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
*ω* is the closed
(*dω = 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 *ω*.
Since *ω* is non-degenerate, this gives rise to an isomorphism between the vector
fields and 1-forms given by *i*_{X}ω + α= 0.
The vector field is said to be Hamiltonian if it corresponds to exact form
*W* satisfying
*[ · , · ]* is actually a supercommutator,
for an arbitrary bivector field
* W = V*^{s} ∧ U^{s} we have
*W = V*^{s} ∧ U^{s} we have
*L*_{Z} denotes the Lie derivative along the vector field *Z*.
According to Liouville's theorem Hamiltonian vector field
preserves *ω*
*W*:
* z*_{s} where
*ω = ω*^{rs}dz_{r} ∧ z_{s} bivector field
*W* has the following form
*W = W*^{rs}∂_{zr} ∧ ∂_{zs} where
*W*^{rs} is matrix inverted to *ω*^{rs}.

i_{Xh}ω + dh = 0,

where
i_{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
i_{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 that
i_{X}i_{Y}L_{Z}ω =
i_{[Z,W]} i_{X}ω ∧ i_{Y}ω,

where the bracket
[X,W] = [X,V^{s}] ∧ U^{s}
+ V^{s} ∧ [X,U^{s}]

Equation (6) is based on the following useful property of the Lie derivative
L_{X}i_{W}ω = i_{[X,W]}ω +
i_{W}L_{X}ω.

Indeed, for an arbitrary bivector field
L_{X}i_{W}ω = L_{X}i_{Vs ∧ Us}ω =
L_{X} i_{Us}i_{Vs}ω

= i_{[X,Us]}i_{Vs}ω +
i_{Us}i_{[X,Vs]}ω +
i_{Us}i_{Vs}L_{X}ω =
i_{[X,W]}ω + i_{W}L_{X}ω

where = i

L_{Xf}ω = 0;

therefore it commutes with
[X_{f} ,W] = 0.

In the local coordinates
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
*X*_{h} satisfying
*E* should satisfy
*[E , X*_{h}] = 0
Indeed,
*[E,X*_{h}] = 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).

i_{Xh}g_{*}(ω) + g_{*}(dh) = 0

For
i_{Xh}ω + dh = 0

Hamilton's equation.
It's easy to show that the vector field
i_{Xh}L_{E}ω + dL_{E}h =
L_{E}(i_{Xh}ω + dh) = 0

since
I^{(k)} = i_{Wk} ω_{E}^{k} k = 1...n,

where
L_{Xh}I^{(1)}
= L_{Xh}(i_{W}ω_{E}) =
i_{[Xh , W]}ω_{E}
+ i_{W}L_{Xh}ω_{E},

where according to Liouville's theorem both terms
i_{W}L_{Xh}L_{E}ω =
i_{W}L_{E}L_{Xh}ω =
0

since
i_{Xh}g_{*}(ω) +
d(g_{*}h + f) = 0

{I^{(k)} , f} = i_{Wk}ω^{k}_{[Xf , E]}

is fulfilled.
As a result conserved quantities associated with Non-Noether symmetries form Lie algebra under
the Poisson bracket.
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" *u*_{s} such that
*i*_{us}ω = 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} + C_{s}u_{s}, we can introduce equivalence class
* X*_{f}^{∗} (then all *u*_{s} belong to
*0*^{∗} ).
The bivector field *W* is also far from being unique, but if
*W*_{1} and *W*_{2} both satisfy
*v*_{s} are some vector fields and
*i*_{us}ω = 0
(in other words when * W*_{1} − W_{2} belongs to the class
*0*^{∗})

i_{X}i_{Y} ω =
i_{W1,2} i_{X}ω ∧ i_{Y}ω,

then
i_{(W1 − W2)} i_{X}ω ∧ i_{Y}ω = 0 ∀X,Y

is fulfilled. It is possible only when
W_{1} − W _{2} = v_{s} ∧ u_{s}

where
I ^{(k)}
= i_{Wk}ω_{E}^{k} k = 1...rank(ω)

(where
L_{Xh∗}I^{(1)}
= L_{Xh∗}(i_{W}ω_{E})
= i_{[Xh∗ , W]}ω_{E} +
i_{W}L_{Xh∗}ω_{E} = 0

The second term vanishes since
A = p_{0}dx_{0} + p_{s}dx_{s}

where
p_{0}ω = (p_{s}dp_{s} ∧ dx_{0} + p_{0}dp_{s} ∧ dx_{s}).

u = p_{0}∂_{x0} + p_{s}∂_{xs}.

One can check that the following non- Hamiltonian vector field
E =p_{0}x_{0}∂_{x0}
+ p_{1}x_{1}∂_{x1} + ⋯ + p_{n}x_{n}∂_{xn}

generates non-Noether symmetry. Indeed,
I^{(1)} = p_{s}

I^{(2)} = p_{r}p_{s}

⋯

I^{(n)} = p_{s}

This example shows that the set of conserved quantities can be obtained from a single
one-parameter group of non-Noether transformations.
I

⋯

I

- S. Hojman, A new conservation law constructed without using either Lagrangians or Hamiltonians, 1992 J. Phys. A: Math. Gen. 25 L291-295
- F. González-Gascón, Geometric foundations of a new conservation law discovered by Hojman, 1994 J. Phys. A: Math. Gen. 27 L59-60
- M. Lutzky, Remarks on a recent theorem about conserved quantities, 1995 J. Phys. A: Math. Gen. 28 L637-638
- M. Lutzky, New derivation of a conserved quantity for Lagrangian systems, 1998 J. Phys. A: Math. Gen. 15 L721-722
- N.M.J. Woodhouse, Geometric Quantization, Claredon, Oxford, 1992.
- G. Chavchanidze, Bi-Hamiltonian structure as a shadow of non-Noether symmetry 2001, math-ph/0106018