushed them off for nine months.

“Then, almost overnight, the electric shocks intensified on my left torso and I started to have pain in my chest. Breast cancer was the first thing that popped into my mind, but before I could make an appointment to get it checked out, my health went downhill—fast.

“I woke up one morning feeling awful—super tired, my back hurt, and I was nauseous—but I went into work anyhow. Once there, I told my coworker how I felt, adding that my chest felt like it was getting crushed. He immediately said, ‘That sounds like a heart attack.’

“I thought he was crazy. I was only 37, I wasn’t overweight, I didn’t have clogged arteries, and I was very fit. Still, he insisted on taking me to the hospital. Once we were there, I had a full-blown heart attack in the waiting room of the ER.

“During the entire ordeal, I never thought “This is it, I’m having a heart attack.” I just felt miserable and knew something was terribly wrong. Looking back, I realise I overlooked one huge risk factor: my family history of heart disease. After I had my heart attack, I found out that nearly everyone on my mom’s side had early heart disease. My grandfather also had a heart attack at 37.

“The whole experience was terrifying and I still feel traumatised remembering it. But in a way, it was a really good wake up call. Now that I know my risks, I can be better prepared. I take heart medication and a baby aspirin every day, and I get tested regularly. I’m back to my favourite sport—rock climbing—but I make sure to tell everyone to ask their parents about their history of heart disease.”

“I thought I was just having a panic attack.”

“I was just 21 years old and a senior in college when I had a heart attack. I was at my apartment with my roommates, getting ready to go to soccer practice when I started having what I thought was anxiety. My mother had just died six weeks earlier (of heart disease!), and I was still very much grieving her loss.

“Every time I took a breath, my chest would hurt when I exhaled. After a few minutes, the pain in my chest increased until it felt like someone was both squeezing and stabbing my heart. It wasn’t the worst pain in my life, but it was extremely uncomfortable. I felt faint and lightheaded. I was cold, but I was sweating and nauseous too.

“But the weirdest part was that my wrists felt numb—almost like my arms were asleep, but the feeling was focused around my wrists. It didn’t matter how I sat, laid, or stood, the pain and the weird feelings wouldn’t go away. That’s when I knew something was really wrong.

“I went to the hospital where doctors discovered a blood clot partially blocking one of my arteries, which had caused a heart attack. They were able to break up the clot with medication, so I didn’t need surgery.

“Today I’m doing great and I only have two restrictions: I am no longer allowed to participate in contact sports (like soccer) to prevent clot movement or formation from injury, and I’m supposed to keep my heart rate below 170 beats per minute as a precaution not to strain my heart.

“My mom was just 58 when she died and I still miss her every day. Her memory reminds me why heart health is so important, even in young, seemingly healthy women.”


Network Working Group                                      S. Boeyen
Request for Comments: 2559                                   Entrust
Updates: 1778                                               T. Howes
Category: Standards Track                                   Netscape
                                                          P. Richard
                                                               Xcert
                                                          April 1999

                Internet X.509 Public Key Infrastructure
                     Operational Protocols - LDAPv2

Status of this Memo

   This document specifies an Internet standards track protocol for the
   Internet community, and requests discussion and suggestions for
   improvements.  Please refer to the current edition of the "Internet
   Official Protocol Standards" (STD 1) for the standardization state
   and status of this protocol.  Distribution of this memo is unlimited.

Copyright Notice

   Copyright (C) The Internet Society (1999).  All Rights Reserved.

1.  Abstract

   The protocol described in this document is designed to satisfy some
   of the operational requirements within the Internet X.509 Public Key
   Infrastructure (IPKI).  Specifically, this document addresses
   requirements to provide access to Public Key Infrastructure (PKI)
   repositories for the purposes of retrieving PKI information and
   managing that same information.  The mechanism described in this
   document is based on the Lightweight Directory Access Protocol (LDAP)
   v2, defined in RFC 1777, defining a profile of that protocol for use
   within the IPKI and updates encodings for certificates and revocation
   lists from RFC 1778. Additional mechanisms addressing PKIX
   operational requirements are specified in separate documents.

   The key words 'MUST', 'REQUIRED', 'SHOULD', 'RECOMMENDED', and 'MAY'
   in this document are to be interpreted as described in RFC 2119.

