Georgian Mathematical Journal Volume 10 (2003), Number 1, 57–61

BI-HAMILTONIAN STRUCTURE AS A SHADOW OF NON-NOETHER SYMMETRY

G. CHAVCHANIDZE

Abstract. The correspondence between non-Noether symmetries and bi-Hamiltonian structures is discussed. We show that in regular Hamiltonian systems the presence of the global bi-Hamiltonian structure is caused by the symmetry of the space of solution. As an example, the well-known bi-Hamiltonian realization of the Korteweg–de Vries equation is discussed.

2000 Mathematics Subject Classification: 70H33, 70H06, 53Z05.

Key words and phrases: Bi-Hamiltonian system, non-Noether symmetry, non-Cartan symmetry, Korteweg–de Vries equation.

The Noether theorem, Lutzky’s theorem, bi-Hamiltonian formalism and bidifferential calculi are often used in generating conservation laws and all these approaches are unified by the same idea – to construct conserved quantities using some invariant geometric object (a generator of the symmetry – a Hamiltonian vector field in the Noether theorem, non-Hamiltonian one in Lutzky’s approach, a closed 2-form in bi-Hamiltonian formalism and an auxiliary differential in the case of bidifferential calculi). There is a close relationship between these three approaches. Some aspects of this relationship were established in [3–4] and [6]. In the present paper it is discussed how the bi-Hamiltonian structure can be interpreted as a manifestation of the symmetry of space of solutions. A good candidate for this role is non-Noether symmetry. Such a symmetry is a group of transformations 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 a regular Hamiltonian system the phase space is equipped with symplectic form ω (closed dω = 0 and nondegenerate i_{X}ω = 0 → X = 0 2-form) and the time evolution is governed by Hamilton’s equation

i_{X_{h}}ω + dh = 0,  (1)

where X_{h} is the vector field tangent to solutions X_{h} = Σ_{i}ṗ_{i}∂_{p_{i}} + q̇_{i}∂_{q_{i}} and i_{X_{h}}ω denotes the contraction of X_{h} and ω. The vector field is said to be (locally) Hamiltonian if it preserves ω. According to Liouville’s theorem, X_{h} defined by (1) automatically preserves ω (indeed, L_{X_{h}}ω = di_{X_{h}}ω + i_{X_{h}}dω = −ddh = 0).

One can show that the group of transformations of a phase space generated by any non-Hamiltonian vector field E

g(a) = e^{aL_{E}}

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

does not preserve action

g_{∗}(A) = g_{∗} (∫ pdq − hdt) = ∫ g_{∗}(pdq − hdt) ≠ 0

because d(L_{E}(pdq − hdt)) = L_{E}ω − dE(h) ∧ dt ≠ 0 (the first term in the right-hand side 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 non-Noether symmetry (since E preserves the 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]:

l_{k} = T r(R^{k}), k = 1, 2, …, n,

where R = ω^{−1}L_{E}ω and n is half-dimension of the phase space.

It is interesting that for any non-Noether symmetry, the triple (h, ω, ω_{E}) carries the bi-Hamiltonian structure (§4.12 in [7], [8–10]). Indeed, ω_{E} is a closed (dω_{E} = dL_{E}ω = L_{E}dω = 0) and an invariant (L_{X_{h}}ω_{E} = L_{X_{h}}L_{E}ω = L_{E}L_{X_{h}}ω = 0) 2-form (but generic ω_{E} is degenerate). So every non-Noether symmetry quite naturally endows a dynamical system with the bi-Hamiltonian structure.

Now let us discuss how non-Noether symmetry can be recovered from a bi-Hamiltonian system. The generic bi-Hamiltonian structure on a phase space consists of a Hamiltonian system h, ω and an auxiliary closed 2- form ω^{•} satisfying L_{X_{h}}ω^{•} = 0. Let us call it the global bi-Hamiltonian structure whenever ω^{•} is exact (there exists a 1-form θ^{•} such that ω^{•} = dθ^{•}) and X_{h} is (globally) a Hamiltonian vector field with respect to ω^{•} (i_{X_{h}}ω^{•} + dh^{•} = 0). In the local coordinates θ^{•} = θ_{i}^{•}dz^{i}. As far as ω is nondegenerate, there exists a vector field E^{•} = E^{•i}∂_{z^{i}} such that i_{E^{•}}ω = θ^{•} (in the local coordinates E^{•i} = (ω^{−1})^{ij}θ_{j}^{•}). By construction,

L_{E^{•}}ω = ω^{•}.

Indeed, L_{E^{•}}ω = di_{E^{•}}ω + i_{E^{•}}dω = dθ^{•} = ω^{•} and

i_{[E^{•},X_{h}]}ω = L_{E^{•}}(i_{X_{h}}ω) − i_{X_{h}}L_{E^{•}}ω = −d(E^{•}(h) − h^{•}) = −dh′.

In other words, [X_{h}, E^{•}] is a Hamiltonian vector field, i.e., [X_{h}, E] = X_{h′}. So E^{•} is not a generator of symmetry since it does not commute with X_{h} but one can construct (locally) the Hamiltonian counterpart of E^{•} (note that E^{•} itself is non-Hamiltonian) – X_{g} with

g(z) = ∫_{0}^{t} h′dt.  (2)

Here the integration along the solution of Hamilton’s equation with fixed origin and end point in z(t) = z is assumed. Note that (2) 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 the generator of non-Noether symmetry, i.e., the non-Hamiltonian vector field E = E^{•} − −X_{g} commuting with X_{h} and satisfying

L_{E}ω = L_{E^{•}}ω − L_{X_{g}}ω = L_{E^{•}}ω = ω^{•}

(thanks to Liouville’s theorem L_{X_{g}}ω = 0). So in the case of a regular Hamiltonian system every global bi-Hamiltonian structure is naturally associated with the (non-Noether) symmetry of space of solutions.

