Georgian Mathematical Journal Volume 8 (2001), Number 1, 27-32

NON-NOETHER SYMMETRIES IN SINGULAR DYNAMICAL SYSTEMS

G. CHAVCHANIDZE

Abstract. The Hojman–Lutzky conservation law establishes a certain correspondence between non-Noether symmetries and conserved quantities. In the present paper the extension of the Hojman–Lutzky theorem to singular dynamical systems is carried out.

2000 Mathematics Subject Classification: 70665, 70645, 70H45, 70H15. Key words and phrases: Conservation laws, symmetry, Lagrangian and Hamiltonian dynamical systems.

1. Introduction

Noether’s theorem associates conservation laws with particular continuous symmetries of the Lagrangian. According to the Hojman’s theorem [1–3] there exist 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 the Hojman–Lutzky theorem to singular dynamical systems is considered.

First let us recall some basic knowledge of the description of regular dynamical systems (see, e.g., [5]). Since the trajectories are solutions of the Euler–Lagrange equations

\frac{d}{dt} \frac{∂L}{∂v^{a}} − \frac{∂L}{∂q^{a}} = 0,

the tangent vector of trajectory satisfies Hamilton’s equation

i_{X_{h}}ω_{L} + dh = 0,

where ω_{L} is the closed (dω_{L} = 0) and non-degenerate (i_{X}ω_{L} = 0 ⇔ X = 0) 2-form, h is the Hamiltonian and i_{X}ω_{L} = X⌋ω_{L} denotes the contraction of X to ω_{L}. In the local coordinates ω_{L} = dθ_{L}, where θ_{L} = \frac{∂L}{∂v^{a}} dq^{a} and h = \frac{∂L}{∂v^{a}} v^{a} − L. Since ω_{L} is non-degenerate, this gives rise to an isomorphism between the vector fields and the 1-forms given by i_{X}ω_{L} + α = 0. The vector field is said to be the Hamiltonian if it corresponds to the exact form

i_{X}_{f}ω_{L} + df = 0.

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The Poisson bracket is defined as follows:

{f, g} = X_{f}g = −X_{g}f = i_{X}_{f}i_{X}_{g}ω.

By introducing a bivector field W satisfying

i_{X}i_{Y} ω = W ⌋i_{X}ω ∧ i_{Y} ω,

the Poisson bracket can be rewritten as

{f, g} = W ⌋df ∧ dg.

It is easy to show that

i_{X}i_{Y} L_{Z}ω = [Z, W]⌋i_{X}ω ∧ i_{Y} ω,  (1)

where the bracket [ , ] is actually a supercommutator (for an arbitrary bivector field W = ∑_{i} V^{i} ∧ U^{i} we have [X, W] = ∑_{i}[X, V^{i}] ∧ U^{i} + ∑_{i} V^{i} ∧ [X, U^{i}]). Equation (1) 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 W = ∑_{i} V^{i} ∧ U^{i} we have

L_{X}i_{W} ω = L_{X}i_{∑_{i} V^{i}∧U^{i}}ω = L_{X} ∑_{i} i_{U^{i}}i_{V^{i}}ω = ∑_{i} i_{[X,U^{i}]}i_{V^{i}}ω

+ ∑_{i} i_{U^{i}}i_{[X,V^{i}]}ω + ∑_{i} i_{U^{i}}i_{V^{i}}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 the Hamiltonian vector field preserves ω_{L}

L_{X}_{f}ω = 0;

therefore it commutes with W:

[X_{f}, W] = 0.

In the local coordinates ξ_{i}, where ω = ω^{ij}dξ_{i} ∧ ξ_{j}, the bivector field W has the form W = W^{ij} \frac{∂}{∂ξ_{i}} ∧ \frac{∂}{∂ξ_{j}}, where W^{ij} is the matrix inverse to ω^{ij}.

