The small oscillations of an arbitrary scleronomous system subject to time-independent non dissipative forces are discussed. The linearized equations of motion are solved by quadratures. As in the conservative case, the general integral is shown to consist of a superposition of harmonic oscillations. A complexification of the resolving algorithm is presented.
We show that the contact dynamics obtained from the Herglotz variational principle can be described as a constrained nonholonomic or vakonomic ordinary Lagrangian system depending on a dissipative variable with an adequate choice of one constraint. As a consequence we obtain the dynamics of contact nonholonomic and vakonomic systems as ordinary variational calculus with constraints on a Lagrangian with a dissipative variable. The variation of the energy and the other dissipative quantities are also obtained giving the usual results.
The trajectories of a qubit dynamics over the two-sphere are shown to be geodesics of certain Riemannian or physically-sound Lorentzian manifolds, both in the non-dissipative and dissipative formalisms, when using action-angle variables. Several aspects of the geometry and topology of these manifolds (qubit manifolds) have been studied for some special physical cases.
We prove that well posed quasilinear equations of parabolic type, perturbed by bounded nondegenerate random forces, are exponentially mixing for a large class of random forces.
We investigate the particle and kinetic-energy densities for $N$ non-interacting fermions confined in a local potential. Using Gutzwillers semi-classical Green function, we describe the oscillating parts of the densities in terms of closed non-periodic classical orbits. We derive universal relations between the oscillating parts of the densities for potentials with spherical symmetry in arbitrary dimensions, and a ``local virial theorem valid also for arbitrary non-integrable potentials. We give simple analytical formulae for the density oscillations in a one-dimensional potential.
We investigate the particle and kinetic-energy densities for a system of $N$ fermions bound in a local (mean-field) potential $V(bfr)$. We generalize a recently developed semiclassical theory [J. Roccia and M. Brack, Phys. Rev. Lett. {bf 100}, 200408 (2008)], in which the densities are calculated in terms of the closed orbits of the corresponding classical system, to $D>1$ dimensions. We regularize the semiclassical results $(i)$ for the U(1) symmetry breaking occurring for spherical systems at $r=0$ and $(ii)$ near the classical turning points where the Friedel oscillations are predominant and well reproduced by the shortest orbit going from $r$ to the closest turning point and back. For systems with spherical symmetry, we show that there exist two types of oscillations which can be attributed to radial and non-radial orbits, respectively. The semiclassical theory is tested against exact quantum-mechanical calculations for a variety of model potentials. We find a very good overall numerical agreement between semiclassical and exact numerical densities even for moderate particle numbers $N$. Using a local virial theorem, shown to be valid (except for a small region around the classical turning points) for arbitrary local potentials, we can prove that the Thomas-Fermi functional $tau_{text{TF}}[rho]$ reproduces the oscillations in the quantum-mechanical densities to first order in the oscillating parts.