Гиперболические поля Руссари с вырожденной квадратичной частью

A local normal form for Roussarie vector fields with degenerate quadratic part is presented.

Stability for the spherically symmetric Einstein–Vlasov system–a coercivity estimate

The stability of static solutions of the spherically symmetric, asymptotically flat Einstein–Vlasov system is studied using a Hamiltonian approach based on energy-Casimir functionals. The main results are a coercivity estimate for the quadratic part of the expansion of the natural energy-Casimir functional about an isotropic steady state, and the linear stability of such steady states. The coercivity bound shows in a quantified way that this quadratic part is positive definite on a class of linearly dynamically accessible perturbations, provided the particle distribution of the steady state is a strictly decreasing function of the particle energy and provided the steady state is not too relativistic. In contrast to the stability theory for isotropic steady states of the gravitational Vlasov-Poisson system the monotonicity of the particle distribution alone does not determine the stability character of the state, a fact which was observed by Ze'ldovitch et al. in the 1960's. The result in an essential way exploits the non-linear structure of the Einstein equations satisfied by the steady state and is not just a perturbation result of the analogous coercivity bounds for the Newtonian case. It should be an essential step in a fully non-linear stability analysis for the Einstein–Vlasov system.

Potential enstrophy in stratified turbulence

Direct numerical simulations are used to investigate potential enstrophy in stratified turbulence with small Froude numbers, large Reynolds numbers, and buoyancy Reynolds numbers ($R{e}_{b} $) both smaller and larger than unity. We investigate the conditions under which the potential enstrophy, which is a quartic quantity in the flow variables, can be approximated by its quadratic terms, as is often done in geophysical fluid dynamics. We show that at large scales, the quadratic fraction of the potential enstrophy is determined by $R{e}_{b} $. The quadratic part dominates for small $R{e}_{b} $, i.e. in the viscously coupled regime of stratified turbulence, but not when $R{e}_{b} \gtrsim 1$. The breakdown of the quadratic approximation is consistent with the development of Kelvin–Helmholtz instabilities, which are frequently observed to grow on the layerwise structure of stratified turbulence when $R{e}_{b} $ is not too small.

Normalisation holomorphe de structures de Poisson

We show that a Poisson structure whose linear part vanishes can be holomorphically normalized in a neighbourhood of its singular point $0\in \Bbb C^n$ if, on the one hand, a Diophantine condition on a Lie algebra associated to the quadratic part is satisfied and, on the other hand, the normal form satisfies some formal conditions.

QUARK LAGRANGIAN DIAGONALIZATION VERSUS NON-DIAGONAL KINETIC TERMS

Loop corrections induce a dependence on the momentum squared of the coefficients of the Standard Model Lagrangian, making highly nontrivial (or even impossible) the diagonalization of its quadratic part. Fortunately, the introduction of appropriate counterterms solves this puzzle.

Canonical Perturbation of a Fast Time-Periodic Hamiltonian via Liapunov-Floquet Transformation

In this study a possible application of time-dependent canonical perturbation theory to a fast nonlinear time-periodic Hamiltonian with strong internal excitation is considered. It is shown that if the time-periodic unperturbed part is quadratic, the Hamiltonian may be canonically transformed to an equivalent form in which the new unperturbed part is time-invariant so that the time-dependent canonical perturbation theory may be successfully applied. For this purpose, the Liapunov-Floquet (L-F) transformation and its inverse associated with the unperturbed time-periodic quadratic Hamiltonian are computed using a recently developed technique. Action-angle variables and time-dependent canonical perturbation theory are then utilized to find the solution in the original coordinates. The results are compared for accuracy with solutions obtained by both numerical integration and by the classical method of directly applying the time-dependent perturbation theory in which the time-periodic quadratic part is treated as another perturbation term. A strongly excited Mathieu-Hill quadratic Hamiltonian with a cubic perturbation and a nonlinear time-periodic Hamiltonian without a constant quadratic part serve as illustrative examples. It is shown that, unlike the classical method in which the internal excitation must be weak, the proposed formulation provides accurate solutions for an arbitrarily large internal excitation.

Canonical Perturbation of a Fast Time-Periodic Hamiltonian via Liapunov-Floquet Transformation

In this study a possible application of time-dependent canonical perturbation theory to a fast nonlinear time-periodic Hamiltonian with strong internal excitation is considered. It is shown that if the time-periodic unperturbed part is quadratic, the Hamiltonian may be canonically transformed to an equivalent form in which the new unperturbed part is time-invariant so that the time-dependent canonical perturbation theory may be successfully applied. For this purpose, the Liapunov-Floquet (L-F) transformation and its inverse associated with the unperturbed time-periodic quadratic Hamiltonian are computed using a recently developed technique. Action-angle variables and time-dependent canonical perturbation theory are then utilized to find the solution in the original coordinates. The results are compared for accuracy with solutions obtained by both numerical integration and by the classical method of directly applying the time-dependent perturbation theory in which the time-periodic quadratic part is treated as another perturbation term. A strongly excited Mathieu-Hill quadratic Hamiltonian with a cubic perturbation and a nonlinear time-periodic Hamiltonian without a constant quadratic part serve as illustrative examples. It is shown that, unlike the classical method in which the internal excitation must be weak, the proposed formulation provides accurate solutions for an arbitrarily large internal excitation.