Non-Noether symmetries in singular dynamical systems
George Chavchanidze
Department of Theoretical Physics,
A. Razmadze Institute of Mathematics,
1 Aleksidze Street, Tbilisi 0193, Georgia
Abstract. 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.
Keywords: Non-Noether symmetry; Conservation laws; Constrained dynamics;
MSC 2000: 70H33, 70H06, 53Z05
Georgian Math. J. 10 (2003) 057-061
1. 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
iXhω + dh = 0,
(1)
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.
(2)
The Poisson bracket is defined as follows:
{f , g} = Xf g = − Xg f = iXf
iXgω.
(3)
By introducing a bivector field W satisfying
iXiYω = iW iXω ∧ iYω,
(4)
Poisson bracket can be rewritten as
{f , g} = iW df ∧ dg.
(5)
It's easy to show that
iXiYLZω =
i[Z,W] iXω ∧ iYω,
(6)
where the bracket [ · , · ] is actually a supercommutator,
for an arbitrary bivector field
W = ∑s Vs ∧ Us we have
[X,W] = ∑s[X,Vs] ∧ Us
+ ∑sVs ∧ [X,Us]
(7)
Equation (6) is based on the following useful property of the Lie derivative
LXiWω = i[X,W]ω +
iWLXω.
(8)
Indeed, for an arbitrary bivector field
W = ∑s Vs ∧ Us we have
LXiWω = LX∑siVs ∧ Usω =
LX∑s iUsiVsω
= ∑s i[X,Us]iVsω +
∑s iUsi[X,Vs]ω +
∑siUsiVsLXω =
i[X,W]ω + iWLXω
(9)
where LZ denotes the Lie derivative along the vector field Z.
According to Liouville's theorem Hamiltonian vector field
preserves ω
LXfω = 0;
(10)
therefore it commutes with W:
[Xf ,W] = 0.
(11)
In the local coordinates zs where
ω = ∑rsωrsdzr ∧ zs bivector field
W has the following form
W = ∑rsWrs∂∂zr ∧ ∂∂zs where
Wrs is matrix inverted to ωrs.
2. Case of regular Lagrangian systems
We can say that a group of transformations
g(z) = ezLE generated by the vector
field E maps the space of solutions of equation onto itself if
iXhg*(ω) + g*(dh) = 0
(12)
For Xh satisfying
iXhω + dh = 0
(13)
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
(14)
since [E,Xh] = 0.
When E is not Hamiltonian,
the group of transformations g(z) = ezLE 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) = iWk ωEk k = 1...n,
(15)
where Wk and ωEk 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,
(16)
where according to Liouville's theorem both terms
[Xh , W] = 0 and
iWLXhLEω =
iWLELXhω =
0(17)
since [E , Xh] = 0 and
LXhω = 0 vanish.
In the same manner one can verify that
LXhI(k) = 0
Remark 1.
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(18)
Remark 2.
Discrete non-Noether symmetries give rise to the conservation of
I(k) = iWkg*(ω)k
where g*(ω) is transformed ω.
Remark 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
[Xh , E]. Namely it is easy to show by taking the Lie
derivative of (15) along vector field E that
{I(k) , f} = iWkωk[Xf , E](19)
is fulfilled.
As a result conserved quantities associated with Non-Noether symmetries form Lie algebra under
the Poisson bracket.
Remark 4.
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
3. Case of irregular Lagrangian systems
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,
(20)
then
i(W1 − W2) iXω ∧ iYω = 0 ∀X,Y
(21)
is fulfilled. It is possible only when
W1 − W 2 = ∑svs ∧ us
(22)
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)
= iWkωEk k = 1...rank(ω)
(23)
(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
(24)
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.
Remark 5.
W is not unique, but I(k) doesn't depend
on choosing representative from the class W∗.
Remark 6.
Theorem is also valid for generators E satisfying
[E , Xh∗] = Xf∗
Example 1.
Hamiltonian description of the relativistic particle leads to the following action
A = ∫ p0dx0 +
∑spsdxs
(25)
where
p0 = (p2 + m2)1/2
with vanishing canonical Hamiltonian and degenerate 2-form defined by
p0ω = ∑s(psdps ∧ dx0 + p0dps ∧ dxs).
(26)
ω possesses the "null vector field"
iuω = 0
u = p0∂∂x0 + ∑sps∂∂xs.
(27)
One can check that the following non- Hamiltonian vector field
E =p0x0∂∂x0
+ p1x1∂∂x1 + ⋯ + pnxn∂∂xn
(28)
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
(29)
This example shows that the set of conserved quantities can be obtained from a single
one-parameter group of non-Noether transformations.
Acknowledgements.
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.
[1]
S. Hojman,
A new conservation law constructed without using either Lagrangians or Hamiltonians,
J. Phys. A: Math. Gen. 25 L291-295,
1992
[2]
F. González-Gascón,
Geometric foundations of a new conservation law discovered by Hojman,
J. Phys. A: Math. Gen. 27 L59-60,
1994
[3]
M. Lutzky,
Remarks on a recent theorem about conserved quantities,
J. Phys. A: Math. Gen. 28 L637-638,
1995
[4]
M. Lutzky,
New derivation of a conserved quantity for Lagrangian systems,
J. Phys. A: Math. Gen. 15 L721-722,
1998
[5]
N.M.J. Woodhouse,
Geometric Quantization,
Claredon, Oxford,
1992.
[6]
G. Chavchanidze,
Bi-Hamiltonian structure as a shadow of non-Noether symmetry,
math-ph/0106018,
2001