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
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.
Bi-Hamiltonian system; Non-Noether symmetry; Non-Cartan symmetry; Korteweg- De Vries equation.
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 $i_Xω = 0 ⇒ X = 0$ 2-form) and time
evolution is governed by Hamilton's equation
$$
i_{X_h}ω + dh = 0
$$
where $X_h$ is Hamiltonian vector field that defines time evolution
$$
\frac{df}{dt} = X_h(f)
$$
for any function $f$ and $i_{X_h}ω$ 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
$$
L_{X_h}ω = di_{X_h}ω + i_{X_h}dω = − ddh = 0
$$
One can show that group of transformations of phase space generated by any non-Hamiltonian vector
field $E$
$$
g(a) = e^{aL_E}
$$
does not preserve action
$$
g_{*}(A) = g_{*}(∫ pdq − hdt) = ∫ g_{*}(pdq − hdt) ≠ 0
$$
because $d(L_E(pdq − hdt)) = L_Eω − dE(h) ∧ dt ≠ 0$ (first term in r.h.s. does not vanish
since $E$ is non-Hamiltonian and as far as $E$ is time independent $L_Eω$ 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]
$$
I^{(k)} = Tr(R^k) k = 1,2 ... n
$$
where $R = ω^{−1}L_Eω$ and $n$ is half-dimension of phase space.
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
($dω_E = dL_Eω = L_Edω = 0$) and invariant
($L_{X_h}ω_E = L_{X_h}L_Eω = L_EL_{X_h}ω = 0$)
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 $L_{X_h}ω^{∗} = 0$. 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
$ω^{∗}$ ($i_{X_h}ω^{∗} + dh^{∗} = 0$).
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
$$
i_{[E^{∗},X_h]}ω =
L_{E^{∗}}(i_{X_h}ω) − i_{X_h}L_{E^{∗}}ω
= − d(E^{∗}(h) − h^{∗}) = − dh'
$$
In other words $[X_h , E^{∗}]$ is 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
$$
g(z) =\stackrev{\stackrel{t}{∫}}{0} h'dτ
$$
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, $X_g$ 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
$$
L_Eω = L_{E^{∗}}ω − L_{X_g}ω = L_{E^{∗}}ω = ω^{∗}
$$
(thanks to Liouville's theorem $L_{X_g}ω = 0$). So in case of regular Hamiltonian system every
global bi-Hamiltonian structure is naturally associated with (non-Noether) symmetry of space of
solutions.
As a toy example one can consider free particle
$$
h = ½ \stackrev{∑}{m} p_m^2
ω = \stackrev{∑}{m} dp_m ∧ dq_m
$$
this Hamiltonian system can be extended to the bi-Hamiltonian one
$$
h, ω, ω^{∗} = \stackrev{∑}{m} p_mdp_m ∧ dq_m
$$
clearly $dω^{∗} = 0$ and $X_h$ preserves
$ω^{∗}$. Conserved quantities $p_m$ are associated with this simple
bi-Hamiltonian structure.
This system can be obtained from the following (non-Noether) symmetry (infinitesimal form)
$$
q_m → (1 + ap_m)q_m\\
p_m → (1 + ap_m)p_m
$$
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
$$
u_t + u_{xxx} + uu_x = 0
$$
(zero boundary conditions for $u$ and its derivatives are assumed) appears to be Hamilton's equation
$$
i_{X_h}ω+ dh = 0
$$
where
$$
X_h = \stackrev{\stackrel{+ ∞}{∫}}{− ∞} dx u_t\frac{δ}{δu}$$
(here $\frac{δ}{δu}$
denotes variational derivative with respect to the field $u(x)$) is the vector field tangent to the
solutions,
$$
ω = \stackrev{\stackrel{+ ∞}{∫}}{− ∞} dx du ∧ dv
$$
is the symplectic form (here $v$ is defined by $v_x = u$) and the function
$$
h = \stackrev{\stackrel{+ ∞}{∫}}{− ∞} dx (\frac{u^3}{3} − u_x^2)
$$
plays the role of Hamiltonian. This dynamical system possesses non-trivial symmetry — one-parameter
group of non-cannonical transformations $g(a) = e^{L_E}$ generated by the non-Hamiltonian vector
field
$$
E = \stackrev{\stackrel{+ ∞}{∫}}{− ∞} dx (u_{xx} + \frac{u^2}{2})\frac{∂}{∂u} + X_F
$$
here first term represents non-Hamiltonian part of the generator of the symmetry, while the second one
is its Hamiltonian counterpart assosiated with
$$
F = \stackrev{\stackrel{+ ∞}{∫}}{− ∞}(\frac{u^2v}{12} + \frac{G}{4} + \frac{3vI⁽^2⁾}{4I⁽^3⁾})dx
$$
($I^{(2,3)}$ are defined in
(22), while
$G$ is defined by $G_x = \frac{u^3}{3} − u_x^2$ .
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:
$$
I^{(1)} = \stackrev{\stackrel{+ ∞}{∫}}{− ∞} u dx\\
I^{(2)} = \stackrev{\stackrel{+ ∞}{∫}}{− ∞} u^2 dx\\
I^{(3)} = \stackrev{\stackrel{+ ∞}{∫}}{− ∞} (\frac{u^3}{3} − u_x^2) dx\\
I^{(4)} = \stackrev{\stackrel{+ ∞}{∫}}{− ∞} (\frac{5}{36}u^4 − \frac{5}{3}uu_x^2
+ u_{xx}^2) dx\\
⋯
$$
and ensures integrability of KdV equation. Second Hamiltonian realization of KdV equation discovered
by F. Magri
[7]
$$
i_{X_{h^{∗}}}ω^{∗} + dh^{∗} = 0
$$
(where $ω^{∗} = L_Eω$ and $h^{∗} = L_Eh$) is a result of
invariance of KdV under aforementioned transformations $g(a)$.
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.
References
-
F. González-Gascón
Geometric foundations of a new conservation law discovered by Hojman
J. Phys. A: Math. Gen. 27 L59-60
1994
-
M. Lutzky
New derivation of a conserved quantity for Lagrangian systems
J. Phys. A: Math. Gen. 15 L721-722
1998
-
M. Crampin, W. Sarlet, G. Thompson
Bi-differential calculi and bi-Hamiltonian systems
J. Phys. A: Math. Gen. 33 No. 22 L177-180
2000
-
P. Guha
A Note on Bidifferential Calculi and Bihamiltonian systems
IHÉS preprint M/64
2000
-
G. Chavchanidze
Non-Noether symmetries in singular dynamical systems
Georgian Math. J. 8 (2001) 027-032
2001
-
N.M.J. Woodhouse
Geometric Quantization
Claredon, Oxford
1992
-
F. Magri
A simple model of the integrable Hamiltonian equation
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
C. R. Acad. Sci. Paris 308, série I, 287-92
1989