George Chavchanidze 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 i_Xhω + dh = 0, where $\omega$ is the closed ($d\omega \; =\; 0$) and non-degenerate ($i$_Xω = 0 ⇒ X = 0) 2-form, $h$ is the Hamiltonian and $i$_Xω 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 $i$_Xω + α= 0. The vector field is said to be Hamiltonian if it corresponds to exact form i_Xfω + df = 0. The Poisson bracket is defined as follows: {f , g} = X_f g = − X_g f = i_Xf i_Xgω. By introducing a bivector field $W$ satisfying i_Xi_Yω = i_W i_Xω ∧ i_Yω, Poisson bracket can be rewritten as {f , g} = i_W df ∧ dg. It's easy to show that i_Xi_YL_Zω = i_[Z,W] i_Xω ∧ i_Yω, where the bracket $[\; ·\; ,\; ·\; ]$ is actually a supercommutator, for an arbitrary bivector field $W\; =\sum sVs\wedge \; Us$ we have [X,W] = ∑s[X,V^s] ∧ U^s + ∑sV^s ∧ [X,U^s] Equation (6) is based on the following useful property of the Lie derivative L_Xi_Wω = i_[X,W]ω + i_WL_Xω. Indeed, for an arbitrary bivector field $W\; =\sum sVs\wedge \; Us$ we have L_Xi_Wω = L_X∑si_{V^s ∧ U^s}ω = L_X∑s i_U^si_V^sω = ∑s i_[X,U^s]i_V^sω + ∑s i_U^si_[X,V^s]ω + ∑si_U^si_V^sL_Xω = i_[X,W]ω + i_WL_Xω where $L$_Z denotes the Lie derivative along the vector field $Z$. According to Liouville's theorem Hamiltonian vector field preserves $\omega$ L_Xfω = 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 $\omega rs$.

Case of regular Lagrangian systems

We can say that a group of transformations $g(z)\; =\; ezL$_E generated by the vector field $E$ maps the space of solutions of equation onto itself if i_Xhg_*(ω) + g_*(dh) = 0 For $X$_h satisfying i_Xhω + dh = 0 Hamilton's equation. It's easy to show that the vector field $E$ should satisfy $[E\; ,\; X$_h] = 0 Indeed, i_XhL_Eω + dL_Eh = L_E(i_Xhω + dh) = 0 since $[E,X$_h] = 0. When $E$ is not Hamiltonian, the group of transformations $g(z)\; =\; ezL$_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: I^(k) = i_W^k ω_E^k       k = 1...n, where $Wk$ and $\omega$_E^k are outer powers of $W$ and $L$_Eω. We have to prove that $I(k)$ is constant along the flow generated by the Hamiltonian. In other words, we should find that $L$_XhI^(k) = 0 is fulfilled. Let us consider $L$_XhI⁽¹⁾ L_XhI⁽¹⁾ = L_Xh(i_Wω_E) = i_{[Xh , W]}ω_E + i_WL_Xhω_E, where according to Liouville's theorem both terms $[X$_h , W] = 0 and i_WL_XhL_Eω = i_WL_EL_Xhω = 0 since $[E\; ,\; X$_h] = 0 and $L$_Xhω = 0 vanish. In the same manner one can verify that $L$_XhI^(k) = 0 Theorem is valid for a larger class of generators $E$ . Namely, if $[E\; ,\; X$_h] = X_f where $X$_f is an arbitrary Hamiltonian vector field, then $I(k)$ is still conserved. Such a symmetries map the solutions of the equation $i$_Xhω + dh = 0 on solutions of i_Xhg_*(ω) + d(g_*h + f) = 0 Discrete non-Noether symmetries give rise to the conservation of $I(k)=\; i$_W^kg_*(ω)^k where $g$_*(ω) is transformed $\omega$. If $I(k)$ is a set of conserved quantities associated with $E$ and $f$ is any conserved quantity, then the set of functions $\{I(k),\; f\}$ (which due to the Poisson theorem are integrals of motion) is associated with $[X$_h , E]. Namely it is easy to show by taking the Lie derivative of (15) along vector field $E$ that {I^(k) , f} = i_W^kω^k_{[Xf , E]} 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 $[[E[E\; ,\; W]]W]\; =\; 0$ Lutzky's conservation laws are in involution [7] $\{Y(l),\; Y(k)\}\; =\; 0$

