Destruction of adiabatic invariance at resonances in slow-fast

Hamiltonian systems

Аnatoly Neishtadt

Space Research Institute, Moscow

aneishta@iki.rssi.ru

Аdiabatic invariant is an approximate first integral of the system with slow and fast variables (slow-fast system).

e B

d v

A l 3 4/ const

l

v

B

2

const d sin const

If a system has enough number of adiabatic invariants then the motion over long time intervals is close toa regular one.

Destruction of adiabatic invariance is one of mechanisms of creation of chaotic dynamics.

( , , ),

( ) ( , , )

x f x

x g x

System with rotating phases:

(slow)

(fast)

0 1

averaging

( ),x F x F f

0

I(x) is a first integral of the averaged system => it is an adiabatic invariant of the original system

x

k x k xn n1 1 0 ( ) ... ( ) - resonant surface

-trajectory of the averaged system

( ) ( ( ),..., ( ))x x xn 1

ki are integer numbers

Slow-fast Hamiltonian system:

H H p q I H p q I

p q I m m

0 1

1 2

( , , ) ( , , , )

( , ) , ( , )

R R T

pH

q

H

q

qH

p

H

p

IH

H

I

H

I

0 2 1

0 2 1

1

0 1

slow variables

fast phases

averaging(adiabatic approximation)

,

pH

qq

H

p

I I

0 0

0 const

I

p

q

resonant surface

I = const

adiabatic trajectory

capture

escape scattering

Two-frequency systems:

( , ), ( , ), ( , )1 2 1 2 1 2I I I

1

2 k k1 1 2 2 0

Effect of each resonance can be studied separately.

А. Partial averaging for given resonance.

Canonical transformation: ( , , , ) ( , , , )I I R J1 2 1 2

k k

l l1 1 2 2

1 1 2 2

,

,

R l I l I

J k I k I

2 1 1 2

2 1 1 2

k l k l1 2 2 1 1

Averaging over J const

Hamiltonian:

H H R J p q H 0 1( , , , )

is the resonant phase

B. Expansion of the Hamiltonian near the resonant surface.

PR R p q

O

p p O q q O

td

d

res

( , )( ),

( ), ( ),

, ( )

R

qp

R R p qres ( , )

pq

qp

Hres

,

0

- resonant flow

Dynamics of (resonant phase) and (deviation from the resonant surface) is described bythe pendulum-like Hamiltonian:

P

pendulum with a torque and slowly varying parameters

Phase portraits of pendulum-like system

P P

Capture:

Probability of capture:

S p q( , ) const

( , )p q const

In-out function:“inner adiabatic invariant” = const

Scattering on resonance.

Value should be treated as a random variable

uniformly distributed on the interval

Results of consequent passages through resonances should be treated as statistically independent according to phaseexpansion criterion.

~ ~ ~

1 1

E E k k r t 0 sin( )

Resonance: k r 0

Example: motion of relativistic charged particle in stationary uniform magnetic field and high-frequency harmonic electrostatic wave (A.Chernikov, G.Schmidt, N., PRL, 1992; A.Itin, A.Vasiliev, N., Phys.D, 2000).

Larmor circle

waveCapture into resonance means capture into regime of surfatron acceleration (T.Katsouleas, J.M.Dawson, 1985)

B

k

x1

x2

x3

B B e A B x e

k x k x t

0 3 0 1 2

0 1 1 3 3

, ,

cos( )

H m c c A ee

c 2 4 2 2| |P

m c c c B x e k x k x te

c2 4 2 2 2

0 12 2

0 1 1 3 3P P P1 2 3( ) cos( )

P P2 0, p Ae

c

Assumptions:

| |~ , ~ , , ~

,

p

mc kc

e

mc

eB

mc

c

c

1 1 1

02

0

After rescaling:

~( ) cos( )H q k q k q tc 1 2 2 2

12

1 1 3 3P P1 3

e

mc

eB

mcc cc

02

0, ,

After transformation:

H

1 2 2 2 2 2 2

12

32

3

k I p q I

k k k k k

p

k c( cos ) sin cos

, sin /

Conjugated variables:

( , ) ( , )p q I 1

I

q

p

Resonant surface: Resonant flow:

p

kk p qccos ( ( / ) ) sin/1 12 1 2 2 2 2 2

kc

sin kc

sin

Hamiltonian of the “pendulum”:

F gP b

bk

q

p q

gk k

p q

c

c

c

02

2

2 2 2 2 2 2

2 2 3 2

2 2 2 2

1

2

1

1

1

cos

cos

sin

( ( / ) )

sin

/

kc

sin Trajectory of the resonant flow is an ellipse.

Capture into resonance and escape from resonance:

kc

sinTrajectory of the resonant flow is a hyperbola.

Condition of acceleration:E

B

kc

kc kc0

0

2 2

21

( / ) sin

( / ) ( / )

Capture into resonance (regime of unlimited surfatron acceleration):

Scattering on resonance: