﻿ ]> Non-Noether symmetries in singular dynamical systems

# Non-Noether symmetries in singular dynamical systems

Department of Theoretical Physics, A. Razmadze Institute of Mathematics, 1 Aleksidze Street, Tbilisi 0193, Georgia
In the present paper geometric aspects of relationship between non-Noether symmetries and conservation laws in Hamiltonian systems is discussed. Case of irregular/constrained dynamical systems on presymplectic and Poisson manifolds is considered.
Non-Noether symmetry; Conservation laws; Constrained dynamics;
70H33, 70H06, 53Z05
Georgian Math. J. 10 (2003) 057-061

## Introduction

Noether's theorem associates conservation laws with particular continuous symmetries of the Lagrangian. According to the Hojman's theorem [1]-[3] there exists the definite correspondence between non-Noether symmetries and conserved quantities. In 1998 M. Lutzky showed that several integrals of motion might correspond to a single one-parameter group of non-Noether transformations [4]. In the present paper, the extension of Hojman-Lutzky theorem to singular dynamical systems is considered.
First of all let us recall some basic knowledge of description of the regular dynamical systems (see, e. g. [5]). In this case time evolution is governed by Hamilton's equation $iXhω+dh=0,$ where $\omega$ is the closed ($d\omega =0$) and non-degenerate (${i}_{X}\omega =0⇒X=0$) 2-form, $h$ is the Hamiltonian and ${i}_{X}\omega$ denotes contraction of $X$ with $\omega$. Since $\omega$ is non-degenerate, this gives rise to an isomorphism between the vector fields and 1-forms given by ${i}_{X}\omega +\alpha =0$. The vector field is said to be Hamiltonian if it corresponds to exact form $iXfω+df=0.$ The Poisson bracket is defined as follows: ${f,g}=Xfg=−Xgf=iXfiXgω.$ By introducing a bivector field $W$ satisfying $iXiYω=iWiXω∧iYω,$ Poisson bracket can be rewritten as ${f,g}=iWdf∧dg.$ It's easy to show that $iXiYLZω=i[Z,W]iXω∧iYω,$ where the bracket $\left[·,·\right]$ is actually a supercommutator, for an arbitrary bivector field $W=\sum _{s}{V}^{s}\wedge {U}^{s}$ we have $[X,W]=∑s[X,Vs]∧Us+∑sVs∧[X,Us]$ Equation (6) is based on the following useful property of the Lie derivative $LXiWω=i[X,W]ω+iWLXω.$ Indeed, for an arbitrary bivector field $W=\sum _{s}{V}^{s}\wedge {U}^{s}$ we have $LXiWω=LX∑siVs∧Usω=LX∑siUsiVsω=∑si[X,Us]iVsω+∑siUsi[X,Vs]ω+∑siUsiVsLXω=i[X,W]ω+iWLXω$ where ${L}_{Z}$ denotes the Lie derivative along the vector field $Z$. According to Liouville's theorem Hamiltonian vector field preserves $\omega$ $LXfω=0;$ therefore it commutes with $W$: $[Xf,W]=0.$ In the local coordinates ${z}_{s}$ where $\omega =\sum _{rs}{\omega }^{rs}d{z}_{r}\wedge {z}_{s}$ bivector field $W$ has the following form $W=\sum _{rs}{W}^{rs}\frac{\partial }{\partial {z}_{r}}\wedge \frac{\partial }{\partial {z}_{s}}$ where ${W}^{rs}$ is matrix inverted to ${\omega }^{rs}$.

## Case of regular Lagrangian systems

We can say that a group of transformations $g\left(z\right)={e}^{z{L}_{E}}$ generated by the vector field $E$ maps the space of solutions of equation onto itself if $iXhg*(ω)+g*(dh)=0$ For ${X}_{h}$ satisfying $iXhω+dh=0$ Hamilton's equation. It's easy to show that the vector field $E$ should satisfy $\left[E,{X}_{h}\right]=0$ Indeed, $iXhLEω+dLEh=LE(iXhω+dh)=0$ since $\left[E,{X}_{h}\right]=0$. When $E$ is not Hamiltonian, the group of transformations $g\left(z\right)={e}^{z{L}_{E}}$ is non-Noether symmetry (in a sense that it maps solutions onto solutions but does not preserve action).
(Lutzky, 1998) If the vector field $E$ generates non-Noether symmetry, then the following functions are constant along solutions: where ${W}^{k}$ and ${\omega }_{E}^{k}$ are outer powers of $W$ and ${L}_{E}\omega$.
We have to prove that ${I}^{\left(k\right)}$ is constant along the flow generated by the Hamiltonian. In other words, we should find that ${L}_{{X}_{h}}{I}^{\left(k\right)}=0$ is fulfilled. Let us consider ${L}_{{X}_{h}}{I}^{\left(1\right)}$ $LXhI(1)=LXh(iWωE)=i[Xh,W]ωE+iWLXhωE,$ where according to Liouville's theorem both terms $\left[{X}_{h},W\right]=0$ and $iWLXhLEω=iWLELXhω=0$ since $\left[E,{X}_{h}\right]=0$ and ${L}_{{X}_{h}}\omega =0$ vanish. In the same manner one can verify that ${L}_{{X}_{h}}{I}^{\left(k\right)}=0$
Theorem is valid for a larger class of generators $E$ . Namely, if $\left[E,{X}_{h}\right]={X}_{f}$ where ${X}_{f}$ is an arbitrary Hamiltonian vector field, then ${I}^{\left(k\right)}$ is still conserved. Such a symmetries map the solutions of the equation ${i}_{{X}_{h}}\omega +dh=0$ on solutions of $iXhg*(ω)+d(g*h+f)=0$
Discrete non-Noether symmetries give rise to the conservation of ${I}^{\left(k\right)}={i}_{{W}^{k}}{g}_{*}\left(\omega {\right)}^{k}$ where ${g}_{*}\left(\omega \right)$ is transformed $\omega$.
If ${I}^{\left(k\right)}$ is a set of conserved quantities associated with $E$ and $f$ is any conserved quantity, then the set of functions $\left\{{I}^{\left(k\right)},f\right\}$ (which due to the Poisson theorem are integrals of motion) is associated with $\left[{X}_{h},E\right]$. Namely it is easy to show by taking the Lie derivative of (15) along vector field $E$ that ${I(k),f}=iWkω[Xf,E]k$ is fulfilled. As a result conserved quantities associated with Non-Noether symmetries form Lie algebra under the Poisson bracket.
If generator of symmetry satisfies Yang-Baxter equation $\left[\left[E\left[E,W\right]\right]W\right]=0$ Lutzky's conservation laws are in involution [7] $\left\{{Y}^{\left(l\right)},{Y}^{\left(k\right)}\right\}=0$

