The result of J. Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multidimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the Poincaré-Melnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian.

We propose a geometric setting for the Hamiltonian description of mechanical systems with a nonholonomic constraint, which may be used for constraints of general type (nonlinear in the velocities, and such that the constraint forces may not obey Chetaev's rule). Such constraints may be realized by servomechanisms; therefore, the corresponding mechanical system may be nonconservative. In that setting, the kinematic properties of the constraint are described by a submanifold of the tangent bundle, mapped, by Legendre's transformation, onto a submanifold (called the Hamiltonian constraint submanifold) of the phase space (i.e., of the cotangent bundle to the configuration manifold). The dynamical properties of the constraint are described by a vector subbundle of the tangent bundle to the phase space along the Hamiltonian constraint submanifold. In order to be able to deal with systems obtained by reduction by a symmetry group, we generalize that setting by using a Poisson struc...

Abstract. We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behaviour of this system. 1.

Dedicated to Vladimir Arnold on his 65th birthday Abstract. As is well-known, there is a variational principle for the Euler–Poincaré equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle for the Lie–Poisson equations on g ∗ , the dual of g, and also to generalize this construction. The more general situation is that in which the original configuration space is not a Lie group, but rather a configuration manifold Q on which a Lie group G acts freely and properly, so that Q → Q/G becomes a principal bundle. Starting with a Lagrangian system on T Q invariant under the tangent lifted action of G, the reduced equations on (T Q)/G, appropriately identified, are the Lagrange–Poincaré equations. Similarly, if we start with a Hamiltonian system on T ∗ Q, invariant under the cotangent lifted action of G, the resulting reduced equations on (T ∗ Q)/G are called the Hamilton–Poincaré equations. Amongst our new results, we derive a variational structure for the Hamilton–Poincaré equations, give a formula for the Poisson structure on these reduced spaces that simplifies previous formulas of Montgomery, and give a new representation for the symplectic structure on the associated symplectic leaves. We illustrate the formalism with a simple, but interesting example, that of a rigid body with internal rotors. 1.

Hamilton-Jacobi Equations, Viscosity Solutions and Asymptotics of Hamiltonian Systems by Diogo Aguiar Gomes Doctor of Philosophy in Mathematics University of California at Berkeley Professor Lawrence C. Evans, Chair The objective of this dissertation is to understand the relations between Hamiltonian dynamics and viscosity solutions of Hamilton-Jacobi equations. By combining ideas from classical mechanics with viscosity solution techniques we study the asymptotic behavior and invariant sets of Hamiltonian systems. Then we consider the problem of slowly varying Hamiltonians and provide a weak interpretation of both the adiabatic invariance of action and the Hannay angle. We apply measure and ergodic theory tools to characterize fine properties of Hamilton-Jacobi partial di#erential equations and show how ergodic properties of the Hamiltonian dynamics control the regularity of viscosity solutions. In particular, we prove a (sharp) partial uniqueness result for the time independent case, study the regularity of solutions and prove several estimates on di#erence quotients. Finally, we prove that the dual of a certain infinite dimensional linear programming problem, that is the core of Aubry-Mather theory, is equivalent to determining viscosity solutions of Hamilton-Jacobi equations. iii To Alexandra. iv Acknowledgements My advisor, Professor Lawrence C. Evans, deserves special thanks by his guidance and enthusiasm about my research. Many parts of this thesis were discussed during our weekly meetings having benefited enormously from his suggestions and advice. I am grateful to Professor Waldyr Oliva, who initiated me to research in Hamiltonian systems, by his support, encouragement and many comments and suggestions about my work. Part of the last chapter of this dissertation...

We perform a bifurcation analysis of normal-internal resonances in parametrised families of quasi-periodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the `backbone' system; forced, the system is a skew-product flow with a quasi-periodic driving with n basic frequencies. The dynamics of the forced system are simplified by averaging over the orbits of a linearisation of the unforced system. The averaged system turns out to have the same structure as in the well-known case of periodic forcing (n = 1); for a real analytic system, the non--integrable part can even be made exponentially small in the forcing strength. We investigate the persistence and the bifurcations of quasi-periodic n-dimensional tori in the averaged system, filling normal-internal resonance `gaps' that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of `gaps within gaps' makes the quasi--periodic case more complicated than the periodic case.

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In this paper we study the stability of integrable Hamiltonian systems under small perturbations, proving a weak form of the KAM/Nekhoroshev theory for viscosity solutions of Hamilton-Jacobi equations. The main advantage of our approach is that only a finite number of terms in an asymptotic expansion are needed in order to obtain uniform control. Therefore there are no convergence issues involved. An application of these results is to show that Diophantine invariant tori and Aubry-Mather sets are stable under small perturbations.

In the present paper we elaborate on the underlying Hamiltonian structure of interconnected energy-conserving physical systems. It is shown that a power-conserving interconnection of port-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltonian system, and a power-conserving partial interconnection to an implicit port-controlled Hamiltonian system. The crucial concept is the notion of a (generalized) Dirac structure, defined on the space of energy-variables or on the product of the space of energy-variables and the space of flow-variables in the port-controlled case. Three natural representations of generalized Dirac structures are treated. Necessary and sufficient conditions for closedness (or integrability) of Dirac structures in all three representations are obtained. The theory is applied to implicit port-controlled generalized Hamiltonian systems, and it is shown that the closedness condition for the Dirac structure leads to strong conditions on the ...

For the n-centre problem of one particle moving in the potential of attracting centres of small mass xed in an arbitrary smooth potential and magnetic field, we prove the existence of periodic and chaotic trajectories shadowing sequences of collision orbits. In particular, we obtain large subshifts of solutions of this type for the circular restricted 3-body problem of celestial mechanics. Poincar e had conjectured existence of the periodic ones and given them the name "second species solutions".