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 *i*_{X}ω = 0 ⇒ X = 0 2-form) and time
evolution is governed by Hamilton's equation
*X*_{h} is Hamiltonian vector field that defines time evolution
*f* and *i*_{Xh}ω denotes contraction of
*X*_{h} and *ω*. Vector field is said to be (locally) Hamiltonian if it preserves *ω*.
According to the Liouville's theorem *X*_{h} defined by (1) automatically preserves *ω*
due to relation

i_{Xh}ω + dh = 0

where
X_{h}(f) = ḟ

for any function
L_{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 vector
field *E*
*d(L*_{E}(pdq − hdt)) = L_{E}ω − 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 *L*_{E}ω and
*dE(h) ∧ 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 tangent
to solutions *L*_{E}(X_{h}) = [E , X_{h}] = 0 it maps the space of solutions onto itself). Any such
symmetry yields the following integrals of motion [1],[2],[4],[5]
*R = ω*^{−1}L_{E}ω and *n* is half-dimension of phase space.

g(a) = e^{aLE}

does not preserve action
g_{*}(A) = g_{*}( pdq − hdt) = g_{*}(pdq − hdt) ≠ 0

because
I^{(k)} = Tr(R^{k}) k = 1,2 ... n

where
It is interesting that for any non-Noether symmetry, triple *(h, ω, ω*_{E}) carries
bi-Hamiltonian structure (§4.12 in [6],[7]-[9]).
Indeed *ω*_{E} is closed
(*dω*_{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 *ω*_{E} 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 *L*_{Xh}ω^{∗} = 0. Let us call it global
bi-Hamiltonian structure whenever *ω*^{∗} is exact (there exists 1-form *θ*^{∗} such that
*ω*^{∗} = dθ^{∗}) and *X*_{h} is (globally) Hamiltonian vector field with respect to
*ω*^{∗} (*i*_{Xh}ω^{∗} + dh^{∗} = 0).
As far as *ω* is nondegenerate there exists vector field
*E*^{∗} such that
*i*_{E∗}ω = θ^{∗}.
By construction

L_{E∗}ω = ω^{∗}

Indeed
L_{E∗}ω = di_{E∗}ω + i_{E∗}dω
= dθ^{∗} = ω^{∗}

And
*[X*_{h} , E^{∗}] is Hamiltonian vector field, i. e., *[X*_{h} , E] = X_{h'}. So
*E*^{∗} is not generator of symmetry since it does not commute with *X*_{h} but one can
construct (locally) Hamiltonian counterpart of *E*^{∗} (note that *E*^{∗} itself is
non-Hamiltonian) — *X*_{g} with
*z(t) = z*,
is assumed. Note that (10) defines *g(z)* only locally and, as a result, *X*_{g} is a locally
Hamiltonian vector field, satisfying, by construction, the same commutation relations as
*E*^{∗} (namely *[X*_{h} , X_{g}] = X_{h'}).
Finally one recovers generator of non-Noether symmetry — non-Hamiltonian vector field
*E = E*^{∗} − X_{g} commuting with *X*_{h} and satisfying
*L*_{Xg}ω = 0). So in case of regular Hamiltonian system every
global bi-Hamiltonian structure is naturally associated with (non-Noether) symmetry of space of
solutions.

i_{[E∗,Xh]}ω =
L_{E∗}(i_{Xh}ω) − i_{Xh}L_{E∗}ω
= − d(E^{∗}(h) − h^{∗}) = − dh'

In other words
g(z) = h'dτ

Here integration along solution of Hamilton's equation, with fixed origin and end point in
L_{E}ω = L_{E∗}ω − L_{Xg}ω = L_{E∗}ω = ω^{∗}

(thanks to Liouville's theorem
h = ½ p_{m}^{2} ω = dp_{m} ∧ dq_{m}

this Hamiltonian system can be extended to the bi-Hamiltonian one
h, ω, ω^{∗} = p_{m}dp_{m} ∧ dq_{m}

clearly
q_{m} → (1 + ap_{m})q_{m}

p_{m} → (1 + ap_{m})p_{m}

p

u_{t} + u_{xxx} + uu_{x} = 0

(zero boundary conditions for
i_{Xh}ω+ dh = 0

where
X_{h} = dx u_{t}

(here
ω = dx du ∧ dv

is the symplectic form (here
h = dx ( − u_{x}^{2})

plays the role of Hamiltonian. This dynamical system possesses non-trivial symmetry — one-parameter
group of non-cannonical transformations
E = dx (u_{xx} + ) + X_{F}

here first term represents non-Hamiltonian part of the generator of the symmetry, while the second one
is its Hamiltonian counterpart assosiated with
F = ( + + )dx

(
I^{(1)} = u dx

I^{(2)} = u^{2} dx

I^{(3)} = ( − u_{x}^{2}) dx

I^{(4)} = (u^{4} − uu_{x}^{2}
+ u_{xx}^{2}) dx

⋯

and ensures integrability of KdV equation. Second Hamiltonian realization of KdV equation discovered
by F. Magri [7]
I

I

I

⋯

i_{Xh∗}ω^{∗} + dh^{∗} = 0

(where - F. González-Gascón, Geometric foundations of a new conservation law discovered by Hojman, 1994 J. Phys. A: Math. Gen. 27 L59-60
- M. Lutzky , New derivation of a conserved quantity for Lagrangian systems, 1998 J. Phys. A: Math. Gen. 15 L721-722
- M. Crampin, W. Sarlet, G. Thompson, Bi-differential calculi and bi-Hamiltonian systems, 2000 J. Phys. A: Math. Gen. 33 No. 22 L177-180
- P. Guha, A Note on Bidifferential Calculi and Bihamiltonian systems, 2000 IHÉS preprint M/64
- G. Chavchanidze, Non-Noether symmetries in singular dynamical systems, 2001 Georgian Math. J. 8 (2001) 027-032
- N.M.J. Woodhouse, Geometric Quantization. Claredon, Oxford, 1992
- F. Magri, A simple model of the integrable Hamiltonian equation, 1978 J. Math. Phys. 19 no.5, 1156-1162, 1978
- A. Das, Integrable models, World Scientific Lecture Notes in Physics, vol. 30, 1989
- R. Brouzet, Sur quelques propriétés géométriques des variétés bihamiltoniennes, 1989 C. R. Acad. Sci. Paris 308, série I, 287-92