IC/66/95

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

APPLICATIONOF HAMILTON-JACOBI THEORY

TO VLASOV'S EQUATION

D. PFIRSCH

1966

PIAZZA OBERDAN

TRIESTE

IC/66/95

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

APPLICATION OF HAMILTON- JACOBI THEORY

TO VLASOV'S EQUATION*

D- PFIRSCH**

TRIESTE

July 1966

* To be submitted to Nuclear Fusion

** Permanent address: Max-Planck-Institut fur Physik und AstrophysUi, Munich, Fed. Rep. Germany

ABSTRACT

Solutions of Vlasov's equation are given in terms of the

Hamilton-Jacob! function S for the characteristics. Such

solutions are useful in order to derive equations for

macroscopic quantities which are of interest for instance

in turbulence theories.

- 1 -

APPLICATION OP HAMILTON-JACOBI THEORY TO VLASOV'S EQUATION

One Important class of problems in plasma physics deals

with the description of plasma turbulence. Because there exist

many microinstabilities which can be the cause for turbulence,

a microscopic treatment is often necessary. The fundamental

equation is in this case Vlasov's equation, which holds if

collisions can be neglected. One can then distinguish 4 problems:

1.) One has first to derive for some macroscopic quantities,

like the fluctuating electric or magnetic fields, nonlinear

equations of the following kind: if C represents the macroscopic

quantities and A,B,D matrices of second, third and fourth rank,

then an equation

A- C t 3:C C f J>; CCC

has to be derived from Vlasov's equation and Maxwell's

equations by some kind of nonsecular perturbation theory

(see for instance (1))

2.) One has to introduce statistical concepts leading to hierarchy

equations for the quantities

where the brackets indicate ensemble averages. One must add

to this a closure prescription, for instance neglect of 4th

order correlations.

J5.) One has to apply asymptotic methods to these equations in

order to obtain kinetic wave equations.

- 2 -

4.) One has to solve these kinetic wave equations.

In this paper I will be concerned with the first of these

points. It is well known that the classical Kami1ton-Jabobi-

theory allows in a straight forwardway a nonsecular perturbation

theory which is similar to perturbation theories in quantum

theory. An example will illustrate the differences between usual

perturbation theory and perturbation theory of the Hamilton-

Jacobi-equation:

The exact solution of the equation

x = (ft t? xwith X(o) * 0 , K(0) * *>C

is

Perturbation theory with the equation of motion yields to first

order in f

X & cc sink t t €°t (t **M- *''»* 0 ,

which expression contains a so called secular term, which restricts

the validity of this solution to times / « ;F . The Hamilton-

Jacob!-theory yields in first order

a; c<L(l-t)St'nh-r~ •This solution is valid for £« / and t «fx ,i.e.; for much

longer times than the first approximate solution. But as can be

seen from the exact first oder Hamilton Jacobi-expression, one

must develop in general a very difficult elimination procedure.

It is an essential point, that in applying Hamilton-Jacobi's

Theory to Vlasov's equation in order to obtain macroscopic

- 3 -

quantities one is not concerned with any elimination problem.

There Is a second essential point shown in this paper namely

that the Hamilton-Jacobi method allows a rather easy and

straightforward nonsecular perturbation theory in such a way

that in order to obtain the equations for the macroscopic

quantities one need not solve equations for the distribution

function as in quasilinear or similar theories (see for instance

(1) and the literature cited there). To gain this advantage,

however, one must sacrifice knowledge about microscopic quantities,

i.e. the distribution function in x,v - space.

1. Solution of Vlasov's Equation in terms of the Hamilton-

Jacobi Function

The Hamilton-Jacobi-equation of a Hamiltonian System with the

Hamiltonian H (£, xt t) is

In the following H is assumed to be the Hamiltonian in Vlasov's

equation

where the brackets indicate Polsson-brackets. A solution of (l)

can be written in the form

+ < . (3)

U, «£are constants of the integration; o& can be a vector with

as many components as X has* o^o can and will be dropped. £ has

the meaning of a generating function for a canonical trans-

formation. tL can be thought of as the new momenta and

2f (4)

as being the new coordinates, whereas the old momenta are

given by fo f&X . The Hamilton-Jacobi-equation expresses the

fact that the new Hamiltonian is identically equal to zero,

thus $£ and p, are constants of the motion.

Since any solution of Vlasov's equation can be written as a

function of the constants of motion, a solution of (2) is

f - / tyS).We are, however, not interested in such a function depending

on the variables <£ and P , because we wish later to calculate

macroscopic quantities which are functions of K and /

Therefore we introduce instead of |L the variable £ , which

can be done by relation (4). This transformation?however, is

not a canonical one. Thus in order to obtain the particle density

i n^ȣ -space we have to multiply (5) still by the Jacobian of

this transformation. This new function is therefore

(6)

With this function one calculates for instance the particle

density

or the current density (in case of a vanishing vector potential

and with e=m=l)

Similar we can find any other macroscopic quantity without per

forming any eliminations.

