Mark Levi

Quasiperiodic motions in superquadratic time-periodic potentials

It is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time-periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most (in a precise sense) motions are in fact quasiperiodic. The class of potentials includes, as very particular examples, the exponential, polynomial and much more. This extends earlier results and gives an answer to a problem posed by Littlewood in the mid 1960s. Along the way machinery is developed for estimating the action-angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived.

Rotational elastic dynamics

Abstract The combined dynamical effects of elasticity and a rotating reference frame are explored for structures in a zero gravity environment. A simple yet general approach to modeling is presented, and this approach is applied to analyze in detail the dynamics of a specific prototypical structure. Energy dissipation is included and its effects are studied in detail in a model problem. Bifurcations and stability are analyzed as well.

Constrained relative motions in rotational mechanics

The dynamical effects of imposing constraints on the relative motions of component parts in a rotating mechanical system or structure are explored. It is noted that various simplifying assumptions in modeling the dynamics of elastic beams imply strain constraints, i.e., that the structure being modeled is rigid in certain directions. In a number of cases, such assumptions predict features in both the equilibrium and dynamic behavior which are qualitatively different from what is seen if the assumptions are relaxed. It is argued that many pitfalls may be avoided by adopting so-called geometrically exact models, and examples from the recent literature are cited to demonstrate the consequences of not doing this. These remarks are brought into focus by a detailed discussion of the nonlinear, nonlocal model of a shear-free, inextensible beam attached to a rotating rigid body. Here it is shown that linearization of the equations of motion about certain relative equilibrium configurations leads to a partial differential equation. Such spatially localized models are not obtained in general, however, and this leaves open questions regarding the way in which the geometry of a complex structure influences computational requirements and the possibility of exploiting parallelism in performing simulations. A general treatment of linearization about implicit solutions to equilibrium equations is presented and it is shown that this approach avoids unintended imposition of constraints on relative motions in the models. Finally, the example of a rotating kinematic chain shows how constraining the relative motions in a rotating mechanical system may destabilize uniformly rotating states.

Stability of the Inverted Pendulum—A Topological Explanation

An explanation and a proof of stability of the inverted pendulum whose suspension point undergoes vertical periodic oscillations is given. The main idea of the argument is topological; as it turns out, existence of stable regimes can be proven with little effort using only very crude qualitative information about the system. More precisely, let n be the number of times the pendulum becomes vertical during one forcing period. If n changes by more than 4 with the change of a parameter

KAM theory for particles in periodic potentials

\mu

Geometry and physics of averaging with applications

, then for an open interval of intermediate values of

Stabilization of the inverted linearized pendulum by high frequency vibrations

\mu

Geometry of Arnold Diffusion

the pendulum will be stable.

A period-adding phenomenon

It is shown that the system of the form x + V ′ ( x ) = p ( t ) with periodic V and p and with ( p ) = 0 is near-integrable for large energies. In particular, most (in the sense of Lebesgue measure) fast solutions are quasiperiodic, provided V ∈ C (5) and p ∈ L 1 ; furthermore, for any solution x ( t ) there exists a velocity bound c for all time: | x ( t )| c for all t ∈ R . For any real number r there exists a solution with that average velocity, and when r is rational, this solution can be chosen to be periodic.

The Poncelet Grid and Billiards in Ellipses

Abstract The main point of this paper is an observation that behind the standard averaging procedure there lies some simple previously unobserved geometry. In particular, the averaged forces in a rapidly forced system are, as it turns out, the constraint forces of an associated auxiliary non-holonomic system. The curvature of these constraints enters the expression for the averaged system. For example, the curvature of the pursuit curve enters the averaged equation of the pendulum with vibrating suspension point. This observation gives a new physical and geometrical insight into the mechanics of the Paul traps and the stability of forced inverted pendula.