Non-autonomous normal forms for Hamiltonian systems

A method for calculating normal forms for non-autonomous periodically perturbed Hamiltonian systems is developed. The solution for an autonomous Hamiltonian normal form is well known, and involves the solution of a homological equation on the vector space of homogeneous scalar polynomials. An algorithm is presented for generating an analogous non-autonomous homological equation using Lie transforms. Solution of this equation will generate a normal form for the non-autonomous Hamiltonian. Although this equation is defined on an infinite-dimensional space, it is shown that the problem can be reduced to an equivalent one on a finite-dimensional space. A solution can then be found in an analogous way to the solution for the autonomous problem. It is also shown that the normal form satisfies invariance properties. Finally, an example problem is presented to illustrate the solution technique.     

Reliability of quasi integrable generalized Hamiltonian systems

The reliability of quasi-integrable generalized Hamiltonian systems is studied. An m-dimensional integrable generalized Hamiltonian system has M Casimir functions C₁,…,C_M and n (n=(m−M)/2) independent first integrals H_M+1,…,H_M+n in involution. When an integrable generalized Hamiltonian system is subjected to light dampings and weakly stochastic excitations, it becomes a quasi-integrable generalized Hamiltonian system. The averaged Itô equations for slowly processes C₁,…,C_M,H_M+1,…,H_M+n can be obtained by using stochastic averaging method, from which a backward Kolmogorov equation governing the conditional reliability function and a Pontryagin equation governing the conditional mean of the first passage time are established. The conditional reliability function and the conditional mean of first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, an example of a 5-dimensional quasi-integrable generalized Hamiltonian system is worked out in detail and the solutions are confirmed by using a Monte Carlo simulation of the original system.     

Feedback stabilization of nonlinear Hamiltonian control systems

Abstract

Conditions for feedback stabilization of smooth affine nonlinear Hamiltonian systems such as robotic manipulators are given. Some results existing in the literature are unified and extended. Application to nonconservative bilinear systems in elasticity is illustrated.     

Hamiltonian orbit structure of the set of paraxial optical systems

In the paraxial regime of three-dimensional optics, two evolution Hamiltonians are equivalent when one can be transformed to the other modulo scale by similarity through an optical system. To determine the equivalence sets of paraxial optical Hamiltonians one requires the orbit analysis of the algebra sp(4, R) of 4 x 4 real Hamiltonian matrices. Our strategy uses instead the isomorphic algebra so(3, 2) of 5 x 5 matrices with metric (+1, +1, +1, -1, -1) to find four orbit regions (strata), six isolated orbits at their boundaries, and six degenerate orbits at their common point. We thus resolve the degeneracies of the eigenvalue classification.     

Stochastic averaging of quasi-integrable Hamiltonian systems under bounded noise excitations

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to bounded noise excitations. Two cases of external resonance only and both internal and external resonances are considered separately. The reduced averaged Fokker–Planck–Kolmogorov equation is solved by using the combination of the finite difference method and the successive over relaxation method. An example of two coupled Duffing oscillators under bounded noise excitations is given to illustrate the application of the proposed procedure. It is shown that the analytical results agree well with those from digital simulation of original system.     

Time-delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems

Innovative procedures for the time-delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems subject to Gaussian white noise excitations are proposed. First, the problem of time-delay stochastic optimal control of quasi-integrable Hamiltonian systems is formulated and converted into the problem of stochastic optimal control without time delay. Then the converted control problem is solved by applying the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The time-delay feedback stabilization of quasi-integrable Hamiltonian systems is formulated as an ergodic control problem with an un-determined cost function which is determined later by minimizing the largest Lyapunov exponent of the controlled system. As an example, a two-degree-of-freedom quasi-integrable Hamiltonian system with time-delay feedback control forces is investigated in detail to illustrate the procedures and their effectiveness.     

TOFI: An isochronous time-of-flight mass spectrometer

A time-of-flight mass spectrometer was constructed to measure the masses of neutron-rich recoils produced in proton-induced fragmentation reactions. This system consists of four sector magnets designed such that the overall flight time is independent of the velocity of the recoil ions and thus depends only on their mass-to-charge ratios.     

Multiple periodic solutions with prescribed minimal period to second-order Hamiltonian systems

A new variational method based on Ekeland’s principle is introduced for the existence, localization and multiplicity of periodic solutions of a prescribed minimal period to second-order Hamiltonian systems. The oscillatory property at zero or infinity of only one component of the gradient of the potential function is sufficient for the existence of infinitely many solutions. Also, oscillating properties of several components of the gradient of the potential function yield sequences of solutions with some of the components tending in norm to zero and others to infinity.     

Error bounds via a new variational principle for classical Hamiltonian systems

Summary An extremum variational principle for classical Hamiltonian systems [9] is used for error estimate of approximate solution for a mechanical system with one degree of freedom. The procedure is developed separately for two-point boundary value problems and for Cauchy problems. Finally, the theory is illustrated by two concrete problems.Research supported by the U.S. NSF, Grant No. Yor 82/062.On leave from the University of Novi Sad, Yugoslavia     

Modal truncation for linear Hamiltonian systems: a physical energy approach

This paper proposes a new criterion to select the eigenmodes to be left out in a modal truncation procedure. This criterion takes account of the dynamics of the system, on the one hand, and the input/output matrices of the system, on the other. Therefore, this approach is satisfactory from a system and control theory point of view. Moreover, this criterion is physically meaningful because its computation is directly linked with the physical energy supplied to each eigenmode.