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Adiabatic Invariance and Passage Through Resonance for Nearly Periodic Hamiltonian Systems
Author(s) -
Kevorkian J.
Publication year - 1982
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm198266295
Subject(s) - adiabatic invariant , hamiltonian (control theory) , adiabatic process , hamiltonian system , singularity , mathematics , canonical transformation , adiabatic quantum computation , coordinate system , covariant hamiltonian field theory , mathematical physics , invariant (physics) , resonance (particle physics) , classical mechanics , physics , mathematical analysis , quantum mechanics , geometry , mathematical optimization , quantum computer , quantum
This is a continuation of previous work on passage through resonance to nearly periodic Hamiltonian systems. We review the classical technique for calculating adiabatic invariants and exhibit the occurrence of zero divisors in the results as a certain critical term evolves slowly through a resonance condition. We then isolate the coordinate associated with the singularity and remove the remaining coordinate from the Hamiltonian to any desired order by successive canonical transformations. This is a variant of the von Zeipel procedure used extensively in celestial mechanics. The momentum conjugate to the cyclic coordinate is an adiabatic invariant, and the reduced Hamiltonian is then solved by constructing and matching three multiple variable expansions which describe the solution before, during, and after resonance passage.