Hamiltonian Trajectories Research Articles

Hamiltonian flows for impulsive control systems

Under appropriate commutativity assumptions, smooth control systems that are affine with respect to controls can be extended into classes of generalized controls that contain impulses. We show how to construct generalized Hamiltonian trajectories for the extended system that lift both the continuous and the discontinuous components of candidate optimal trajectories into the cotangent bundle. This construction gives useful insights into the structure of generalized extremal trajectories. lt is also useful from the computational point of view.An example is discussed.

Tunneling decay in a magnetic field

We provide a semiclassical theory of tunneling decay in a magnetic field and a three-dimensional potential of a general form. Because of broken time-reversal symmetry, the standard WKB technique has to be modified. The decay rate is found from the analysis of the set of the particle Hamiltonian trajectories in complex phase space and time. In a magnetic field, the tunneling particle comes out from the barrier with a finite velocity and behind the boundary of the classically allowed region. The exit location is obtained by matching the decaying and outgoing WKB waves at a caustic in complex configuration space. Different branches of the WKB wave function match on the switching surface in real space, where the slope of the wave function sharply changes. The theory is not limited to tunneling from potential wells which are parabolic near the minimum. For parabolic wells, we provide a bounce-type formulation in a magnetic field. The theory is applied to specific models which are relevant to tunneling from correlated two-dimensional electron systems in a magnetic field parallel to the electron layer.

Tunneling Transverse to a Magnetic Field and Its Occurrence in Correlated 2D Electron Systems

We investigate tunneling decay in a magnetic field. Because of broken time-reversal symmetry, the standard WKB technique does not apply. The decay rate and the outcoming wave packet are found from the analysis of the set of the particle Hamiltonian trajectories and its singularities in complex space. The results are applied to tunneling from a strongly correlated 2D electron system in a magnetic field parallel to the layer. We show in a simple model that electron correlations strongly affect the tunneling rate.

A note on the construction of Hamiltonian trajectories along heteroclinic chains

We provide a short, simple proof of the existence of Hamiltonian trajectories arbitrarily close to a given chain of heteroclinic orbits connecting ''codimension-one, KAM, whiskered tori''.

A Numerical Method for Solution of the Generalized Liouville Equation

A numerical method for the time evolution of systems described by Liouville-type equations is derived. The algorithm uses a lattice of numerical markers, which follow exactly Hamiltonian trajectories, to represent the operatord/dt in moving (i.e., Lagrangian) coordinates. However, nonconservative effects such as particle drag, creation, and annihilation are allowed in the evolution of the physical distribution function, which is itself represented according to a δfdecomposition. Further, the method is suited to the study of a general class of systems involving the resonant interaction of energetic particles with plasma waves. Detailed results are presented for both the classic bump-on-tail problem and the beam-driven TAE instability. In both cases, the algorithm yields exceptionally smooth, low-noise evolution of wave energy, especially in the linear regime. Phenomena associated with the nonlinear regime are also described.

Hamiltonian Trajectories and Duality in the Optimal Control of Linear Systems with Convex Costs

A duality theorem is proved for problems of optimal control of linear dynamical systems in continuous time subject to linear constraints and convex costs, such as penalties. Optimality conditions are stated in terms of a “minimaximum principle” in which the primal and dual control vectors satisfy a saddle point condition at almost every instant of.time. This principle is shown to be equivalent to a generalized Hamiltonian differential equation in the primal and dual state variables, along with a transversality condition that likewise is in Hamiltonian form.

Periodic solutions of Hamilton’s equations and local minima of the dual action

The dual action is a functional whose extremals lead to solutions of Hamilton’s equations. Up to now, extremals of the dual action have been obtained either through its global minimization or through application of critical point theory. A new methodology is introduced in which local minima of the dual action are found to exist. Applications are then made to the existence of Hamiltonian trajectories having prescribed period.

On a theorem by Ekeland and Lasry concerning the number of periodic Hamiltonian trajectories

This note deals with the existence of iz distinct modes of vibration for Hamihonian systems with n degrees of freedom on a prescribed energy surface. This problem has been studied in a recent remarkable paper by Ekeland and Lasry [5], who have given an important contribution to the complete extension in the large of the local results of Liapunov [ri]? Weinstein [ 111 and Moser [S]. The purpose of the present paper is to give a new proof of the result of [5], jointly with a slight improvement.

Hamiltonian trajectories having prescribed minimal period

Hamiltonian trajectories having prescribed minimal period

On the Convergence of Sequences of Convex Sets in Finite Dimensions

Four types of convergence for sequences of convex sets are investigated. Their interrelationships are explored.