Exact invariants for time-dependent Hamiltonian systems with one degree-of-freedom

1978 ◽  
Vol 11 (5) ◽  
pp. 843-854 ◽  
Author(s):  
W Sarlet
Keyword(s):  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Time Dependent

The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom

1998 ◽  
Vol 18 (4) ◽  
pp. 1007-1018 ◽  
Author(s):  
RAFAEL ORTEGA
Keyword(s):  
Fixed Point ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Time Dependent ◽  
Stable Fixed Point ◽  
Stable Periodic ◽  
Area Preserving ◽  
Fixed Period

Let $F:{\Bbb R}^2 \to {\Bbb R}^2$ be a mapping that is analytic and area preserving. If $F\neq \hbox{\it identity}$, then every stable fixed point is isolated.This result can be applied to prove that the number of stable periodic solutions of a fixed period of certain Hamiltonian systems is finite.

scholarly journals PERIODIC FIRST INTEGRALS FOR HAMILTONIAN SYSTEMS OF LIE TYPE

2011 ◽  
Vol 08 (06) ◽  
pp. 1169-1177 ◽  
Author(s):  
RUBEN FLORES ESPINOZA
Keyword(s):  
Hamiltonian Systems ◽  
Nonlinear Oscillator ◽  
Original System ◽  
First Integrals ◽  
Time Dependent ◽  
Existence Problem ◽  
Adjoint Group ◽  
Killing Form ◽  
Modern Physics ◽  
Existence Of Periodic Solutions

In this paper, we study the existence problem of periodic first integrals for periodic Hamiltonian systems of Lie type. From a natural ansatz for time-dependent first integrals, we refer their existence to the existence of periodic solutions for a periodic Euler equation on the Lie algebra associated to the original system. Under different criteria based on properties for the Killing form or on exponential properties for the adjoint group, we prove the existence of Poisson algebras of periodic first integrals for the class of Hamiltonian systems considered. We include an application for a nonlinear oscillator having relevance in some modern physics applications.

scholarly journals Phase space structure and fractal trajectories in 1½ degree of freedom Hamiltonian systems whose time dependence is quasiperiodic

1998 ◽  
Vol 5 (2) ◽  
pp. 69-74 ◽  
Author(s):  
M. G. Brown
Keyword(s):  
Phase Space ◽  
Time Dependence ◽  
Hamiltonian Systems ◽  
Fluid Flows ◽  
Space Structure ◽  
Degree Of Freedom ◽  
Periodic Functions ◽  
Phase Space Structure ◽  
Fractal Properties ◽  
Incompressible Fluid Flows

Abstract. We consider particle motion in nonautonomous 1 degree of freedom Hamiltonian systems for which H(p,q,t) depends on N periodic functions of t with incommensurable frequencies. It is shown that in near-integrable systems of this type, phase space is partitioned into nonintersecting regular and chaotic regions. In this respect there is no different between the N = 1 (periodic time dependence) and the N = 2, 3, ... (quasi-periodic time dependence) problems. An important consequence of this phase space structure is that the mechanism that leads to fractal properties of chaotic trajectories in systems with N = 1 also applies to the larger class of problems treated here. Implications of the results presented to studies of ray dynamics in two-dimensional incompressible fluid flows are discussed.

Time-dependent constrained Hamiltonian systems and Dirac brackets

1996 ◽  
Vol 29 (21) ◽  
pp. 6843-6859 ◽  
Author(s):  
Manuel de León ◽  
Juan C Marrero ◽  
David Martín de Diego
Keyword(s):  
Hamiltonian Systems ◽  
Time Dependent ◽  
Constrained Hamiltonian Systems ◽  
Dirac Brackets
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