## Case of irregular Lagrangian systems

The singular Lagrangian (Lagrangian with vanishing Hessian) leads to degenerate 2-form $\omega$ and we no longer have isomorphism between vector fields and 1-forms. Since there exists a set of "null vectors" ${u}_{s}$ such that every Hamiltonian vector field is defined up to linear combination of vectors ${u}_{s}$. By identifying ${X}_{f}$ with ${X}_{f}+\sum _{s}{C}_{s}{u}_{s},$ we can introduce equivalence class ${X}_{f}^{\ast }$ (then all ${u}_{s}$ belong to ${0}^{\ast }$ ). The bivector field $W$ is also far from being unique, but if ${W}_{1}$ and ${W}_{2}$ both satisfy $iXiYω=iW1,2iXω∧iYω,$ then is fulfilled. It is possible only when $W1−W2=∑svs∧us$ where ${v}_{s}$ are some vector fields and ${i}_{{u}_{s}}\omega =0$ (in other words when ${W}_{1}-{W}_{2}$ belongs to the class ${0}^{\ast }$)
If the non-Hamiltonian vector field $E$ satisfies $\left[E,{X}_{h}^{\ast }\right]={0}^{\ast }$ commutation relation (generates non-Noether symmetry), then the functions (where ${\omega }_{E}={L}_{E}\omega$) are constant along trajectories.
Let's consider ${I}^{\left(1\right)}$ $LXh∗I(1)=LXh∗(iWωE)=i[Xh∗,W]ωE+iWLXh∗ωE=0$ The second term vanishes since $\left[E,{X}_{h}^{\ast }\right]={0}^{\ast }$ and ${L}_{{X}_{h}^{\ast }}\omega =0$. The first one is zero as far as $\left[{X}_{h}^{\ast },{W}^{\ast }\right]={0}^{\ast }$ and $\left[E,{0}^{\ast }\right]={0}^{\ast }$ are satisfied. So ${I}^{\left(1\right)}$ is conserved. Similarly one can show that ${L}_{{X}_{h}}{I}^{\left(k\right)}=0$ is fulfilled.
$W$ is not unique, but ${I}^{\left(k\right)}$ doesn't depend on choosing representative from the class ${W}^{\ast }$.
Theorem is also valid for generators $E$ satisfying $\left[E,{X}_{h}^{\ast }\right]={X}_{f}^{\ast }$
Hamiltonian description of the relativistic particle leads to the following action $A=∫p0dx0+∑spsdxs$ where ${p}_{0}=\left({p}^{2}+{m}^{2}{\right)}^{1/2}$ with vanishing canonical Hamiltonian and degenerate 2-form defined by $p0ω=∑s(psdps∧dx0+p0dps∧dxs).$ $\omega$ possesses the "null vector field" ${i}_{u}\omega =0$ $u=p0∂∂x0+∑sps∂∂xs.$ One can check that the following non- Hamiltonian vector field $E=p0x0∂∂x0+p1x1∂∂x1+⋯+pnxn∂∂xn$ generates non-Noether symmetry. Indeed, $E$ satisfies $\left[E,{X}_{h}^{\ast }\right]={0}^{\ast }$ because of ${X}_{h}^{\ast }={0}^{\ast }$ and $\left[E,u\right]=u$. Corresponding integrals of motion are combinations of momenta: $I(1)=∑spsI(2)=∑r>sprps⋯I(n)=∏sps$ This example shows that the set of conserved quantities can be obtained from a single one-parameter group of non-Noether transformations.
Author is grateful to Z. Giunashvili and M. Maziashvili for constructive discussions and particularly grateful to George Jorjadze for invaluable help. This work was supported by INTAS (00-00561) and Scholarship from World Federation of Scientists.
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