Of course if we wanted to know the particle density in x,£ -

space we would have to eliminate vi by use of the relation

A * at * w h i c h w o u l d D e l n general very difficult.

From (6) we may derive an interesting result as to the meaning

of the Haml1ton-Jacobi-Function in the framework of Vlasov's

equation: One take as the function f

(9)

so that

Thus, this Jacobian is just the probability to find a particle

at point 2L a t time t if its orbits are characterised by the

constant of motion *c 9 ^ . This corresponds closely to the

meaning of tP (ift'clg)^!}^ ̂ 1 In quantum theory, where in this case

cC0 are quantum numbers. A simple example is given by the

one dimensional motion of a particle in a time dependent potential,

The Hamilton-Jacobi equation for this problem is

* FmE * Fm

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We can solve this equation in the form

Vflt - - -t «*and

From this we calculate the Jacobian

(12)

2* fix \rM\where T/'(X,oC\ is the velocity of the particle characterised by

the constant of motion oC . It is well known that \/lit(*>*Q\ is Just

the prabability about which we have spoken.

To conclude this section I will give an example for the function

f. The example is a particle in a time varying but spatially

constant field of force g (t) say with g (t<to) = o, t0 finite.

Putting the mass m = 1 we have the Hamilton-Jacobi-eauation

ii i( X = 0 .(14)

A solution of this equation can be found in the form

(15)

with

4(0and

0

\ dr <j (r)"OO

(16)

-7-

& I 4 (18)

(20)

From this we obtain according to (6)

or if we are interested in a spatially homogeneous solution:

(22)

It is remarkable that this function is exactly time-independent,

We will have to do later with similar features.

Clearly (22) yields a time-independent density, whereas the

current density is given by

(W \fjffj

In the usual V- representation we would have instead of (22)

(24)

eliminating 06 we obtain a time-dependent function. This

fact will also be of importance below.

-8-

2. Perturbation theory

Clearly it is not possible in general to solve the Hamilton-

Jacobi-equation exactly. But as mentioned in the introduction

and as was illustrated by an example, the Hamilton-Jacobi-theory

allows a rather easy and straight-forward nonsecular perturbation

theory, which will be presented here.

Assume the Hamiltonian H can be decomposed into two parts

where Ho describes a problem which can be solved exactly, where-

as H, is a small perturbation. This means we assume to have

solved the equation

in the form

X = X (*,*,

by which the gi 5 are defined.

If we make the ansatz

(28)

where S,, S2j'.are higher and higher order contributions to S in

the sense of the smallness of H,, then we obtain for these

9-

quantities equations of the form

9s, u f K

- ' ' *~ . (30)

and in general

Of Jtf OfZzH / ^f ZgJL - expression containing (So, ..i

S ) -e n i/ ^f. ZgJL - expression containing (So, ..i,

there M/fa ? *%fin '" indicate 0'%(fl, Xj/)fJfl\ *

All these equations can be solved by integrating along the

characteristics, which are given by

M

These equations are just the equations for the unperturbed

paths of the particles characterised by the constants o£ .

These paths can be described by a time parameter Z~ as

From this it follows for instance

•-0Q

/f / ^ ' (2*)

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The lower limit ~ot> implies in general an adiabatic turning on

of the perturbation. For all that however, it can happen that

the integral diverges at the lower limit, because it could have

a nonvanishing meanvalue for /-*-<*> .In this case one has to

proceed in the following way:

We first decompose S, or in general S into two parts

- rOJ ,(35)

(2)

/7+J ft

=• - *n (36)

(2)where we define S v ' by the equation

is defined in eq. (31), for instance e--= Hj. Thus

n )

where the integral runs along the particle path I1 = 1 (x1)

Inserting this into equation (31) we obtain the equation

( 3 8 )

Now it is possible in general to solve this equation in the

form

(39) •

where we will not encounter secularities.

J>. Application to the one-dimensional one-component plasma

The simplest case that one can treat is a one-dimensional one-

component plasma which is spatially homogeneous in the unper-

turbed state. Putting m = e=l we have as zero order equation

it

with the solution

C = - I

From this i t follows

Thus

is the exact form of the distribution f describing a plasma

which is homogeneous in the unperturbed state. I want to

emphasize again that f (<*) is exactly independent of time.

