We study the approach to near-equipartition in the N-dimensional Fermi-Pasta-Ulam Hamiltonian with quartic (hard spring) nonlinearity. We investigate numerically the time evolution of orbits with initial energy in some few low-frequency linear modes. Our results indicate a transition where, above a critical energy which is independent of N, one can reach equipartition if one waits for a time proportional to N(2). Below this critical energy the time to equipartition is exponentially long. We develop a theory to determine the time evolution and the excitation of the nonlinear modes based on a resonant normal form treatment of the resonances among the oscillators. Our theory predicts the critical energy for equipartition, the time scale to equipartition, and the form of the nonlinear modes below equipartition, in qualitative agreement with the numerical results. (c) 1995 American Institute of Physics.

New asymptotically exact amplitude equations are derived for a dissipative system near a Hopf bifurcation. Unlike the usual coupled complex Ginzburg-Landau equations these are valid for O(1) group velocities.

We adapt the formally-defined Fokker action into a variational principle for the electromagnetic two-body problem. We introduce properly defined boundary conditions to construct a Poincarèinvariant-action-functional of a finite orbital segment into the reals. The boundary conditions for the variational principle are an endpoint along each trajectory plus the respective segment of trajectory for the other particle inside the lightcone of each endpoint. We show that the conditions for an extremum of our functional are the mixed-type-neutral-equations with implicit state-dependentdelay of the electromagnetic-two-body problem. We put the functional on a natural Banach space and show that the functional is Frechét-differentiable. We develop a method to calculate the second variation for C 2 orbital perturbations in general and in particular about circular orbits of large enough radii. We prove that our functional has a local minimum at circular orbits of large enough radii, at variance with the limiting Kepler action that has a minimum at circular orbits of arbitrary radii. Our results suggest a bifurcation at some O(1) radius below which the circular orbits become saddle-point extrema. We give a precise definition for the distributional-like integrals of the Fokker action and discuss a generalization to a Sobolev space H 2 0 of trajectories where the equations of motion are satisfied almost everywhere. Last, we discuss the existence of solutions for the statedependent delay equations with slightly perturbated arcs of circle as the boundary conditions and the possibility of nontrivial solenoidal orbits.

We study a stiff quasiperiodic orbit of the electromagnetic two-body problem of Dirac's electrodynamics of point charges. The delay equations of motion are expanded about circular orbits to obtain the variational equations up to nonlinear terms. The three-frequency orbit involves two harmonic modes of the variational dynamics with a period of the order of the time for light to travel the interparticle distance. In the atomic magnitude, these harmonic modes have a frequency that is fast compared with the circular rotation. The quasiperiodic orbit has three frequencies: the frequency of circular rotation (slow) and the two fast frequencies of two mutually orthogonal harmonic modes. Poynting's theorem gives a mechanism for a beat of the mutually orthogonal fast modes to cancel the radiation of the unperturbed circular motion by interference. The nonradiation condition for this destructive interference is that the two fast frequencies beat at the circular frequency. The resonant orbits have magnitudes in qualitative and quantitative agreement with quantum electrodynamics (QED), as follows: (i) the orbital angular momenta are integer multiples of Planck's constant to a good approximation, (ii) the orbital frequencies agree with a corresponding emission line of QED within a few percent on average, (iii) the orbital frequencies are given by a difference of two linear eigenvalues, viz., the frequencies of the mutually orthogonal fast modes, and (iv) the angular momentum of gyration of the variational motion about a resonant circular orbit is of the order of Planck's constant.

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