Non-Noether symmetries in singular dynamical systems

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 $$i_{X_h}\omega +dh=0,$$ where $\omega$ is the closed ($d\omega =0$) and non-degenerate ($i_X\omega =0\Rightarrow X=0$) 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 $$i_{X_f}\omega +df=0.$$ The Poisson bracket is defined as follows: $$\{f,g\}=X_{f}g=-X_{g}f=i_{X_f}{i}_{X_g}\omega .$$ By introducing a bivector field $W$ satisfying $$i_X{i}_Y\omega =i_W{i}_X\omega \wedge i_Y\omega ,$$ Poisson bracket can be rewritten as $$\{f,g\}=i_{W}df\wedge dg.$$ It's easy to show that $$i_X{i}_Y{L}_Z\omega =i_{[Z,W]}{i}_X\omega \wedge i_Y\omega ,$$ where the bracket $[·,·]$ is actually a supercommutator, for an arbitrary bivector field $W=\sum _s{V}^s\wedge U^s$ we have $$[X,W]=\sum _s[X,V^s]\wedge U^s+\sum _s{V}^s\wedge [X,U^s]$$ Equation (6) is based on the following useful property of the Lie derivative $$L_X{i}_W\omega =i_{[X,W]}\omega +i_W{L}_X\omega .$$ Indeed, for an arbitrary bivector field $W=\sum _s{V}^s\wedge U^s$ we have $$\begin{gathered} L_X{i}_W\omega =L_X\sum _s{i}_{V^s\wedge U^s}\omega =L_X\sum _s{i}_{U^s}{i}_{V^s}\omega \\ =\sum _s{i}_{[X,U^s]}{i}_{V^s}\omega +\sum _s{i}_{U^s}{i}_{[X,V^s]}\omega +\sum _s{i}_{U^s}{i}_{V^s}{L}_X\omega =i_{[X,W]}\omega +i_W{L}_X\omega \end{gathered}$$ where $L_Z$ denotes the Lie derivative along the vector field $Z$. According to Liouville's theorem Hamiltonian vector field preserves $\omega$ $$L_{X_f}\omega =0;$$ therefore it commutes with $W$: $$[X_f,W]=0.$$ 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 ${\omega }^{rs}$.

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 $$i_{X_h}{g}_{*}\left(\omega \right)+g_{*}\left(dh\right)=0$$ For $X_h$ satisfying $$i_{X_h}\omega +dh=0$$ Hamilton's equation. It's easy to show that the vector field $E$ should satisfy $[E,X_h]=0$ Indeed, $$i_{X_h}{L}_E\omega +d{L}_{E}h=L_E(i_{X_h}\omega +dh)=0$$ since $[E,X_h]=0$. 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).

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 $i_{u_s}\omega =0s=1,2...n-rank\left(\omega \right),$ 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 $$i_X{i}_Y\omega =i_{W_{1,2}}{i}_X\omega \wedge i_Y\omega ,$$ then $$i_{(W_1-W_2)}{i}_X\omega \wedge i_Y\omega =0\forall X,Y$$ is fulfilled. It is possible only when $$W_1-W_2=\sum _s{v}_s\wedge u_s$$ 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 $0^{\ast }$)