Integrable Hamiltonians from close approximations to invariant tori

1996 ◽  
pp. 93-102
Author(s):  
M Kaasalainen ◽  
J Binney
Keyword(s):  
Invariant Tori

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

Parabolic invariant tori in quasi-periodically forced skew-product maps

2021 ◽  
Vol 277 ◽  
pp. 234-274
Author(s):  
Xinyu Guan ◽  
Jianguo Si ◽  
Wen Si
Keyword(s):  
Invariant Tori ◽  
Skew Product ◽  
Product Maps
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