2. The Hojman–Lutzky Theorem for Regular Lagrangian Systems

We can say that a group of transformations g(a) = e^{aL_{E}} generated by the vector field E maps the space of solutions of the equation onto itself if

i_{X_{h}}g_{∗}(ω_{L}) + g_{∗}(dh) = 0.  (2)

For X_{h} satisfying

i_{X_{h}}ω_{L} + dh = 0

Hamilton’s equation. It is easy to show that the vector field E should satisfy [E, X_{h}] = 0 (Indeed, i_{X_{h}}L_{E}ω_{L} +dL_{E}h = L_{E}(i_{X_{h}}ω_{L} +dh) = 0 since [E, X_{h}] = 0).

When E is not Hamiltonian, the group of transformations g(a) = e^{aL_{E}} is non-Noether symmetry (in a sense that it maps solutions onto solutions but does not preserve action).

Theorem (Lutzky, 1998). If the non-Hamiltonian vector field E generates non-Noether symmetry, then the following functions are constant along solutions:

I^{(k)} = W^{k}⌋ω_{E}^{k}, k = 1, …, n,

where W^{k} and ω_{E}^{k} are the outer powers of W and L_{E}ω.

Proof. We have to prove that I^{(k)} is constant along the flow generated by the Hamiltonian. In other words, we should find that L_{X_{h}}I^{(k)} = 0 is fulfilled. Let us consider

L_{X_{h}}I^{(1)} = L_{X_{h}}(W ⌋ω_{E}) = [X_{h}, W]⌋ω_{E} + W ⌋L_{X_{h}}ω_{E},

where, according to Liouville’s theorem, both terms ([X_{h}, W] = 0 and W ⌋L_{X_{h}}L_{E}ω = W ⌋L_{E}L_{X_{h}}ω = 0 since [E, X_{h}] = 0 and L_{X_{h}}ω = 0) vanish. In the same manner one can verify that L_{X_{h}}I^{(k)} = 0.

Note 1. The 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_{X_{h}}ω_{L} +dh = 0 on the solutions of the equation i_{X_{h}}g_{∗}(ω) + d(g_{∗}h + f) = 0.

Note 2. Discrete non-Noether symmetries give rise to the conservation of I^{(k)} = W^{k}⌋g_{∗}(ω)^{k} where g_{∗}(ω) is transformed ω.

Note 3. 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 (2) along vector field E that {I^{(k)}, f} = i_{[W,X_{h}]^{k}}ω_{E}^{k} is fulfilled. As a result, the conserved quantities associated with Non-Noether symmetries form a Lie algebra under the Poisson bracket.

3. The Case of Irregular Lagrangian Systems

The singular Lagrangian (a Lagrangian with the vanishing Hessian det \frac{∂L}{∂v^{i}∂v^{j}} = 0) leads to the degenerate 2-form ω_{L} and we no longer have an isomorphism between the vector fields and the 1-forms. Since there exists a set of “null vectors” u^{k} such that i_{u^{k}}ω = 0, k = 1, 2, …, n − rank(ω), every Hamiltonian vector field is defined up to a linear combination of vectors u^{k}. By identifying X_{f} with X_{f} + C_{k}u^{k}, we can introduce an equivalence class X̃_{f} (then all u^{k} belong to ˜0 ). The bivector field W is also far from being unique, but if W_{1} and W_{2} both satisfy

i_{X}i_{Y} ω = W_{1,2}⌋i_{X}ω ∧ i_{Y} ω,

then

(W_{1} − W_{2})⌋i_{X}ω ∧ i_{Y} ω = 0 ∀X, Y

is fulfilled. This is possible only when

W_{1} − W_{2} = v_{k} ∧ u^{k},

where v_{k} are some vector fields and i_{u^{k}}ω = 0 (in other words, when W_{1} − W_{2} belongs to the class ˜0.

Theorem. If the non-Hamiltonian vector field E satisfies the commutation relation [E, X̃_{h}] = ˜0 (generates non-Noether symmetry), then the functions

I^{(k)} = W^{k}⌋ω_{E}^{k}, k = 1, …, rank(ω),

(where ω_{E} = L_{E}ω) are constant along the trajectories.

