Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia

Georgian Math. J. 10 (2003) 057-061

Noether's theorem associates conservation laws with particular continuous symmetries ofthe Lagrangian. According to the Hojman's theorem [1]-[3] there exists the definite correspondence betweennon-Noether symmetries and conserved quantities. In 1998 M. Lutzky showed that several integrals ofmotion 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 equationi_{Xh}ω + dh = 0, where $\omega $ is the closed($d\omega \; =\; 0$) and non-degenerate($i$_{X}ω = 0 ⇒ X = 0) 2-form,$h$ is the Hamiltonian and$i$_{X}ω denotes contraction of$X$ with $\omega $.Since $\omega $ is non-degenerate, this gives rise to an isomorphism between the vectorfields and 1-forms given by $i$_{X}ω + α= 0.The vector field is said to be Hamiltonian if it corresponds to exact formi_{Xf}ω + df = 0. The Poisson bracket is defined as follows:{f , g} = X_{f} g = − X_{g} f = i_{Xf}i_{Xg}ω. By introducing a bivector field $W$ satisfyingi_{X}i_{Y}ω = i_{W} i_{X}ω ∧ i_{Y}ω, Poisson bracket can be rewritten as{f , g} = i_{W} df ∧ dg. It's easy to show thati_{X}i_{Y}L_{Z}ω =i_{[Z,W]} i_{X}ω ∧ i_{Y}ω, where the bracket $[\; \xb7\; ,\; \xb7\; ]$ is actually a supercommutator,for an arbitrary bivector field$W\; =\sum sVs\wedge \; Us$ we have[X,W] = ∑ s [X,V^{s}] ∧ U^{s}+ ∑ s V^{s} ∧ [X,U^{s}] Equation (6) is based on the following useful property of the Lie derivativeL_{X}i_{W}ω = i_{[X,W]}ω +i_{W}L_{X}ω. Indeed, for an arbitrary bivector field$W\; =\sum sVs\wedge \; Us$ we haveL_{X}i_{W}ω = L_{X}∑ s i_{Vs ∧ Us}ω =L_{X}∑ s i_{Us}i_{Vs}ω= ∑ s i_{[X,Us]}i_{Vs}ω +∑ s i_{Us}i_{[X,Vs]}ω +∑ s i_{Us}i_{Vs}L_{X}ω =i_{[X,W]}ω + i_{W}L_{X}ω where $L$_{Z} denotes the Lie derivative along the vector field $Z$.According to Liouville's theorem Hamiltonian vector fieldpreserves $\omega $L_{Xf}ω = 0; therefore it commutes with $W$:[X_{f} ,W] = 0. In the local coordinates $z$_{s} where$\omega \; =\sum rs\omega rsdz$_{r} ∧ z_{s} bivector field$W$ has the following form$W\; =\sum rsWrs\partial \partial z$_{r} ∧ ∂ ∂z_{s} where$Wrs$ is matrix inverted to $\omega rs$.

We can say that a group of transformations$g(z)\; =\; ezL$_{E} generated by the vectorfield $E$ maps the space of solutions of equation onto itself ifi_{Xh}g_{*}(ω) + g_{*}(dh) = 0 For $X$_{h} satisfyingi_{Xh}ω + dh = 0 Hamilton's equation.It's easy to show that the vector field $E$ should satisfy$[E\; ,\; X$_{h}] = 0Indeed,i_{Xh}L_{E}ω + dL_{E}h =L_{E}(i_{Xh}ω + dh) = 0 since $[E,X$_{h}] = 0. When $E$ is not Hamiltonian,the group of transformations $g(z)\; =\; ezL$_{E} is non-Noethersymmetry (in a sense that it maps solutions onto solutions but does not preserve action).

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$i$_{us}ω = 0 s = 1,2 ... n − rank(ω),every Hamiltonian vector field isdefined up to linear combination of vectors $u$_{s}. By identifying $X$_{f}with $X$_{f} + ∑ s C_{s}u_{s}, we can introduce equivalence class$X$_{f}^{∗} (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 satisfyi_{X}i_{Y} ω =i_{W1,2} i_{X}ω ∧ i_{Y}ω, theni_{(W1 − W2)} i_{X}ω ∧ i_{Y}ω = 0 ∀X,Y is fulfilled. It is possible only whenW_{1} − W _{2} = ∑ s v_{s} ∧ u_{s} where $v$_{s} are some vector fields and$i$_{us}ω = 0(in other words when $W$_{1} − W_{2} belongs to the class$0\ast $)

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