Hamiltonian formalism for nonlinear waves
V.E.Zakharov
1
and E.A.Kuznetsov
2
Landau Institute for Theoretical Physics,
2 Kosygin str., Moscow 117334, Russia
Abstract
Hamiltonian description for nonlinear waves in plasma, hydrodynamics and
magnetohydrodynamics is presented. The main attention is paid to the prob-
lem of canonical variables introducing. The connection with other approaches
of the Hamiltonian structure introducing is presented, in particular, with the
help of the Poisson brackets expressed in terms of natural variables. It is shown
that the degeneracy of the noncanonical Poisson brackets is connected for the
system of hydrodynamic type with the specific symmetry, namely, with the
relabeling transformations of the Lagrangian markers of fluid particles. All
known theorems about the vorticity conservation (the Ertel’s, Cauchy’s and
Kelvin’s theorems, the frosenness of vorticity and conservation of the topo-
logical Hopf invariant) are a sequence of this symmetry. The canonical vari-
ables are introduced into the collisionless plasma kinetics and into the Benney
equation. The problem of Hamiltonian structures is discussed for surface and
internal waves as well as for the Rossby waves. The Hamiltonian structure also
is introduced for the Davey-Stuartson equation, describing the interaction of
quasimonochromatic waves with the induced low-frequency medium motion.
At the end of this survey a general method for investigation of weak nonlin-
ear waves is considered, based on both the classical perturbation theory and
reduction of Hamiltonians.
Contents
1 Introduction 1
2 General Remarks 5
3 Hamiltonian Formalism in Continuous Media 9
4 Canonical Variables in Hydrodynamics 12
1
e-mail: zakharov@itp.ac.ru
2
e-mail: kuznetso@itp.ac.ru
1
5 Noncanonical Poisson Brackets 19
6 Ertel’s Theorem 23
7 Gauge Symmetry - Relabeling Group 27
8 The Hopf Invariant and Degeneracy of the Poisson Brackets 34
9 Inhomogeneous Fluid and Surface Waves 38
10 Hamiltonian Formalism for Plasma and Magnetohydrodynamics 44
11 The Hamiltonian Formalism in Kinetics 53
12 Classical Perturbation Theory and Reduction of Hamiltonians 57
1 Introduction
The equations of hydrodynamics and their generalizations are among the most basic
tools for description of nonlinear waves in macroscopic physics. In studying them, an
important question is whether these equations, in the case of absence of dissipation,
have a Hamiltonian structure. This problem is primarily important in connection
with the problem of quantization. However, also in the classical case, establishing
that a given system is Hamiltonian allows one to hope (although this is not always
a simple matter) to introduce explicitly canonical variables, after which all the vari-
ants of perturbation theory are considerably simplified and standardized (cf., for
example,[1, 2, 3, 4]). In particular, this approach gives an opportunity to consider
all nonlinear processes from the general point of view without fixing their proper
peculiarities connected with a given concrete medium. The Hamiltonian approach
gives also certain advantages when approximations must be performed. Classical
example of this is a description of well-separated space or/and time scales, in partic-
ular, of high-frequency and low-frequency waves (for review, see the remarkable book
of Whitham [5]). For the Hamiltonian continuous systems, the stability problem for
stationary solutions as cnoidal waves, solitons, vortices, etc. is formulated more or
less at the same manner and it can be solved by studying the quadratic Hamiltonian
for small perturbations or by taking the Hamiltonian with combination with another
integrals (numbers of particles, momentum, etc.) as the Lyapunov functional if one
treats the nonlinear stability (cf., for instance, [6, 7, 8]).
Besides hydrodynamics, the equations of the hydrodynamic type are widely used
for description of various processes in plasma physics as well as in magnetohydrody-
namics. They combine the equation of medium motion and the Maxwell equations
for electromagnetic field. These models play also an essential role for solid state
physics and nonlinear optics.
2
The problem of the Hamiltonian structure for hydrodynamic equations has a
long history. There are two traditional approaches to answer it. First, one can try,
for some system or other, to directly guess a complete set of canonical variables.
