COMPLETELY AND PARTIALLY INTEGRABLE HAMILTONIAN SYSTEMS IN THE NONCOMPACT CASE

2004 ◽  
Vol 01 (03) ◽  
pp. 167-183 ◽  
Author(s):  
EMANUELE FIORANI
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Integrable Hamiltonian System ◽  
Time Dependent ◽  
Integrable Hamiltonian Systems ◽  
Invariant Submanifold ◽  
Completely Integrable ◽  
Completely Integrable Hamiltonian System

Action-angle coordinates are shown to exist around an instantly compact invariant submanifold of a time-dependent completely integrable Hamiltonian system. Partially integrable Hamiltonian systems are also considered in the noncompact case; a comparison is made with other possible approaches. Results on symplectically complete foliations contained in the Appendix A can be used to give alternative proofs of some propositions.

Invariant incompatible Poisson structures

1995 ◽  
Vol 450 (1940) ◽  
pp. 723-730 ◽  
Keyword(s):  
Hamiltonian System ◽  
Poisson Structure ◽  
Integrable Hamiltonian System ◽  
Complete Classification ◽  
Degenerate Case ◽  
Poisson Structures ◽  
Completely Integrable ◽  
Completely Integrable Hamiltonian System ◽  
Invariant Submanifolds

Supplemental invariant Poisson structures P c which are incompatible with the original Poisson structure P 1 are discovered for an arbitrary completely integrable Hamiltonian system. For the non-degenerate case, the complete classification of the invariant Poisson structures P c is obtained provided that the invariant submanifolds of the integrable Hamiltonian system are compact. The instability of the property of compatibility of any supplemental invariant non-degenerate Poisson structure P 2 with P 1 is established.

scholarly journals Isomonodromic deformation of Fuchsian projective connections on elliptic curves

2003 ◽  
Vol 171 ◽  
pp. 127-161 ◽  
Author(s):  
Shingo Kawai
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Elliptic Curves ◽  
Integrable Hamiltonian System ◽  
Second Order ◽  
Isomonodromic Deformation ◽  
Isomonodromic Deformations ◽  
Completely Integrable ◽  
Completely Integrable Hamiltonian System ◽  
Fuchsian Differential Equations

AbstractWe consider isomonodromic deformations of second-order Fuchsian differential equations on elliptic curves. The isomonodromic deformations are described as a completely integrable Hamiltonian system.

Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems

1996 ◽  
Vol 63 (2) ◽  
pp. 493-500 ◽  
Author(s):  
W. Q. Zhu ◽  
Y. Q. Yang
Keyword(s):  
Hamiltonian System ◽  
Stationary Solution ◽  
Hamiltonian Systems ◽  
Stochastic Systems ◽  
Integrable Hamiltonian System ◽  
Stationary Solutions ◽  
Integrable Hamiltonian Systems ◽  
Exact Stationary Solution ◽  
Special Cases ◽  
Action Variables

It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegrable Hamiltonian system, the exact stationary solution is a functional of the Hamiltonian and has the property of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and α combinations of phase angles in resonant case with α (1 ≤ α ⩽ n – 1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian systems, which are further generalized to account for the modification of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.

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