2.  Introduction

   This specification is part of a multi-part standard for development
   of a Public Key Infrastructure (PKI) for the Internet. This
   specification addresses requirements to provide retrieval of X.509
   PKI information, including certificates and CRLs from a repository.
   This specification also addresses requirements to add, delete and

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RFC 2559          PKIX Operational Protocols - LDAPv2         April 1999

   modify PKI information in a repository. A profile based on the LDAP
   version 2 protocol is provided to satisfy these requirements.

3.  Model

   The PKI components, as defined in PKIX Part 1, which are involved in
   PKIX operational protocol interactions include:

      -  End Entities
      -  Certification Authorities (CA)
      -  Repository

   End entities and CAs using LDAPv2, retrieve PKI information from the
   repository using a subset of the LDAPv2 protocol.

   CAs populate the repository with PKI information using a subset of
   the LDAPv2 protocol.

4.  Lightweight Directory Access Protocol (LDAP)

   The following sections examine the retrieval of PKI information from
   a repository and management of PKI information in a repository. A
   profile of the LDAPv2 protocol is defined for providing these
   services.

   Section 5 satisfies the requirement to retrieve PKI information (a
   certificate, CRL, or other information of interest) from an entry in
   the repository, where the retrieving entity (either an end entity or
   a CA) has knowledge of the name of the entry. This is termed
   "repository read".

   Section 6 satisfies the same requirement as 5 for the situation where
   the name of the entry is not known, but some other related
   information which may optionally be used as a filter against
   candidate entries in the repository, is known.  This is termed
   "repository search".

   Section 7 satisfies the requirement of CAs to add, delete and modify
   PKI information information (a certificate, CRL, or other information
   of interest)in the repository. This is termed "repository modify".

   The subset of LDAPv2 needed to support each of these functions is
   described below.  Note that the repository search service is a
   superset of the repository read service in terms of the LDAPv2
   functionality needed.

   Note that all tags are implicit by default in the ASN.1 definitions
   that follow.

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5.  LDAP Repository Read

   To retrieve information from an entry corresponding to the subject or
   issuer name of a certificate, requires a subset of the following
   three LDAP operations:

     BindRequest (and BindResponse)
     SearchRequest (and SearchResponse)
     UnbindRequest

   The subset of each REQUIRED operation is given below.

5.1.  Bind

5.1.1.  Bind Request

   The full LDAP v2 Bind Request is defined in RFC 1777.

   An application providing a LDAP repository read service MUST
   implement the following subset of this operation:

      BindRequest ::=
        [APPLICATION 0] SEQUENCE {
           version      INTEGER (2),
           name         LDAPDN, -- MUST accept NULL LDAPDN
           simpleauth [0] OCTET STRING  -- MUST accept NULL simple
           }

   An application providing a LDAP repository read service MAY implement
   other aspects of the BindRequest as well.

   Different services may have different security requirements.  Some
   services may allow anonymous search, others may require
   authentication. Those services allowing anonymous search may choose
   only to allow search based on certain criteria and not others.

   A LDAP repository read service SHOULD implement some level of
   anonymous search access. A LDAP repository read service MAY implement
   authenticated search access.

5.1.2.  Bind Response

   The full LDAPv2 BindResponse is described in RFC 1777.

   An application providing a LDAP repository read service MUST
   implement this entire protocol element, though only the following
   error codes may be returned from a Bind operation:

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       success                      (0),
       operationsError              (1),
       protocolError                (2),
       authMethodNotSupported       (7),
       noSuchObject                 (32),
       invalidDNSyntax              (34),
       inappropriateAuthentication  (48),
       invalidCredentials           (49),
       busy                         (51),
       unavailable                  (52),
       unwillingToPerform           (53),
       other                        (80)

5.2.  Search

5.2.1.  Search Request

   The full LDAPv2 SearchRequest is defined in RFC 1777.

   An application providing a LDAP repository read service MUST
   implement the following subset of the SearchRequest.

      SearchRequest ::=
        [APPLICATION 3] SEQUENCE {
           baseObject     LDAPDN,
           scope             ENUMERATED {
                             baseObject   (0),
                                        },
           derefAliases   ENUMERATED {
                             neverDerefAliases   (0),
                                     },
           sizeLimit      INTEGER (0),
           timeLimit      INTEGER (0),
           attrsOnly      BOOLEAN, -- FALSE only
           filter         Filter,
           attributes     SEQUENCE OF AttributeType
                               }

      Filter ::=
        CHOICE {
          present        [7] AttributeType, -- "objectclass" only
                 }

   This subset of the LDAPv2 SearchRequest allows the LDAPv2 "read"
   operation: a base object search with a filter testing for the
   existence of the objectClass attribute.

