destruction of adiabatic invariance at resonances in slow-fast hamiltonian systems Аnatoly...
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Destruction of adiabatic invariance at resonances in slow-fast
Hamiltonian systems
Аnatoly Neishtadt
Space Research Institute, Moscow
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
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: