The Euler-Jacobi-Lie integrability theorem

2013 ◽  
Vol 18 (4) ◽  
pp. 329-343 ◽  
Author(s):  
Valery V. Kozlov
Keyword(s):  
Integrability Theorem

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.

scholarly journals Around Jouanolou non-integrability theorem

2000 ◽  
Vol 11 (2) ◽  
pp. 239-254 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Jean Moulin Ollagnier ◽  
Andrzej Nowicki ◽  
Jean-Marie Strelcyn
Keyword(s):  
Integrability Theorem

A higher integrability theorem from a reverse weighted inequality

2019 ◽  
Vol 51 (6) ◽  
pp. 967-977 ◽  
Author(s):  
Samir Saker ◽  
Donal O'Regan ◽  
Ravi Agarwal
Keyword(s):  
Weighted Inequality ◽  
Integrability Theorem ◽  
Higher Integrability

scholarly journals INTRINSIC FORMULATION OF GEOMETRIC INTEGRABILITY AND ASSOCIATED RICCATI SYSTEM GENERATING CONSERVATION LAWS

2009 ◽  
Vol 06 (05) ◽  
pp. 825-837 ◽  
Author(s):  
PAUL BRACKEN
Keyword(s):  
Conservation Laws ◽  
Infinite Number ◽  
Integrability Theorem ◽  
Geometric Integrability

An intrinsic version of the integrability theorem for the classical Bäcklund theorem is presented. It is characterized by a one-form which can be put in the form of a Riccati system. It is shown how this system can be linearized. Based on this result, a procedure for generating an infinite number of conservation laws is given.

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