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   An application providing a LDAP repository read service MAY implement
   other aspects of the SearchRequest as well.

5.2.2.

   The full LDAPv2 SearchResponse is defined in RFC 1777.

   An application providing a LDAP repository read service over LDAPv2
   MUST implement the full SearchResponse.

   Note that in the case of multivalued attributes such as
   userCertificate a SearchResponse containing this attribute will
   include all values, assuming the requester has sufficient access
   permissions.  The application/relying party may need to select an
   appropriate value to be used. Also note that retrieval of a
   certificate from a named entry does not guarantee that the
   certificate will include that same Distinguished Name (DN) and in
   some cases the subject DN in the certificate may be NULL.

5.3.  Unbind

   The full LDAPv2 UnbindRequest is defined in RFC 1777.

   An application providing a LDAP repository read service MUST
   implement the full UnbindRequest.

6.  LDAP Repository Search

   To search, using arbitrary criteria, for an entry in a repository
   containing a certificate, CRL, or other information of interest,
   requires a subset of the following three LDAP operations:

     BindRequest (and BindResponse)
     SearchRequest (and SearchResponse)
     UnbindRequest

   The subset of each operation REQUIRED is given below.

6.1.  Bind

   The BindRequest and BindResponse subsets needed are the same as those
   described in Section 5.1.

   The full LDAP v2 Bind Request is defined in RFC 1777.

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6.2.  Search

6.2.1.  Search Request

   The full LDAPv2 SearchRequest is defined in RFC 1777.

   An application providing a LDAP repository search service MUST
   implement the following subset of the SearchRequest protocol unit.

      SearchRequest ::=
        [APPLICATION 3] SEQUENCE {
           baseObject     LDAPDN,
           scope          ENUMERATED {
                               baseObject     (0),
                               singleLevel    (1),
                               wholeSubtree   (2)
                                     },
           derefAliases   ENUMERATED {
                               neverDerefAliases     (0),
                                     },
           sizeLimit      INTEGER (0 .. maxInt),
           timeLimit      INTEGER (0 .. maxInt),
           attrsOnly      BOOLEAN,  -- FALSE only
           filter         Filter,
           attributes     SEQUENCE OF AttributeType
                                }

   All aspects of the SearchRequest MUST be supported, except for the
   following:

   - Only the neverDerefAliases value of derefAliases needs to be
     supported

   - Only the FALSE value for attrsOnly needs to be supported

   This subset provides a more general search capability.  It is a
   superset of the SearchRequest subset defined in Section 5.2.1. The
   elements added to this service are:

   - singleLevel and wholeSubtree scope needs to be supported

   - sizeLimit is included

   - timeLimit is included

   - Enhanced filter capability

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   An application providing a LDAP repository search service MAY
   implement other aspects of the SearchRequest as well.

6.2.2.  Search Response

   The full LDAPv2 SearchResponse is defined in RFC 1777.

   An application providing a LDAP repository search service over LDAPv2
   MUST implement the full SearchResponse.

6.3.  Unbind

   An application providing a LDAP repository search service MUST
   implement the full UnbindRequest.

7.  LDAP Repository Modify

   To add, delete and modify PKI information in a repository  requires a
   subset of the following LDAP operations:

     BindRequest (and BindResponse)
     ModifyRequest (and ModifyResponse)
     AddRequest (and AddResponse)
     DelRequest (and DelResponse
     UnbindRequest

   The subset of each operation REQUIRED is given below.

7.1.  Bind

   The full LDAP v2 Bind Request is defined in RFC 1777.

   An application providing a LDAP repository modify service MUST
   implement the following subset of this operation:

      BindRequest ::=
        [APPLICATION 0] SEQUENCE {
           version      INTEGER (2),
           name         LDAPDN,
           simpleauth [0] OCTET STRING
           }

   A LDAP repository modify service MUST implement authenticated access.

   The BindResponse subsets needed are the same as those described in
   Section 5.1.2.

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7.2.  Modify

7.2.1.  Modify Request

   The full LDAPv2 ModifyRequest is defined in RFC 1777.

