Discrete resonances

2015 ◽  
Vol 29 (35n36) ◽  
pp. 1530016
Author(s):  
Franco Vivaldi
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Discrete Space ◽  
Arithmetic Dynamics ◽  
Countable Number ◽  
Elliptic Orbits ◽  
Recent Developments ◽  
Linear And Nonlinear ◽  
Salient Features ◽  
Discrete Spaces

The concept of resonance has been instrumental to the study of Hamiltonian systems with divided phase space. One can also define such systems over discrete spaces, which have a finite or countable number of points, but in this new setting the notion of resonance must be re-considered from scratch. I review some recent developments in the area of arithmetic dynamics which outline some salient features of linear and nonlinear stable (elliptic) orbits over a discrete space, and also underline the difficulties that emerge in their analysis.

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.

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