Then the problem of calculating Poisson brackets between any physical quantities is
automatically solved, and one succeeds also in writing down a variational principle.
Usually the Hamiltonian variables are expressed in terms of the natural variables
(velocity, pressure) in a by no means trivial fashion.
An alternative path is to find directly expressions for the Poisson brackets in
”natural” variables. This does not enable one to introduce a variational principle,
but for many physical problems, including the problem of quantization, it appears
to be sufficient. The equations of hydrodynamic type have the same degree of non-
linearity (quadratic in the velocities) as the energy integral. It then follows that the
expression for the Poisson bracket must b e linear with respect to the variables (the
velocity, the density, etc.) that enter these equations. It is easy to show that all such
brackets are brackets of the Berezin-Kirillov-Kostant type on certain Lie groups.
This quite important fact was understood relatively recently, apparently first by
V.I.Arnold [9], [10] (see also, [11]) although Poisson brackets between velocity com-
ponents were already calculated in connection with the problem of quantization in
a paper of L.D.Landau [12]. Also devoted to these notions were some papers of
I.E.Dzyaloshinskii and G.E.Volovik [13], and S.P.Novokov. In For the equations of
magnetohydrodynamics the noncanonical Poisson brackets were calculated first by
Greene and Morrison [15] and for the Vlasov-Maxwell equations for a plasma they
were obtained by Morrison [16].
As for canonical variables, for the ideal hydrodynamics of a homogeneous in-
compressible fluid they were already found in the previous century by Clebsch (cf.,
for example, [17]). The topological meaning of these variables was clarified in the
paper of Kuznetsov and Mikhailov [18]. In 1932, H.Bateman [19], and later indepen-
dently B.I.Davydov [20], extended the result of Clebsch to a compressible barotropic
liquid. In 1952 for nonbarotropic flows of ideal liquid variables were found by
I.M.Khalatnikov [21]. Later this result was rediscovered in a set of other articles
(see, for example, [22]).
From these results one can obtain the canonical variables for an incompressible
fluid of variable density, including fluids with a free boundary, as was done by Kon-
torovich, Kravchik and Time [23]. However, the extremely important problem from
the point of view of surface waves, of the Hamiltonian description of a fluid with free
surface, was solved earlier by one of the authors of the present book (V.E.Zakharov).
The canonical variables were introduced without proof in 1966 [24], and the complete
proof was published in 1968 in [25]. In these papers only potential flows of the fluid
were considered. A partial transfer of the results to the case of nonpotential flow was
accomplished by Voronovich [26] and Goncharov [27], who also solved the problem of
the Hamiltonian description of internal waves in the ocean. A presentation of these
results can be found in the monograph of Yu.Z.Miropolskii [28] as well as a recent
3
book by Goncharov and Pavlov [29], both written entirely from the point of view of
Hamiltonian formalism.
Of especial interest is the Hamiltonian formalism for the Benney equations, de-
scribing nonpotential long waves on shallow water. The system of Benney equations
is completely integrable [30],[31], and the Hamiltonian formalism for them was for-
mulated (in the language of Poisson brackets between moments of the longitudinal
velocity) in a paper of Manin and Kupershmidt [32].
Canonical variables enabling one to calculate Poisson brackets between any quan-
tities, were found for the Benney equations in [30]. This question turned out un-
expectedly to be related to the question of the Hamiltonian description of plasma,
which had attracted attention earlier. A Hamiltonian description of magnetohydro-
dynamics was achieved by the authors of the present book in 1970 [33]. Canonical
variables in a two-fluid hydrodynamic model were introduced in [34], and were used
later in various papers describing nonlinear processes in plasma (cf., for example
[35]). This did not solve the question of introducing canonical variables in the col-
lisionless kinetics of a plasma, although, after paper [31], it became clear that such
variables must exist. In the present survey we introduce such variables, using the
equivalence of the Vlasov equations to an infinite system of hydrodynamic equations.
This equivalence, which was noted by one of the authors (E.A.Kuznetsov) is estab-
lished by a Radon transform, and was essentially used in [30], where it was shown
that the Benney equations are equivalent to one variant of the Vlasov equations.
