Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia

Georgian Math. J. 10 (2003) 057-061

Noether theorem, Lutzky's theorem, bi-Hamiltonian formalism and bidifferential calculi are often used in generating conservation laws and allthis approaches are unified by the single idea — to construct conserved quantities out of some invariantgeometric object (generator of the symmetry — Hamiltonian vector field in Noether theorem, non-Hamiltonian one in Lutzky's approach, closed 2-form in bi-Hamiltonian formalism and auxiliarydifferential in case of bidifferential calculi). There is close relationship between later three approaches.Some aspects of this relationship has been uncovered in [3],[4]. In the present paper it isdiscussed how bi-Hamiltonian structure can be interpreted as a manifestation of symmetry of space ofsolutions. Good candidate for this role is non-Noether symmetry. Such a symmetry is a group oftransformation that maps the space of solutions of equations of motion onto itself, but unlike theNoether one, does not preserve action.

In the case of regular Hamiltonian system phase space is equipped with symplectic form $\omega $(closed $d\omega \; =\; 0$ and nondegenerate $i$_{X}ω = 0 ⇒ X = 0 2-form) and timeevolution is governed by Hamilton's equationi_{Xh}ω + dh = 0 where $X$_{h} is Hamiltonian vector field that defines time evolutiondf dt = X_{h}(f) for any function $f$ and $i$_{Xh}ω denotes contraction of$X$_{h} and $\omega $. Vector field is said to be (locally) Hamiltonian if it preserves $\omega $.According to the Liouville's theorem $X$_{h} defined by (1) automatically preserves $\omega $ due to relationL_{Xh}ω = di_{Xh}ω + i_{Xh}dω = − ddh = 0

One can show that group of transformations of phase space generated by any non-Hamiltonian vectorfield $E$g(a) = e^{aLE} does not preserve actiong_{*}(A) = g_{*}(∫ pdq − hdt) = ∫ g_{*}(pdq − hdt) ≠ 0 because $d(L$_{E}(pdq − hdt)) = L_{E}ω − dE(h) ∧ dt ≠ 0 (first term in r.h.s. does not vanishsince $E$ is non-Hamiltonian and as far as $E$ is time independent $L$_{E}ω and $dE(h)\; \wedge \; dt$ are linearly independent 2-forms). As a result every non-Hamiltonian vector field $E$commuting with $X$_{h} leads to the non-Noether symmetry (since $E$ preserves vector field tangentto solutions $L$_{E}(X_{h}) = [E , X_{h}] = 0 it maps the space of solutions onto itself). Any suchsymmetry yields the following integrals of motion [1],[2],[4],[5]I^{(k)} = Tr(R^{k}) k = 1,2 ... n where $R\; =\; \omega -1L$_{E}ω and $n$ is half-dimension of phase space.

It is interesting that for any non-Noether symmetry, triple $(h,\; \omega ,\; \omega $_{E}) carries bi-Hamiltonian structure (§4.12 in [6],[7]-[9]). Indeed $\omega $_{E} is closed ($d\omega $_{E} = dL_{E}ω = L_{E}dω = 0) and invariant ($L$_{Xh}ω_{E} = L_{Xh}L_{E}ω = L_{E}L_{Xh}ω = 0) 2-form (but generic $\omega $_{E} is degenerate). So every non-Noethersymmetry quite naturally endows dynamical system with bi-Hamiltonian structure.

Now let's discuss how non-Noether symmetry can be recovered from bi-Hamiltonian system. Generic bi-Hamiltonian structure on phase space consists of Hamiltonian system $h,\; \omega $ and auxiliaryclosed 2- form $\omega \ast $ satisfying $L$_{Xh}ω^{∗} = 0. Let us call it global bi-Hamiltonian structure whenever $\omega \ast $ is exact (there exists 1-form $\theta \ast $ such that$\omega \ast =\; d\theta \ast $) and $X$_{h} is (globally) Hamiltonian vector field with respect to$\omega \ast $ ($i$_{Xh}ω^{∗} + dh^{∗} = 0). As far as $\omega $ is nondegenerate there exists vector field $E\ast $ such that $i$_{E∗}ω = θ^{∗}. By constructionL_{E∗}ω = ω^{∗} Indeed L_{E∗}ω = di_{E∗}ω + i_{E∗}dω= dθ^{∗} = ω^{∗}

Andi_{[E∗,Xh]}ω = L_{E∗}(i_{Xh}ω) − i_{Xh}L_{E∗}ω= − d(E^{∗}(h) − h^{∗}) = − dh' In other words $[X$_{h} , E^{∗}] is Hamiltonian vector field, i. e., $[X$_{h} , E] = X_{h'}. So$E\ast $ is not generator of symmetry since it does not commute with $X$_{h} but one canconstruct (locally) Hamiltonian counterpart of $E\ast $ (note that $E\ast $ itself is non-Hamiltonian) — $X$_{g} with g(z) =t ∫ 0 h'dτ Here integration along solution of Hamilton's equation, with fixed origin and end point in $z(t)\; =\; z$,is assumed. Note that (10) defines $g(z)$ only locally and, as a result, $X$_{g} is a locallyHamiltonian vector field, satisfying, by construction, the same commutation relations as $E\ast $ (namely $[X$_{h} , X_{g}] = X_{h'}). Finally one recovers generator of non-Noether symmetry — non-Hamiltonian vector field $E\; =\; E\ast -\; X$_{g} commuting with $X$_{h} and satisfyingL_{E}ω = L_{E∗}ω − L_{Xg}ω = L_{E∗}ω = ω^{∗} (thanks to Liouville's theorem $L$_{Xg}ω = 0). So in case of regular Hamiltonian system everyglobal bi-Hamiltonian structure is naturally associated with (non-Noether) symmetry of space ofsolutions.

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