All the time-dependence and also all the x-dependence in a

perturbed state, is given by the Jacobian. This is similar to

the behaviour of the distribution function for the example in

section 1. Eliminating o£ by virtue of V^

-12-

could however, obtain a time and space dependent function

The quantities we are interested in are the particle

density and the current density?00

— oo

i (45)

Putting the velocity of light equal to one, we can write the• •

Maxwellian equation JO e - r^V as

£- *4 \ Str [&;*«• <«>Thus in order to obtain for E an equation of third degree we

have to calculate [ %/}to third order in E, which can be done

by the perturbation theory described in section 2. In the present

context this perturbation theory is valid if the following two

conditions are fulfilled

(A) The velocity of a thermal particle must not be changed

appreciably during one period of the electric field,

i.e.

in 4/ ™

or equivalents, J£ *, I mih} tt 7JT

(B) The time needed by a particle travelling with the phase

velocity to move through a wave packet must hr small

compared to the oscillation period of a partiexe trapped

by the potential of the wave packet. If //£-# is

-15-

the width of the wave packet and jf̂ s=

is the amplitude of the electric field then we must

have

4^ 0 ^ \ *r9&

or, equivalently,

Fir / ^ / /

The case in which the following third condition is also satis

fied, is often of interest:

(C) The inverse of the growthrate >*" is small compared

to the oscillation period of a trapped particle,i.e.

1

or, equivalently,

FT & 4-m ̂

Now performing the perturbation theory, we first write equation

(46) to third order:

Before determining all these expressions I will say something

about the second order and third order term. In our perturbation

-14-

theory in section 2 equation (36) reads for n=2

IT (48)

Inserting this relation into the second order term in (47) we

are left with

instead of

is easily calculated from (38):

(*9)

Similarly we have for n = 3 from (36)

( 5 0 )

Thus we have again only to consider oCfSj /*%* which is similar

to (49) given by

06 W* W\T» Tx)\±

-15-

In this expression 0->i IJX is the fall expression, being the

sum of (48) and (49).

We can now write down very easily the interesting quantities

of first, second and third order

CK>

(52)

?(53)

Wi fa

6

3.1. Fourier transformation with respect to space and time

With regard to weak turbulence and some kinds of strong turbulence

it is useful to Fourier transform E (x,t) with respect to both

space and time:

Here we have introduced a periodicity length L by which the

TV

wavevectorsK are given by K = n -j- , n = O , ± l , ....* We can

then write e.g. (47) in terms of the first, second and third

-16

order contributions j ^ J2, j , to the current density. With

(52) to (54) we find easily

fa-Q-U } '(56)

oo

/ (58)

It is of interest to observe in which way secular terms are

avoided in these expressions. Secular terms would show up here

as terms leading to singularities at CJ = o. Because of the

left hand side of equation (47) which after Fourier transformation

is - iCJ E K (03) t such singularities would be given by terms in

the •ji's not vanishing for 03= o. Since for K = 0 the left hand

side is equal to zero also the right hand side vanishes identi-

cally In 0); then looking only for K=0 we find j., which is just

the usual first order current density, is exactly zero for £J=o.

The same is obviously true for the quantities j^ and j , because of

the factors CO in front.

I want to point out the fact that (56) to (58) are the exact

Fourier-transforms. There is no slow time-dependence left in

-17-

&s in the qualilinear theory and therefore is also

no need to solve an additional equation for some quantity like

f (oC) ; f (cC) is here given exactly by the initial unperturbed

state of the plasma for which aC n a s the meaning of the velocity

of the particles. For later times oC is in general a very

complicated function of x,v,t, which however, we need not know.

Another point of interest is the fact that (T, and CTi+ )

can be expressed by linear combinations of o~ • This means

again a simplification over current theories.

Using (56) to (58) one can do now the weak turbulence theory as

described for instance in Kadomtsev's book (1), chapter II, or

the strong turbulence theory in the sense of Kraichnan (2), or

in a slightly modified way as also described in Kadomtsev's

book chapter III.

4. Conclusions

Through application of the Hamilton Jacobi-theory to the

Vlasov equation it was possible to find solutions which are

appropriate for calculating macroscopic quantities. For this

purpose one need not develop an elimination procedure of the

sort necessary for finding the motion of a particle; any

such procedure would generally be very difficult. In regard

to turbulence theory, for instance, it is an interesting

feature that this kind of formalism leads to a simple, straight

forward nonsecular perturbation theory, by which one easily finds

results not yet obtained with present theories. However, one

gains this advantage only by sacrificing knowledge about the

distribution function in x,v- space. On the other hand,

+) This was pointed out to me by Dr. Janussis.

-18-

•,-. „ ;

at least for the case of a plasma which is homogeneous in the

unperturbed state, there exists a similar method for obtaining

an expression for the homogeneous part of the distribution function,

This method consists in solving the Vlasov equation by means of

the following equation:

where S is a function

and tfSfJLa.YQ constants of the motion, and x is given by

Preliminary results of this theory are similar to result*

obtained with the quasi-linear theory.

ACKNOWLEDGEMENTS

The author thanks Prof. Abdus Salam and the IAEA for their

hospitality at the ICTP, Trieste.

This work was sponsored by the Max-Planck-Institut fUr Physik

und Astrophysik, Munich, Germany.

REFERENCES

(1) B.B. Kadomtsev, Plasma Turbulence, Academic Press. London.New York (1965)

(2) R.H. Kraichnan, J. Fluid Mech. 5, 497 (1959)

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