Bi-Hamiltonian structure as a shadow of non-Noether symmetry

George Chavchanidze
Department of Theoretical Physics,A. Razmadze Institute of Mathematics,1 Aleksidze Street, Tbilisi 0193, Georgia
abstract. In the present paper correspondence between non-Noether symmetries and bi-Hamiltonian structuresis disscussed. We show that in regular Hamiltonian systems presence of the global bi-Hamiltonianstructure is caused by symmetry of the space of solution. As an example well known bi-Hamiltonianrealisation of Korteweg-De Vries equation is disscussed.
keywords. Bi-Hamiltonian system; Non-Noether symmetry; Non-Cartan symmetry; Korteweg- De Vries equation.
msc. 70H33, 70H06, 53Z05
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 ω(closed dω = 0 and nondegenerate iXω = 0 ⇒ X = 0 2-form) and timeevolution is governed by Hamilton's equationiXhω + dh = 0where Xh is Hamiltonian vector field that defines time evolutiondfdt = Xh(f) for any function f and iXhω denotes contraction ofXh and ω. Vector field is said to be (locally) Hamiltonian if it preserves ω.According to the Liouville's theorem Xh defined by (1) automatically preserves ω due to relationLXhω = diXhω + iXhdω = − ddh = 0
One can show that group of transformations of phase space generated by any non-Hamiltonian vectorfield Eg(a) = eaLEdoes not preserve actiong*(A) = g*(∫ pdq − hdt) = ∫ g*(pdq − hdt) ≠ 0because d(LE(pdq − hdt)) = LEω − 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 LEω and dE(h) ∧ dt are linearly independent 2-forms). As a result every non-Hamiltonian vector field Ecommuting with Xh leads to the non-Noether symmetry (since E preserves vector field tangentto solutions LE(Xh) = [E , Xh] = 0 it maps the space of solutions onto itself). Any suchsymmetry yields the following integrals of motion [1],[2],[4],[5]I(k) = Tr(Rk)        k = 1,2 ... nwhere R = ω−1LEω and n is half-dimension of phase space.
It is interesting that for any non-Noether symmetry, triple (h, ω, ωE) carries bi-Hamiltonian structure (§4.12 in [6],[7]-[9]). Indeed ωE is closed (dωE = dLEω = LEdω = 0) and invariant (LXhωE = LXhLEω = LELXhω = 0) 2-form (but generic ωE 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, ω and auxiliaryclosed 2- form ω∗ satisfying LXhω∗ = 0. Let us call it global bi-Hamiltonian structure whenever ω∗ is exact (there exists 1-form θ∗ such thatω∗ = dθ∗) and Xh is (globally) Hamiltonian vector field with respect toω∗ (iXhω∗ + dh∗ = 0). As far as ω is nondegenerate there exists vector field E∗ such that iE∗ω = θ∗. By constructionLE∗ω = ω∗Indeed LE∗ω = diE∗ω + iE∗dω= dθ∗ = ω∗
Andi[E∗,Xh]ω = LE∗(iXhω) − iXhLE∗ω= − d(E∗(h) − h∗) = − dh'In other words [Xh , E∗] is Hamiltonian vector field, i. e., [Xh , E] = Xh'. SoE∗ is not generator of symmetry since it does not commute with Xh but one canconstruct (locally) Hamiltonian counterpart of E∗ (note that E∗ itself is non-Hamiltonian) — Xg with g(z) =t∫0 h'dτHere 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, Xg is a locallyHamiltonian vector field, satisfying, by construction, the same commutation relations as E∗ (namely [Xh , Xg] = Xh'). Finally one recovers generator of non-Noether symmetry — non-Hamiltonian vector field E = E∗ − Xg commuting with Xh and satisfyingLEω = LE∗ω − LXgω = LE∗ω = ω∗(thanks to Liouville's theorem LXgω = 0). So in case of regular Hamiltonian system everyglobal bi-Hamiltonian structure is naturally associated with (non-Noether) symmetry of space ofsolutions.
example. As a toy example one can consider free particleh = ½ ∑m pm2       ω = ∑m dpm ∧ dqmthis Hamiltonian system can be extended to the bi-Hamiltonian oneh, ω, ω∗ = ∑m pmdpm ∧ dqmclearly dω∗ = 0 and Xh preserves ω∗. Conserved quantities pm are associated with this simple bi-Hamiltonian structure.This system can be obtained from the following (non-Noether) symmetry (infinitesimal form)qm       →       (1 + apm)qmpm        →       (1 + apm)pm
example. The earliest and probably the most well known bi-Hamiltonian structure is the onediscovered by F. Magri and assosiated with Korteweg- De Vries integrable hierarchy. The KdV equationut + uxxx + uux = 0(zero boundary conditions for u and its derivatives are assumed) appears to be Hamilton's equationiXhω+ dh = 0where Xh = + ∞∫− ∞ dx utδδu (here δδu denotes variational derivative with respect to the field u(x)) is the vector field tangent to thesolutions,ω = + ∞∫− ∞ dx du ∧ dvis the symplectic form (here v is defined by vx = u) and the functionh = + ∞∫− ∞ dx (u33 − ux2)plays the role of Hamiltonian. This dynamical system possesses non-trivial symmetry — one-parametergroup of non-cannonical transformations g(a) = eLE generated by the non-Hamiltonian vectorfieldE = + ∞∫− ∞ dx (uxx + u22)∂∂u + XFhere first term represents non-Hamiltonian part of the generator of the symmetry, while the second oneis its Hamiltonian counterpart assosiated withF = + ∞∫− ∞(u2v12 + G4 + 3vI⁽2⁾4I⁽3⁾)dx(I(2,3) are defined in (22), while G is defined by Gx = u33 − ux2 . The physical origin of this symmetry is unclear, however thesymmetry seems to be very important since it leads to the celebrated infinite sequence of conservationlaws in involution:I(1) = + ∞∫− ∞ u dxI(2) = + ∞∫− ∞ u2 dxI(3) = + ∞∫− ∞ (u33 − ux2) dxI(4) = + ∞∫− ∞ (536u4 − 53uux2 + uxx2) dx⋯and ensures integrability of KdV equation. Second Hamiltonian realization of KdV equation discoveredby F. Magri [7]iXh∗ω∗ + dh∗ = 0(where ω∗ = LEω and h∗ = LEh) is a result of invariance of KdV under aforementioned transformations g(a).
acknowledgements. Author is grateful to Z. Giunashvili for constructive discussions and to G.Jorjadze for support. This work was supported by INTAS (00-00561) and Scholarship from WorldFederation of Scientists.

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