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 $i$_usω = 0       s = 1,2 ... n − rank(ω), every Hamiltonian vector field is defined up to linear combination of vectors $u$_s. By identifying $X$_f with $X$_f + ∑sC_su_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 satisfy i_Xi_Y ω = i_W1,2 i_Xω ∧ i_Yω, then i_{(W1 − W2)} i_Xω ∧ i_Yω = 0       ∀X,Y is fulfilled. It is possible only when W₁ − W ₂ = ∑sv_s ∧ u_s where $v$_s are some vector fields and $i$_usω = 0 (in other words when $W$₁ − W₂ belongs to the class $0\ast$) If the non-Hamiltonian vector field $E$ satisfies $[E\; ,\; X$_h^∗] = 0^∗ commutation relation (generates non-Noether symmetry), then the functions I ^(k) = i_W^kω_E^k        k = 1...rank(ω) (where $\omega$ _E = L_Eω) are constant along trajectories. Let's consider $I(1)$ L_Xh^∗I⁽¹⁾ = L_Xh^∗(i_Wω_E) = i_{[Xh^∗ , W]}ω_E + i_WL_Xh^∗ω_E = 0 The second term vanishes since $[E\; ,\; X$_h^∗] = 0^∗ and $L$_Xh^∗ω = 0. The first one is zero as far as $[X$_h^∗ , W^∗] = 0^∗ and $[E\; ,\; 0\ast ]\; =\; 0\ast$ are satisfied. So $I(1)$ is conserved. Similarly one can show that $L$_XhI^(k) = 0 is fulfilled. $W$ is not unique, but $I(k)$ doesn't depend on choosing representative from the class $W\ast$. Theorem is also valid for generators $E$ satisfying $[E\; ,\; X$_h^∗] = X_f^∗ Hamiltonian description of the relativistic particle leads to the following action A = ∫ p₀dx₀ + ∑sp_sdx_s where $p$₀ = (p² + m²)^1/2 with vanishing canonical Hamiltonian and degenerate 2-form defined by p₀ω = ∑s(p_sdp_s ∧ dx₀ + p₀dp_s ∧ dx_s). $\omega$ possesses the "null vector field" $i$_uω = 0 u = p₀∂∂x₀ + ∑sp_s∂∂x_s. One can check that the following non- Hamiltonian vector field E =p₀x₀∂∂x₀ + p₁x₁∂∂x₁ + ⋯ + p_nx_n∂∂x_n generates non-Noether symmetry. Indeed, $E$ satisfies $[E\; ,\; X$_h^∗] = 0^∗ because of $X$_h^∗ = 0^∗ and $[E,u]\; =\; u$. Corresponding integrals of motion are combinations of momenta: I⁽¹⁾ = ∑sp_s I⁽²⁾ = ∑r > s p_rp_s ⋯ I⁽ⁿ⁾ = ∏sp_s 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. S. Hojman A new conservation law constructed without using either Lagrangians or Hamiltonians J. Phys. A: Math. Gen. 25 L291-295 1992 F. González-Gascón Geometric foundations of a new conservation law discovered by Hojman J. Phys. A: Math. Gen. 27 L59-60 1994 M. Lutzky Remarks on a recent theorem about conserved quantities J. Phys. A: Math. Gen. 28 L637-638 1995 M. Lutzky New derivation of a conserved quantity for Lagrangian systems J. Phys. A: Math. Gen. 15 L721-722 1998 N.M.J. Woodhouse Geometric Quantization Claredon, Oxford 1992. G. Chavchanidze Bi-Hamiltonian structure as a shadow of non-Noether symmetry math-ph/0106018 2001