Finite Hamiltonian systems on phase space

2011 ◽  
pp. 141-163
Author(s):  
Kurt Wolf
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems

scholarly journals COVARIANT DESCRIPTION OF PARAMETRIZED NONRELATIVISTIC HAMILTONIAN SYSTEMS

2004 ◽  
Vol 19 (15) ◽  
pp. 2473-2493 ◽  
Author(s):  
MAURICIO MONDRAGÓN ◽  
MERCED MONTESINOS
Keyword(s):  
Phase Space ◽  
Gauge Transformation ◽  
Hamiltonian Systems ◽  
Symplectic Structure ◽  
Canonical Formalism ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Algebra Of Observables ◽  
Covariant Description ◽  
Constraint Surface

The various phase spaces involved in the dynamics of parametrized nonrelativistic Hamiltonian systems are displayed by using Crnkovic and Witten's covariant canonical formalism. It is also pointed out that in Dirac's canonical formalism there exists a freedom in the choice of the symplectic structure on the extended phase space and in the choice of the equations that define the constraint surface with the only restriction that these two choices combine in such a way that any pair (of these two choices) generates the same gauge transformation. The consequence of this freedom on the algebra of observables is also discussed.

scholarly journals Phase Space Homogenization of Noisy Hamiltonian Systems

2018 ◽  
Vol 19 (4) ◽  
pp. 1081-1114 ◽  
Author(s):  
Jeremiah Birrell ◽  
Jan Wehr
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems

scholarly journals Non-Smooth Geodesic Flows and Classical Mechanics

1969 ◽  
Vol 12 (2) ◽  
pp. 209-212 ◽  
Author(s):  
J. E. Marsden
Keyword(s):  
Phase Space ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Configuration Space ◽  
Smooth Function ◽  
Cotangent Bundle ◽  
Classical Mechanics ◽  
Riemannian Metric ◽  
Geodesic Flows ◽  
Image Position

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by

Hamiltonian Dynamics and Phase Space

2019 ◽  
pp. 83-108
Author(s):  
David D. Nolte
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Hamiltonian Dynamics ◽  
Orbital Dynamics ◽  
Legendre Transform ◽  
Lagrange Equations ◽  
Conservation Of Energy ◽  
Hamiltonian Equations ◽  
Generalized Coordinates ◽  
Conservation Of Momentum

Hamiltonian dynamics are derived from the Lagrange equations through the Legendre Transform that expresses the equations of dynamics in terms of the Hamiltonian, which is a function of the generalized coordinates and of their conjugate momenta. Consequences of the Lagrangian and Hamiltonian equations of dynamics are conservation of energy and conservation of momentum, with applications to collisions and orbital dynamics. Action-angle coordinates can be defined for integrable Hamiltonian systems and reduce all dynamical motions to phase space trajectories on a hyperdimensional torus.

Collapse of hierarchical phase space and mixing rates in Hamiltonian systems

2019 ◽  
Vol 530 ◽  
pp. 121568
Author(s):  
Tulio M. Oliveira ◽  
Roberto Artuso ◽  
Cesar Manchein
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Mixing Rates

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.

TO ESCAPE OR NOT TO ESCAPE, THAT IS THE QUESTION — PERTURBING THE HÉNON–HEILES HAMILTONIAN

2012 ◽  
Vol 22 (06) ◽  
pp. 1230010 ◽  
Author(s):  
FERNANDO BLESA ◽  
JESÚS M. SEOANE ◽  
ROBERTO BARRIO ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Survival Probability ◽  
Crucial Role ◽  
Physical Space ◽  
Periodic Forcing

In this work, we study the Hénon–Heiles Hamiltonian, as a paradigm of open Hamiltonian systems, in the presence of different kinds of perturbations as dissipation, noise and periodic forcing, which are very typical in different physical situations. We focus our work on both the effects of these perturbations on the escaping dynamics and on the basins associated to the phase space and to the physical space. We have also found, in presence of a periodic forcing, an exponential-like decay law for the survival probability of the particles in the scattering region where the frequency of the forcing plays a crucial role. In the bounded regions, the use of the OFLI2 chaos indicator has allowed us to characterize the orbits. We have compared these results with the previous ones obtained for the dissipative and noisy case. Finally, we expect this work to be useful for a better understanding of the escapes in open Hamiltonian systems in the presence of different kinds of perturbations.

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