Example 1. As a toy example one can consider a free particle

h = \frac{1}{2} ∑_{i} p_{i}^{2}, ω = ∑_{i} dp_{i} ∧ dq_{i}.

This Hamiltonian system can be extended to the bi-Hamiltonian one

h, ω, ω^{•} = ∑_{i} p_{i}dp_{i} ∧ dq_{i}.

Clearly, dω^{•} = 0 and X_{h} = ∑_{i} p_{i}∂_{q_{i}} preserves ω^{•}. The conserved quantities p_{i} are associated with this simple bi-Hamiltonian structure. This system can be obtained from the following (non-Noether) symmetry (infinitesimal form):

q_{i} → (1 + ap_{i})q_{i},
p_{i} → (1 + ap_{i})p_{i}

generated by E = ∑_{i} p_{i}q_{i}∂_{q_{i}} + ∑_{i} p_{i}^{2}∂_{p_{i}}

Example 2. The earliest and probably the most well-known bi-Hamiltonian structure is the one discovered by F. Magri and associated with the Korteweg–de Vries integrable hierarchy. The KdV equation

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

(zero boundary conditions for u and its derivatives are assumed) appears to be Hamilton’s equation

i_{X_{h}}ω + dh = 0,

where X_{h} = ∫_{−∞}^{+∞} dx u_{t}∂_{u} (here ∂_{u} denotes a variational derivative with respect to the field u(x)) is the vector field tangent to the solutions,

ω = ∫_{−∞}^{+∞}dx du ∧ dv

is a symplectic form (here v = ∂_{x}^{−1}u) and the function

h = ∫_{−∞}^{+∞}dx (\frac{u^{3}}{3} − u_{x}^{2})

plays the role of a Hamiltonian. This dynamical system possesses a non-trivial symmetry – a one-parameter group of non-cannonical transformations g(a) = e^{L_{E}} generated by the non-Hamiltonian vector field

E = ∫_{−∞}^{+∞}dx (u_{xx} + \frac{1}{2} u^{2}) ∂_{u} + X_{F};

here the first term represents the non-Hamiltonian part of the generator of symmetry, while the second one is its Hamiltonian counterpart associated with

F = ∫_{−∞}^{+∞}(\frac{1}{12} u^{2}v + \frac{1}{4} ∂_{x}^{−1} (\frac{u^{3}}{3} − u_{x}^{2}) + \frac{3}{4} v \frac{l_{3}}{l_{2}} ) dx

(l_{2,3} are defined in (3)). The physical origin of this symmetry is unclear, however the symmetry seems to be very important since it leads to the well-known infinite sequence of conservation laws in involution:

l_{1} = ∫_{−∞}^{+∞}udx,

l_{2} = ∫_{−∞}^{+∞}u^{2}dx,

l_{3} = ∫_{−∞}^{+∞}(\frac{u^{3}}{3} − u_{x}^{2}) dx,

l_{4} = ∫_{−∞}^{+∞}(\frac{5}{36} u^{4} − \frac{5}{3} uu_{x}^{2} + u_{xx}^{2}) dx,  (3)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and ensures the integrability of the KdV equation. The second Hamiltonian realization of the KdV equation discovered by F. Magri [8]

i_{X_{h^{•}}}ω^{•} + dh^{•} = 0

(where ω^{•} = L_{E}ω and h^{•} = L_{E}h) is a result of the invariance of the KdV equation under the aforementioned transformations g(a).

Acknowledgements

The 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 the World Federation of Scientists.

References

1. F. González-Gascón, Geometric foundations of a new conservation law discovered by Hojman. J. Phys. A 27(1994), No. 2, L59–L60.

2. M. Lutzky, New derivation of a conserved quantity for Lagrangian systems. J. Phys. A 31(1998), No. 43, L721–L722.

3. M. Crampin, W. Sarlet, and G. Thompson, Bi-differential calculi and bi-Hamilto-nian systems. J. Phys. A 33(2000), No. 20, L177–L180.

4. P. Guha, A note on bidifferential calculi and bihamiltonian systems. IHÈS preprint M/64, 2000.

5. G. Chavchanidze, Non-Noether symmetries in singular dynamical systems. Georgian Math. J. 8(2001), No. 1, 27–32.

6. G. Chavchanidze, Remark on non-Noether symmetries and bidifferential calculi. Mathph/0106010, 2001.

7. N. M. J. Woodhouse, Geometric quantization. 2nd ed. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992.

8. F. Magri, A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19(1978), No. 5, 1156–1162.

9. A. Das, Integrable models. World Scientific Lecture Notes in Physics 30, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.

10. R. Brouzet, Sur quelques propriétés géométriques des variétés bihamiltoniennes. C. R. Acad. Sci. Paris Sér. I Math. 308(1989), No. 10, 287–292.

(Received 21.03.2002)

Author’s Address:

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: gch@rmi.acnet.ge