Proof. Let us consider I^{(1)}:

LX̃_{h}I^{(1)} = LX̃_{h}(W ⌋ω_{E}) = [X̃_{h}, W]⌋ω_{E} + W ⌋LX̃_{h}ω_{E} = 0

The second term vanishes since [E, X̃_{h}] = ˜0 and LX̃_{h}ω = 0. The first one is zero as far as [X̃_{h}, W̃] = ˜0 and [E, ˜0] = ˜0 are satisfied. So I^{(1)} is conserved. Similarly, one can show that L_{X_{h}}I^{(k)} = 0 is fulfilled.

Note 1. W is not unique, but I^{(k)} does not depend on choosing a representative from the class W̃.

Note 2. Theorem is also valid for the generators E satisfying [E, X̃_{h}] = X̃_{f}.

Note 3. Theorem can be applied to irregular Hamiltonian systems (Hamiltonian systems with degenerate ω).

Example. Hamiltonian description of the relativistic particle leads to the action

∫ \sqrt{p^{2} + m^{2}}dx_{0} + p_{k}dx^{k}

with the vanishing canonical Hamiltonian and degenerate 2-form

\frac{1}{\sqrt{p^{2} + m^{2}}} (p^{k}dp_{k} ∧ dx_{0} + \sqrt{p^{2} + m^{2}}dp_{k} ∧ dx^{k}).

ω possesses the “null vector field” i_{u}ω = 0,

u = \sqrt{p^{2} + m^{2}} \frac{∂}{∂x_{0}} + p_{k} \frac{∂}{∂x^{k}} .

One can check that the non-Hamiltonian vector field

E = \sqrt{p^{2} + m^{2}}x_{0} \frac{∂}{∂x_{0}} + p_{1}x^{1} \frac{∂}{∂x^{1}} + ⋯ + p_{n}x^{n} \frac{∂}{∂x^{n}}

generates non-Noether symmetry. Indeed, E satisfies [E, X̃_{h}] = ˜0 because of X̃_{h} = ˜0 and [E, u] = u. The corresponding integrals of motion are combinations of momenta:

I^{(1)} = \sqrt{p^{2} + m^{2}} + p_{1} + ⋯ + p_{n} = ∑_{μ} p_{μ}; I^{(2)} = ∑_{μν} p_{μ}p_{ν}; … I^{(n)} = ∏_{μ} p_{μ}.

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

4. The Case of Dynamical Systems on Poisson Manifold

The previous two sections dealt with dynamical systems on symplectic and presymplectic manifolds. Now let us consider the case of dynamical systems on the Poisson manifold. In general, the Poisson manifold is an even-dimensional manifold equipped with the Poisson bracket which can be defined by means of the bivector field W satisfying [W, W] = 0 as follows:

{f, g} = i_{W} df ∧ dg

Due to skewsymmetry of the Poisson bracket W is also skewsymmetric and, in general, it is degenerate. The commutation relation [W, W] = 0 (where [ , ] denotes the supercommutator of vector fields) ensures that the Poisson bracket satisfies the Jacobi identity. Like in the case of symplectic (presymplectic) manifold, we have the correspondence between the vector fields and the 1-forms

β(X) + α ∧ β(W) = 0 ∀β ∈ Ω^{1}(M).

The vector field is called the Hamiltonian if it corresponds to exact the 1-form

β(X_{h}) + dh ∧ β(W) = 0 ∀β ∈ Ω^{1}(M).

According to Liouville’s theorem such a vector field preserves bivector field W satisfying [X_{h}, W] = 0. Now let us consider the one-parameter group of transformations g(a) = e^{aL_{E}} generated by the vector field E. When W is nondegenerate, it is easy to show that every vector field E satisfying [E, X_{h}] = 0 generates the symmetry of Hamilton’s equation (maps the space of solutions onto itself). In the case of degenerate W there exists a subspace of 1-forms A such that

α ∧ β(W) = 0 ∀α ∈ A ∀β ∈ Ω^{1}(M).

