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 structures
is disscussed. We show that in regular Hamiltonian systems presence of the global bi-Hamiltonian
structure is caused by symmetry of the space of solution. As an example well known bi-Hamiltonian
realisation of Korteweg-De Vries equation is disscussed.
Keywords: Bi-Hamiltonian system; Non-Noether symmetry; Non-Cartan symmetry; Korteweg- De Vries equation.
MSC 2000: 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 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
(1)
where Xh is Hamiltonian vector field that defines time evolution
dfdt = Xh(f)
(2)
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
(3)
One can show that group of transformations of phase space generated by any non-Hamiltonian vector
field E
g(a) = eaLE
(4)
does not preserve action
g*(A) = g*(∫ pdq − hdt) = ∫ g*(pdq − hdt) ≠ 0
(5)
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(Rk) k = 1,2 ... n
(6)
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∗
(7)
Indeed
LE∗ω = diE∗ω + iE∗dω
= dθ∗ = ω∗(8)
And
i[E∗,Xh]ω =
LE∗(iXhω) − iXhLE∗ω
= − d(E∗(h) − h∗) = − dh'
(9)
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) =t∫0 h'dτ
(10)
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∗
(11)
(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 1.
As a toy example one can consider free particle
h = ½ ∑m pm2
ω = ∑m dpm ∧ dqm
(12)
this Hamiltonian system can be extended to the bi-Hamiltonian one
h, ω, ω∗ = ∑m pmdpm ∧ dqm
(13)
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
(14)
Example 2.
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
(15)
(zero boundary conditions for u and its derivatives are assumed) appears to be Hamilton's equation
iXhω+ dh = 0
(16)
where
Xh = + ∞∫− ∞ dx utδδu(17)
(here δδu
denotes variational derivative with respect to the field u(x)) is the vector field tangent to the
solutions,
ω = + ∞∫− ∞ dx du ∧ dv
(18)
is the symplectic form (here v is defined by vx = u) and the function
h = + ∞∫− ∞ dx (u33 − ux2)
(19)
plays the role of Hamiltonian. This dynamical system possesses non-trivial symmetry — one-parameter
group of non-cannonical transformations g(a) = eLE generated by the non-Hamiltonian vector
field
E = + ∞∫− ∞ dx (uxx + u22)∂∂u + XF
(20)
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
(21)
(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
⋯
(22)
and ensures integrability of KdV equation. Second Hamiltonian realization of KdV equation discovered
by F. Magri [7]
iXh∗ω∗ + dh∗ = 0
(23)
(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 World
Federation of Scientists.
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