An essential aspect of a generally covariant system is the invariance of its action under reparametrizations; this means that the label that parametrizes the trajectories of the system is not the time but a physically irrelevant parameter. As a consequence, the system is constrained to remain on the hypersurface of the phase space where the Hamiltonian is null. In fact, since the “evolution” ge...

We discuss a prototype Fortran 90 separable Hamiltonian system solver and present a template illustrating its use. The solver permits a wide choice of symplectic and non-symplectic integrators, and xed and error-controlled integration step sizes. It is designed to permit insertion of new formulas as they become available.

In this paper we study, using variational methods, a Hamiltonian system of the form −u′′ + u = h(t)V (u), where h and V are differentiable, h is positive, bounded, and bounded away from zero, and V is a “superquadratic” potential. That is, V behaves like q to a power greater than 2, so |V (q)| = o(|q|) for |q| small and V (q) > O(|q|) for |q| large. To prove that a solution homoclinic to zero e...

In this paper, we study the following first-order nonperiodic Hamiltonian system ż = JHz(t, z), where H ∈ C1(R× R ,R) is the form H(t, z) = 1 2 L(t)z · z + R(t, z). Under weak superquadratic condition on the nonlinearitiy. By applying the generalized Nehari manifold method developed recently by Szulkin and Weth, we prove the existence of homoclinic orbits, which are ground state solutions for a...

The existence of a Hamiltonian vector field in which trajectories of the invariant set of a dissipative hyperbolic chaotic system are embedded will be proved (see notation below). Evidence of this with an example concerning the Lorenz system will be provided. Also, a constructive method of designing a Hamiltonian for the Lorenz attractor with a universal approximator will be introduced. The pre...

1 Problem It is generally considered that systems with friction are not part of Hamiltonian dynamics, but this is not always the case. Show that a (nonrelativistic) damped harmonic oscillator can be described by a Hamiltonian (and by a Lagrangian), with the implication that Liouville's theorem applies here. Consider motion in coordinate x of a particle of mass m with equation of motion, m¨x + β...

The system of a closed vortex filament is an integrable Hamiltonian one, namely, a Hamiltonian system with an infinite sequense of constants of motion in involution. An algebraic framework is given for the aim of describing differential geometry of this system. A geometrical structure related to the integrability of this system is revealed. It is not a bi-Hamiltonian structure but similar one. ...

This paper is concerned with state-space realizations of the adjoints of the variationals of Hamiltonian control systems. It will be shown that the variational systems of a class of Hamiltonian systems have self-adjoint state-space realizations, that is, the variational system and its adjoint have the same state-space realizations. This implies that the inputoutput mapping of the adjoint of the...