Adiabatic invariant in light of Hamilton’s principle

• Published in: Chinese journal of physics (Taipei) Vol. 67; pp. 253 - 264
• Main Author: Chen, Yih-Yuh
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2020
• Subjects: Adiabatic invariant; Extended phase space; Hamilton’s principle; Trial trajectory; Trial trajectory; Hamilton’s principle; Adiabatic invariant; Extended phase space
• ISSN: 0577-9073
• DOI: 10.1016/j.cjph.2020.06.013

•Adiabatic invariants can be proved by directly applying Hamilton’s principle.•Explicit incorporation of the small perturbation parameter in the trial trajectory is the key.•We can use it to prove a simple recurrent property of the action variable. Adiabatic invariants are specific physical quantities which do not change appreciably even after a very long time when the Hamiltonian of a mechanical system undergoes a slow change in time. Existing proofs of this nice feature range from sophistication, and typically resort to a sort of averaging principle using Hamilton’s equations of motion. We show that a much simpler argument based directly on Hamilton’s principle per se is possible. Furthermore, this approach readily reveals an interesting local recurrent property of the adiabatic invariants that is rarely emphasized in the existing literature. We also show how our simpler approach can be easily generalized to derive the time dependence of the adiabatic invariant.