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 ω is the closed (dω = 0) and non-degenerate (iXω = 0 ⇒ X = 0) 2-form, h is the Hamiltonian and iXω denotes contraction of X with ω. Since ω is non-degenerate, this gives rise to an isomorphism between the vector fields and 1-forms given by iXω + α= 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} = Xf g = − Xg f = iXf iXgω. By introducing a bivector field W satisfying iXiYω = iW iXω ∧ iYω, Poisson bracket can be rewritten as {f , g} = iW df ∧ dg. It's easy to show that iXiYLZω = i[Z,W] iXω ∧ iYω, where the bracket [ · , · ] is actually a supercommutator, for an arbitrary bivector field W = ∑s V^s ∧ U^s 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 LXiWω = i[X,W]ω + iWLXω. Indeed, for an arbitrary bivector field W = ∑s V^s ∧ U^s we have LXiWω = LX∑siV^s ∧ U^sω = LX∑s iU^siV^sω = ∑s i[X,U^s]iV^sω + ∑s iU^si[X,V^s]ω + ∑siU^siV^sLXω = i[X,W]ω + iWLXω where LZ denotes the Lie derivative along the vector field Z. According to Liouville's theorem Hamiltonian vector field preserves ω LXfω = 0; therefore it commutes with W: [Xf ,W] = 0. In the local coordinates zs where ω = ∑rsω^rsdzr ∧ zs bivector field W has the following form W = ∑rsW^rs∂∂zr ∧ ∂∂zs where W^rs is matrix inverted to ω^rs.

We can say that a group of transformations g(z) = e^zLE generated by the vector field E maps the space of solutions of equation onto itself if iXhg*(ω) + g*(dh) = 0 For Xh satisfying iXhω + dh = 0 Hamilton's equation. It's easy to show that the vector field E should satisfy [E , Xh] = 0 Indeed, iXhLEω + dLEh = LE(iXhω + dh) = 0 since [E,Xh] = 0. When E is not Hamiltonian, the group of transformations g(z) = e^zLE is non-Noether symmetry (in a sense that it maps solutions onto solutions but does not preserve action).

Theorem 1. (Lutzky, 1998) If the vector field E generates non-Noether symmetry, then the following functions are constant along solutions: I^(k) = iW^k ωE^k       k = 1...n, where W^k and ωE^k are outer powers of W and LEω.

Proof. We have to prove that I^(k) is constant along the flow generated by the Hamiltonian. In other words, we should find that LXhI^(k) = 0 is fulfilled. Let us consider LXhI⁽¹⁾ LXhI⁽¹⁾ = LXh(iWωE) = i[Xh , W]ωE + iWLXhωE, where according to Liouville's theorem both terms [Xh , W] = 0 and iWLXhLEω = iWLELXhω = 0 since [E , Xh] = 0 and LXhω = 0 vanish. In the same manner one can verify that LXhI^(k) = 0 ∎

Theorem is valid for a larger class of generators E . Namely, if [E , Xh] = Xf where Xf is an arbitrary Hamiltonian vector field, then I^(k) is still conserved. Such a symmetries map the solutions of the equation iXhω + dh = 0 on solutions of iXhg*(ω) + d(g*h + f) = 0

Discrete non-Noether symmetries give rise to the conservation of I^(k) = iW^kg*(ω)^k where g*(ω) is transformed ω.

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 [Xh , E]. Namely it is easy to show by taking the Lie derivative of (15) along vector field E that {I^(k) , f} = iW^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

The singular Lagrangian (Lagrangian with vanishing Hessian) leads to degenerate 2-form ω and we no longer have isomorphism between vector fields and 1-forms. Since there exists a set of "null vectors" us such that iusω = 0       s = 1,2 ... n − rank(ω), every Hamiltonian vector field is defined up to linear combination of vectors us. By identifying Xf with Xf + ∑sCsus, we can introduce equivalence class Xf^∗ (then all us belong to 0^∗ ). The bivector field W is also far from being unique, but if W1 and W2 both satisfy iXiY ω = iW1,2 iXω ∧ iYω, then i(W1 − W2) iXω ∧ iYω = 0       ∀X,Y is fulfilled. It is possible only when W1 − W2 = ∑svs ∧ us where vs are some vector fields and iusω = 0 (in other words when W1 − W2 belongs to the class 0^∗)

Theorem 2. If the non-Hamiltonian vector field E satisfies [E , Xh^∗] = 0^∗ commutation relation (generates non-Noether symmetry), then the functions I^(k) = iW^kωE^k        k = 1...rank(ω) (where ω E = LEω) are constant along trajectories.

Proof. Let's consider I⁽¹⁾ LXh^∗I⁽¹⁾ = LXh^∗(iWωE) = i[Xh^∗ , W]ωE + iWLXh^∗ωE = 0 The second term vanishes since [E , Xh^∗] = 0^∗ and LXh^∗ω = 0. The first one is zero as far as [Xh^∗ , W^∗] = 0^∗ and [E , 0^∗] = 0^∗ are satisfied. So I ⁽¹⁾ is conserved. Similarly one can show that LXhI^(k) = 0 is fulfilled. ∎

W is not unique, but I^(k) doesn't depend on choosing representative from the class W^∗.

Theorem is also valid for generators E satisfying [E , Xh^∗] = Xf^∗

Example. Hamiltonian description of the relativistic particle leads to the following action A = ∫ p0dx0 + ∑spsdxs where p0 = (p² + m²)^1/2 with vanishing canonical Hamiltonian and degenerate 2-form defined by p0ω = ∑s(psdps ∧ dx0 + p0dps ∧ dxs). ω possesses the "null vector field" iuω = 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 [E , Xh^∗] = 0^∗ because of Xh^∗ = 0^∗ and [E,u] = u. Corresponding integrals of motion are combinations of momenta: I⁽¹⁾ = ∑sps I⁽²⁾ = ∑r > s prps ⋯ 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.