]> Non-Noether symmetries in singular dynamical systems

Non-Noether symmetries in singular dynamical systems

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


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=0X=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}=Xfg=Xgf=iXfiXgω. By introducing a bivector field W satisfying iXiYω=iWiXωiYω, Poisson bracket can be rewritten as {f,g}=iWdfdg. It's easy to show that iXiYLZω=i[Z,W]iXωiYω, where the bracket [·,·] is actually a supercommutator, for an arbitrary bivector field W=sVsUs we have [X,W]=s[X,Vs]Us+sVs[X,Us] 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=sVsUs we have LXiWω=LXsiVsUsω=LXsiUsiVsω=si[X,Us]iVsω+siUsi[X,Vs]ω+siUsiVsLXω=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=rsωrsdzrzs bivector field W has the following form W=rsWrszrzs where Wrs is matrix inverted to ωrs.

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 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)=ezLE 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)=iWkωEk      k=1...n, where Wk and ωEk are outer powers of W and LEω.
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)=iWkg*(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}=iWkω[Xf,E]k 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 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...nrank(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 iXiYω=iW1,2iXωiYω, then i(W1W2)iXωiYω=0      X,Y is fulfilled. It is possible only when W1W2=svsus where vs are some vector fields and iusω=0 (in other words when W1W2 belongs to the class 0)
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) (where ωE=LEω) are constant along trajectories.
Let's consider I(1) LXhI(1)=LXh(iWωE)=i[Xh,W]ωE+iWLXhωE=0 The second term vanishes since [E,Xh]=0 and LXhω=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
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(psdpsdx0+p0dpsdxs). ω possesses the "null vector field" iuω=0 u=p0x0+spsxs. One can check that the following non- Hamiltonian vector field E=p0x0x0+p1x1x1++pnxnxn 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)=spsI(2)=r>sprpsI(n)=sps 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.
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