In the present survey we also give a systematic description of the result recalled
above. In addition we discuss the interesting question of Hamiltonian structure
for a two-dimensional incompressible hydrodynamics, and for the Charny-Obukhov-
Hasegawa-Mima equation describing Rossby waves. In these systems, one has not
succeeded so far in introducing suitable canonical variables, although the existence
of a Hamiltonian structure is a proven fact. Recently Piterbarg [39], generalizing the
results of the papers [38], turned out to prove that the noncanonical Poisson brackets
for such systems for arbitrary flows with closed stream lines can be reduced to the
Gardner-Zakharov-Faddeev brackets appearing at first for the integrable equations
[40] and to suggest a constructive scheme of canonical basis finding. Finally we
consider some general properties of Hamiltonian systems with a continuous number
of degrees of freedom.
The basis of this survey was the paper of the authors [1], published in 1986 in
English in a sufficiently rare journal and therefore b ecame unknown for a wide audi-
ence both Russian and in abroad. Against [1] the text of this survey is revised and
broaden significantly. First of all, both revision and addition were subject for the
problems of the noncanonical Poisson brackets and their degeneracy. For system of
the hydrodynamic type this degeneracy is connected with a hidden symmetry of the
equations, being in fact the gauge one. This symmetry has the Lagrangian origin; it
relates to the group of transformations relabeling the Lagrangian variables marked
each fluid particle. Evidently no changes in markers must influence on the system
4
dynamics. First this fact was understood sufficiently completely by R.Salmon [41] in
1982 although Eckart in 1938 and then in 1960 [44] and later Newcomb understood
the role of this symmetry. In particular, all known theorems about the vorticity con-
servation, i.e., the Ertel theorem about the existence of the Lagrangian (material)
invariants [42] (see also [53], pp 31-32), the Cauchy theorem of frozenness of vortic-
ity into a fluid [17] and the Kelvin theorem about the conservation of the velocity
circulation (see, for instance, [53]), as well as the conservation of the topological
Hopf invariant [55, 56] characterized the flow knottiness, are a consequence of this
symmetry. This symmetry is connected also with introducing the canonical Clebsch
variables and their gauge symmetry.
One should note that introducing canonical variables of the Clebsch kind into the
systems of the hydrodynamic type allows one to find expressions for all noncanon-
ical Poisson brackets known up to now, starting from the canonical one. This fact
first was demonstrated by the authors of the given survey [48, 1] for the equations
of ideal hydrodynamics and for the kinetic Vlasov-Maxwell equations for plasma.
However, passing to the opposite direction, i.e., finding canonical brackets from the
noncanonical one, in the general situation entails some difficulties, connected with
the noncanonical brackets degeneracy.
In this survey we consider in more details all these questions for the hydrodynamic
equations. Here we don’t discuss the role of this symmetry for another models, except
the MHD equations (about this subject see the recent paper [49]). Now this question
for systems of the hydrodynamic type is far well studied and requires additional
investigations. In our opinion, it has a principle meaning for understanding many
nonlinear phenomena which take place in fluids and plasma. First of all these are the
processes of reconnection of vortex lines for fluids or magnetic field lines in plasma,
namely, the processes which change the system topology.
2 General Remarks
We recall some elementary facts. The most naive definition of a finite-dimensional
Hamiltonian system reads as follows. One considers a system of an even number
of differential equations for the time-dependent functions q
k
(t), p
k
(t)(k = 1, ..., N),
having the form
·
q
k
=
∂H
∂p
k
,
.
p
k
= −
∂H
∂q
k
. (2.1)
Here H(p
1
, ..., p
N
, q
1
, ..., q
N
), which is a given function of the variables, is the Hamil-
tonian.
The definition presented here is far from being always satisfactory, since it as-
sumes implicitly that the domain of variation of the p
i
and q
i
(the phase space) is a
domain in the real vector space R
2N
. However, already for the case of the mathemat-
ical pendulum, where the generalized coordinate is an angle, one must identify its
5