This set of 1-forms gives rise to the set of constraints

α ∧ dh(W) = 0 ∀α ∈ A.

These constraints will be preserved if A is an invariant subspace under the action of g, in other words, if L_{E}α ∈ A ∀α ∈ A. So if the vector field E satisfies

[E, X_{h}] = 0 and L_{E}α ∈ A ∀α ∈ A,

then it generates the symmetry of Hamilton’s equation. Now let us consider the correspondence between such symmetries and the conservation laws.

Theorem. If the non-Hamiltonian vector field E generates the symmetry of Hamilton’s equation, then the set of functions I^{(k)} = i_{W^{k}}ω_{E}^{k} is conserved.

Here ω is the 2-form defined by W^{k}(ω) = W^{k−1} and ω_{E} = L_{E}ω; such a form always exists, but in the case of degenerate W it is far from being unique.

Proof. Let us consider I^{(1)}. We have to prove that L_{X_{h}}I^{(1)} = 0 . Indeed,

L_{X_{h}}I^{(1)} = L_{X_{h}}i_{W} ω_{E} = i_{[X_{h},W]}ω_{E} + i_{W} L_{X_{h}}ω_{E}.

In this expression the first term vanishes since according to Liouville’s theorem [X_{h}, W] = 0. Using [E, X_{h}] = 0, the second term can be rewritten as i_{W} L_{X_{h}}L_{E}ω = i_{W} L_{E}L_{X_{h}}ω. Now from the definition of ω one can show that L_{X_{h}}ω(W^{k}) = 0. This means that L_{X_{h}}ω can be expressed in the following form:

L_{X_{h}}ω = ∑_{i} α_{i} ∧ β_{i}, α_{i} ∈ A, β_{i} ∈ Ω^{(1)}(M).

As far as A is invariant under the action of E we have

i_{W} L_{E}L_{X_{h}}ω = i_{W} L_{E} ∑_{i} α_{i} ∧ β_{i} = ∑_{i} i_{W} L_{E}(α_{i} ∧ β_{i}) = ∑_{i} i_{W} (L_{E}(α_{i}) ∧ β_{i})

− ∑_{i} i_{W} (α_{i} ∧ L_{E}(β_{i})) = ∑_{i} i_{W} α̃_{i} ∧ β_{i} − ∑_{i} i_{W} α_{i} ∧ β̃_{i} = 0

because of α, α̃ ∈ A. The proof of L_{X_{h}}I^{(k)} = 0 is similar.

Note 1. The 2-form ω is far from being unique, but if ω_{1} and ω_{2} both satisfy W^{k}(ω) = W^{k−1}, then ω_{1} − ω_{2} can be expressed as L_{X_{h}}ω = ∑_{i} α_{i} ∧ β_{i}, α_{i} ∈ A, β_{i} ∈ Ω^{(1)}(M), and therefore it does not contribute in I^{(k)}.

Note 2. The theorem is valid for a larger class of generators E satisfying

[E, X_{h}] = 0 and L_{E}α ∈ A ∀α ∈ A.

Acknowledgements

The author is grateful to Z. Giunashvili and M. Maziashvili for constructive discussions and particularly grateful to George Jorjadze for invaluable help.

References

1. S. A. Hojman, A new conservation law constructed without using either Lagrangians or Hamiltonians. J. Phys. A 25(1992), L291–295.

2. F. González-Gascón, Geometric foundations of a new conservation law discovered by Hojman. J. Phys. A 27(1994), L59–60.

3. M. Lutzky, Remarks on a recent theorem about conserved quantities. J. Phys. A 28(1995), L637–638.

4. M. Lutzky, New derivation of a conserved quantity for Lagrangian systems. J. Phys. A 15(1998), L721–722.

5. N. M. J. Woodhouse, Geometric quantization. 2nd ed. Clarendon Press, Oxford, 1992.

(Received 27.12.2000; revised 19.02.2001)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, Aleksidze St., Tbilisi 380093, Georgia

E-mail: chaffinch@apexmail.com