   An application providing a LDAP repository modify service MUST
   implement the following subset of the ModifyRequest protocol unit.

      ModifyRequest ::=
        [APPLICATION 6] SEQUENCE {
       object         LDAPDN,
       modification   SEQUENCE OF SEQUENCE {
                        operation     ENUMERATED {
                                        add     (0),
                                        delete  (1)
                                      },
                        modification  SEQUENCE {
                                      type   AttributeType,
                                      values SET OF
                                             AttributeValue
                                      }
                      }
        }

   All aspects of the ModifyRequest MUST be supported, except for the
   following:

   - Only the add and delete values of operation need to be supported

7.2.2.  Modify Response

   The full LDAPv2 ModifyResponse is defined in RFC 1777.

   An application providing a LDAP repository modify service MUST
   implement the full ModifyResponse.

7.3.  Add

7.3.1.  Add Request

   The full LDAPv2 AddRequest is defined in RFC 1777.

   An application providing a LDAP repository modify service MUST
   implement the full AddRequest.

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7.3.2.  Add Response

   The full LDAPv2 AddResponse is defined in RFC 1777.

   An application providing a LDAP repository modify service MUST
   implement the full AddResponse.

7.4.  Delete

7.4.1.  Delete Request

   The full LDAPv2 DelRequest is defined in RFC 1777.

   An application providing a LDAP repository modify service MUST
   implement the full DelRequest.

7.4.2.  Delete Response

   The full LDAPv2 DelResponse is defined in RFC 1777.

   An application providing a LDAP repository modify service MUST
   implement the full DelResponse.

7.5.  Unbind

   An application providing a LDAP repository modify service MUST
   implement the full UnbindRequest.

8.  Non-standard attribute value encodings

   When conveyed in LDAP requests and results, attributes defined in
   X.500 are to be encoded using string representations defined in RFC
   1778, The String Representation of Standard Attribute Syntaxes.
   These string encodings were based on the attribute definitions from
   X.500(1988).  Thus, the string representations of the PKI information
   elements are for version 1 certificates and version 1 revocation
   lists.  Since this specification uses version 3 certificates and
   version 2 revocation lists, as defined in X.509(1997), the RFC 1778
   string encoding of these attributes is inappropriate.

   For this reason, these attributes MUST be encoded using a syntax
   similar to the syntax "Undefined" from section 2.1 of RFC 1778:
   values of these attributes are encoded as if they were values of type
   "OCTET STRING", with the string value of the encoding being the DER-
   encoding of the value itself.  For example, when writing a
   userCertificate to the repository, the CA generates a DER-encoding of
   the certificate and uses that encoding as the value of the
   userCertificate attribute in the LDAP Modify request.This encoding

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   style is consistent with the encoding scheme proposed for LDAPv3,
   which is now being defined within the IETF.

   Note that certificates and revocation lists will be transferred using
   this mechanism rather than the string encodings in RFC 1778 and
   client systems which do not understand this encoding may experience
   problems with these attributes.

9.  Transport

   An application providing a LDAP repository read service, LDAP
   repository search service, or LDAP repository modify service MUST
   support LDAPv2 transport over TCP, as defined in Section 3.1 of RFC
   1777.

   An application providing a LDAP repository read service, LDAP
   repository search service, or LDAP repository modify service MAY
   support LDAPv2 transport over other reliable transports as well.

10.  Security Considerations

   Since the elements of information which are key to the PKI service
   (certificates and CRLs) are both digitally signed pieces of
   information, additional integrity service is NOT REQUIRED.  As
   neither information element need be kept secret and anonymous access
   to such information, for retrieval purposes is generally acceptable,
   privacy service is NOT REQUIRED for information retrieval requests.

   CAs have additional requirements, including modification of PKI
   information.  Simple authentication alone is not sufficient for these
   purposes. It is RECOMMENDED that some stronger means of
   authentication and/or (if simple authentication is used) some means
   of protecting the privacy of the password is used, (e.g.  accept
   modifications only via physically secure networks, use IPsec, use SSH
   or TLS or SSL tunnel). Without such authentication, it is possible
   that a denial-of-service attack could occur where the attacker
   replaces valid certificates with bogus ones.

