Noether theorem, Lutzky's theorem, bi-Hamiltonian formalism and bidifferential calculi are often used in generating conservation laws and all this approaches are unified by the single idea — to construct conserved quantities out of some invariant geometric 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 auxiliary differential 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 is discussed how bi-Hamiltonian structure can be interpreted as a manifestation of symmetry of space of solutions. Good candidate for this role is non-Noether symmetry. Such a symmetry is a group of transformation that maps the space of solutions of equations of motion onto itself, but unlike the Noether 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 time evolution is governed by Hamilton's equation iXhω + dh = 0 where Xh is Hamiltonian vector field that defines time evolution dfdt = Xh(f) for any function f and iXhω denotes contraction of Xh 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 relation LXhω = diXhω + iXhdω = − ddh = 0

One can show that group of transformations of phase space generated by any non-Hamiltonian vector field E g(a) = e^aLE does not preserve action g*(A) = g*(∫ pdq − hdt) = ∫ g*(pdq − hdt) ≠ 0 because d(LE(pdq − hdt)) = LEω − dE(h) ∧ dt ≠ 0 (first term in r.h.s. does not vanish since 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 E commuting with Xh leads to the non-Noether symmetry (since E preserves vector field tangent to solutions LE(Xh) = [E , Xh] = 0 it maps the space of solutions onto itself). Any such symmetry yields the following integrals of motion [1],[2],[4],[5] I^(k) = Tr(R^k)        k = 1,2 ... n where 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-Noether symmetry 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 auxiliary closed 2- form ω^∗ satisfying LXhω^∗ = 0. Let us call it global bi-Hamiltonian structure whenever ω^∗ is exact (there exists 1-form θ^∗ such that ω^∗ = dθ^∗) and Xh is (globally) Hamiltonian vector field with respect to ω^∗ (iXhω^∗ + dh^∗ = 0). As far as ω is nondegenerate there exists vector field E^∗ such that iE^∗ω = θ^∗. By construction LE^∗ω = ω^∗ Indeed LE^∗ω = diE^∗ω + iE^∗dω = dθ^∗ = ω^∗

And i[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'. So E^∗ is not generator of symmetry since it does not commute with Xh but one can construct (locally) Hamiltonian counterpart of E^∗ (note that E^∗ itself is non-Hamiltonian) — Xg with g(z) =∫0t 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, Xg is a locally Hamiltonian 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 satisfying LEω = LE^∗ω − LXgω = LE^∗ω = ω^∗ (thanks to Liouville's theorem LXgω = 0). So in case of regular Hamiltonian system every global bi-Hamiltonian structure is naturally associated with (non-Noether) symmetry of space of solutions.

Example. As a toy example one can consider free particle h = ½ ∑m pm²       ω = ∑m dpm ∧ dqm this Hamiltonian system can be extended to the bi-Hamiltonian one h, ω, ω^∗ = ∑m pmdpm ∧ dqm clearly 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)qm pm        →       (1 + apm)pm

Example. The earliest and probably the most well known bi-Hamiltonian structure is the one discovered by F. Magri and assosiated with Korteweg- De Vries integrable hierarchy. The KdV equation ut + uxxx + uux = 0 (zero boundary conditions for u and its derivatives are assumed) appears to be Hamilton's equation iXhω+ dh = 0 where Xh = ∫− ∞+ ∞ dx utδδu (here δδu denotes variational derivative with respect to the field u(x)) is the vector field tangent to the solutions, ω = ∫− ∞+ ∞ dx du ∧ dv is the symplectic form (here v is defined by vx = u) and the function h = ∫− ∞+ ∞ dx u³3 − ux² plays the role of Hamiltonian. This dynamical system possesses non-trivial symmetry — one-parameter group of non-cannonical transformations g(a) = e^LE generated by the non-Hamiltonian vector field E = ∫− ∞+ ∞ dx uxx + u²2∂∂u + XF here first term represents non-Hamiltonian part of the generator of the symmetry, while the second one is its Hamiltonian counterpart assosiated with F = ∫− ∞+ ∞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 the symmetry seems to be very important since it leads to the celebrated infinite sequence of conservation laws in involution: I⁽¹⁾ = ∫− ∞+ ∞ u dx I⁽²⁾ = ∫− ∞+ ∞ u² dx I⁽³⁾ = ∫− ∞+ ∞ u³3 − ux² dx I⁽⁴⁾ = ∫− ∞+ ∞ 536u⁴ − 53uux² + uxx² dx ⋯ and ensures integrability of KdV equation. Second Hamiltonian realization of KdV equation discovered by 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).