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Destruction of adiabatic invariance at resonances in slow-fast Hamiltonian systems

Destruction of adiabatic invariance at resonances in slow-fast Hamiltonian systems. А natoly Neishtadt Space Research Institute, Moscow aneishta@iki.rssi.ru. А diabatic invariant is an approximate first integral of the system with slow and fast variables (slow-fast system). B. e. l. d.

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Destruction of adiabatic invariance at resonances in slow-fast Hamiltonian systems

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  1. Destruction of adiabatic invariance at resonances in slow-fast Hamiltonian systems Аnatoly Neishtadt Space Research Institute, Moscow aneishta@iki.rssi.ru

  2. Аdiabatic invariant is an approximate first integral of the system with slow and fast variables (slow-fast system). B e l d

  3. If a system has enough number of adiabatic invariants then the motion over long time intervals is close to a regular one. Destruction of adiabatic invariance is one of mechanisms of creation of chaotic dynamics.

  4. System with rotating phases: (slow) (fast) averaging I(x) is a first integral of the averaged system => it is an adiabatic invariant of the original system

  5. - resonant surface x -trajectory of the averaged system are integer numbers

  6. Slow-fast Hamiltonian system: slow variables fast phases

  7. averaging (adiabatic approximation)

  8. resonant surface I I = const adiabatic trajectory q p escape scattering capture

  9. Two-frequency systems: Effect of each resonance can be studied separately.

  10. А.Partial averaging for given resonance. Canonical transformation: is the resonant phase Averaging over Hamiltonian:

  11. B.Expansion of the Hamiltonian near the resonant surface. R p q

  12. - resonant flow

  13. Dynamics of (resonant phase) and (deviation from the resonant surface) is described by the pendulum-like Hamiltonian: pendulum with a torque and slowly varying parameters

  14. Phase portraits of pendulum-like system P P

  15. Capture: Probability of capture:

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

  17. Scattering on resonance. Value should be treated as a random variable uniformly distributed on the interval

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

  19. 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 Resonance: wave Capture into resonance means capture into regime of surfatron acceleration (T.Katsouleas, J.M.Dawson, 1985)

  20. B k Assumptions:

  21. After rescaling: After transformation: Conjugated variables:

  22. Resonant surface: Resonant flow: I q p

  23. Hamiltonian of the “pendulum”:

  24. Trajectory of the resonant flow is an ellipse.

  25. Capture into resonance and escape from resonance:

  26. Trajectory of the resonant flow is a hyperbola. Condition of acceleration:

  27. Capture into resonance (regime of unlimited surfatron acceleration):

  28. Scattering on resonance:

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