   For the LDAP repository modify service, profiled in section 7, there
   are some specific security considerations with respect to access
   control. These controls apply to a repository which is under the same
   management control as the CA. Organizations operating directories are
   NOT REQUIRED to provide external CAs access permission to their
   directories.

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RFC 2559          PKIX Operational Protocols - LDAPv2         April 1999

   The CA MUST have access control permissions allowing it to:

      For CA entries:
         - add, modify and delete all PKI attributes for its own
           directory entry;
         - add, modify and delete all values of these attributes.

      For CRL distribution point entries (if used):
         - create, modify and delete entries of object class
           cRLDistributionPoint immediately subordinate to its own
           entry;
         - add, modify and delete all attributes, and all values of
           these attributes for these entries.

      For subscriber (end-entity) entries:
         - add, modify and delete the attribute userCertificate and all
           values of that attribute, issued by this CA to/from these
           entries.

   The CA is the ONLY entity with these permissions.

   An application providing LDAP repository read, LDAP repository
   search, or LDAP repository modify service as defined in this
   specification is NOT REQUIRED to implement any additional security
   features other than those described herein, however an implementation
   SHOULD do so.

11.  References

   [1]  Yeong, Y., Howes, T. and S. Kille, "Lightweight Directory Access
        Protocol", RFC 1777, March 1995.

   [2]  Howes, T., Kille, S., Yeong, W. and C. Robbins, "The String
        Representation of Standard Attribute Syntaxes", RFC 1778, March
        1995.

   [3]  Bradner, S., "Key Words for use in RFCs to Indicate Requirement
        Levels", BCP 14, RFC 2119, March 1997.

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RFC 2559          PKIX Operational Protocols - LDAPv2         April 1999

12.  Authors' Addresses

   Sharon Boeyen
   Entrust Technologies Limited
   750 Heron Road
   Ottawa, Ontario
   Canada K1V 1A7

   EMail: sharon.boeyen@entrust.com

   Tim Howes
   Netscape Communications Corp.
   501 E. Middlefield Rd.
   Mountain View, CA 94043
   USA

   EMail: howes@netscape.com

   Patrick Richard
   Xcert Software Inc.
   Suite 1001, 701 W. Georgia Street
   P.O. Box 10145
   Pacific Centre
   Vancouver, B.C.
   Canada V7Y 1C6

   EMail: patr@xcert.com

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RFC 2559          PKIX Operational Protocols - LDAPv2         April 1999

13.  Full Copyright Statement

   Copyright (C) The Internet Society (1999).  All Rights Reserved.

   This document and translations of it may be copied and furnished to
   others, and derivative works that comment on or otherwise explain it
   or assist in its implementation may be prepared, copied, published
   and distributed, in whole or in part, without restriction of any
   kind, provided that the above copyright notice and this paragraph are
   included on all such copies and derivative works.  However, this
   document itself may not be modified in any way, such as by removing
   the copyright notice or references to the Internet Society or other
   Internet organizations, except as needed for the purpose of
   developing Internet standards in which case the procedures for
   copyrights defined in the Internet Standards process must be
   followed, or as required to translate it into languages other than
   English.

   The limited permissions granted above are perpetual and will not be
   revoked by the Internet Society or its successors or assigns.

   This document and the information contained herein is provided on an
   "AS IS" basis and THE INTERNET SOCIETY AND THE INTERNET ENGINEERING
   TASK FORCE DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING
   BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION
   HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF
   MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

Boeyen, et al.              Standards Track                    [Page 13]


// stack_introspect.h

/*    Copyright 2010 10gen Inc.
 *
 *    Licensed under the Apache License, Version 2.0 (the "License");
 *    you may not use this file except in compliance with the License.
 *    You may obtain a copy of the License at
 *
 *    http://www.apache.org/licenses/LICENSE-2.0
 *
 *    Unless required by applicable law or agreed to in writing, software
 *    distributed under the License is distributed on an "AS IS" BASIS,
 *    WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *    See the License for the specific language governing permissions and
 *    limitations under the License.
 */

#pragma once

namespace mongo {

    /**
     * checks up call tree
     * if any method on top of me is a constructor, return true
     * may do internal caching
     * probably slow, use with care
     * if not implemented for a platform, returns false
     */
    bool inConstructorChain( bool printOffending=false );

    /**
     * @return if supported on platform, compile options may still prevent it from working
     */
    bool inConstructorChainSupported();

}