On mechanisms of destruction of adiabatic invariance in slow–fast Hamiltonian systems

Abstract

In many problems of classical mechanics and theoretical physics dynamics can be described as a slow evolution of periodic or quasi-periodic processes. Adiabatic invariants are approximate first integrals for such a dynamics. Existence of adiabatic invariants makes dynamics close to regular. Destruction of adiabatic invariance leads to chaotic dynamics. We present a review of some mechanisms of destruction of adiabatic invariance in slow-fast Hamiltonian systems with examples from charged particle dynamics.

Quasistatic transfer protocols for atomtronic superfluid circuits

“…From this description one should understand that the sweep-related mechanism is not related to chaos, but to the bifurcation of the stability island. Consequently it obeys the Kruskal-Neishtadt-Henrard theorem [13][14][15][16][17][18][19][20][21][22][23], namely the cloud is drained into the emerging stability island. This is confirmed by our simulations, see left panels of Fig.…”

“…For such system the adiabatic picture fails miserly [9][10][11][12], because the variation of the control parameter is associated with structural changes in phase space topology: tori merge into chaos, and new sets of tori are formed later on. This can be regarded as the higher-dimensional version of separatrix crossing [13][14][15][16][17][18][19][20][21][22][23], where the so-called Kruskal-Neishtadt-Henrard theorem is followed.…”

Stochastic modeling of spreading and dissipation in mixed-chaotic systems that are driven quasistatically

“…Namely, the phasespace structure varies with the control parameter: tori are destroyed; chaotic corridors are opened allowing migration between different regions in phasespace [14,15]; stochastic regions merge into chaos; sticky regions are formed [16][17][18][19]; sets of tori re-appear or emerge. Some of those issues can be regarded as a higher-dimensional version of non-linear scenarios that are relate to bifurcations of fixed points, notably swallow-tail loops [20][21][22][23][24], or as a higher-dimensional version of the well-studied separatrix crossing [25][26][27][28][29][30][31][32][33][34][35][36], where the Kruskal-Neishtadt-Henrard theorem is followed.…”

Anatolii Iserovish Neishtadt (on his 70th birthday)

“…Захвату в различные области можно приписать корректно определенные вероятности, которые могут быть вычислены. Эти идеи и результаты были обобщены и развиты А. И. Нейштадтом в последующих работах (см., например, обзоры [24], [26], [27]). При переходе через сепаратрису происходит квазислучайный (чувствительный к выбору начальных условий) малый скачок адиабатического инварианта, асимптотическая формула для которого была получена А. И. Нейштадтом в работах [17] (для системы с полутора степенями свободы) и [18] (для системы с быстрыми и медленными движениями).…”

“…А. И. Нейштадтом было дано асимптотическое описание этих явлений в двухчастотных системах. Указанные результаты описаны, например, в работах [22], [23] и обзорах [30], [25], [27].…”