We show that the exact integrator for the classical Kepler motion, recently found by Kozlov (J. Phys. A: Math. Theor. 40 (2007) 4529-4539), can be derived in a simple natural way (using well known exact discretization of the harmonic oscillator). We also turn attention on important earlier references, where the exact discretization of the 4-dimensional isotropic harmonic oscillator has been app...

A classical problem of mechanics involves a projectile fired from a given point with a given velocity whose direction is varied. This results in a family of trajectories whose envelope defines the border of a “safe” domain. In the simple cases of a constant force, harmonic potential, and Kepler or Coulomb motion, the trajectories are conic curves whose envelope in a plane is another conic secti...

Does a minimal harmonic function h remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes D ⊂ R of variable width and minimal harmonic functions h corresponding to the boundary point of D “at infinity.” Suppose f(u) is the width of the tube u units away from its endpoint and f is a Lipschitz function. The answer to the que...

Low-harmonic formulas for closed relativistic strings are given. General parametrizations are presented for the addition of second-and third-harmonic waves to the fundamental wave. The method of determination of the parametriza-tions is based upon a product representation found for the finite Fourier series of string motion in which the constraints are automatically satisfied. The construction ...

INTRODUCTION ........................................................................................................ 263 OVERVIEW .................................................................................................................. 265 DYNAMICS ETHODOLOGY ................................................................................ 268 Molecular Dynamics ..........................

There is a close relationship between harmonic functions { which have recently been proposed for path planning { and hitting probabilities for random processes. The hitting probabilities for random walks can be cast as a Dirichlet problem for harmonic functions, in much the same way as in path planning. This equivalence has implications both for uncertainty in motion planning and in the applica...

A nonlinear model of two-beam free-electron laser (FEL) in the absence of slippage is presented. The two beams are assumed to be cold with different energies and the fundamental resonance of the higher energy beam is at the third harmonic of lower energy beam. By using Maxwell’s equations and full Lorentz force equations of motion for the electron beams, coupled differential equations are deriv...

The N x N Fourier matrix is one distinguished element within the group U(N) of all N x N unitary matrices. It has the geometric property of being a fourth root of unity and is close to the dynamics of harmonic oscillators. The dynamical correspondence is exact only in the N-->infinity contraction limit for the integral Fourier transform and its fractional powers. In the finite-N case, several o...

A master equation for the deformed quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived in the microscopic model by using perturbation theory. The coefficients of the master equation and of equations of motion for observables depend on the deformation function. The steady state solution of the equation for the density matrix in th...

This paper studies synchronization of coupled second-order linear harmonic oscillators with local interaction.We analyze convergence conditions over, respectively, directed fixed and switching network topologies by using tools from algebraic graph theory,matrix theory, and nonsmooth analysis. It is shown that the coupled harmonic oscillators can be synchronized under mild network connectivity c...