G. Birkhoff’s transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian

AbstractDirect 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.

Гиперболические поля Руссари с вырожденной квадратичной частью

Успехи математических наук ◽

10.4213/rm9893 ◽

2021 ◽

Vol 76 (2(458)) ◽

pp. 183-184

Author(s):

Наталья Геннадьевна Павлова ◽

Natalia Gennadievna Pavlova ◽

Алексей Олегович Ремизов ◽

Alexey Olegovich Remizov

Keyword(s):

Normal Form ◽

Vector Fields ◽

Quadratic Part

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

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

Volume 1C: 16th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc97/vib-4107 ◽

1997 ◽

Author(s):

Eric A. Butcher ◽

S. C. Sinha

Keyword(s):

Perturbation Theory ◽

Classical Method ◽

Time Dependent ◽

Fast Time ◽

Quadratic Part ◽

Canonical Perturbation ◽

Internal Excitation ◽

Quadratic Hamiltonian ◽

Unperturbed Part ◽

Time Periodic

Abstract 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.

On the Birkhoff transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian

Regular and Chaotic Dynamics ◽

10.1134/s1560354715030077 ◽

2015 ◽

Vol 20 (3) ◽

pp. 309-316 ◽

Cited By ~ 3

Author(s):

Anatoly P. Markeev

Keyword(s):

Quadratic Part

On potential wells and stability in nonlinear elasticity

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055006 ◽

1978 ◽

Vol 84 (1) ◽

pp. 177-190 ◽

Cited By ~ 7

Author(s):

R. J. Knops ◽

L. E. Payne

Keyword(s):

Potential Energy ◽

Equilibrium Solution ◽

Three Dimensional ◽

Energy Criterion ◽

Simple Extension ◽

Potential Well ◽

Potential Wells ◽

Quadratic Part ◽

Non Linear ◽

Three Dimensional Elasticity

This paper is devoted to a preliminary discussion of potential wells in non-linear three-dimensional elasticity. Our interest in the subject arises from the role that potential wells play in the justification of the energy criterion as a sufficient condition for stability of an elastic equilibrium solution. It will be recalled that the energy criterion, which is a simple extension of the original Lagrange-Dirichlet version for finite-dimensional systems, states that an equilibrium solution is stable provided the potential energy achieves its minimum on the solution. No proof of this statement as it applies to three-dimensional elasticity is yet forthcoming, although when the notion of a minimum is replaced by that of a potential well, several authors have proved that the criterion thus modified is sufficient for the Liapunov stability of the equilibrium solution with respect to appropriate measures. Indeed, the proofs are applicable to many other continuum theories, apart from elasticity. (See, for instance, Coleman(5), Gurtin(11) and Koiter(21).) It thus becomes important to determine what constitutive and other conditions, if any, ensure the existence of a potential well. While we present two such conditions, our main purpose is to describe examples in support of the conjecture that non-existence rather than existence of a potential well is likely to be the generic property. In these examples, particular forms of the potential energy are chosen which have positive-definite quadratic part and yet in any W1, P-neighbourhood (1 ≤ p ≤ ∞) of the origin, have a non-positive infimum, thus violating a condition for existence of a potential well in the, Sobolev space W1, P (1 ≤ p ≤ ∞). Related results are also reported by Ball, Knops and Marsden(3) and by Knops(17) for the space W1, ∞. Another conclusion to be drawn from these examples, is that a potential well cannot be ensured by restricting only the quadratic part of the potential energy. Indeed, in our example, we show that the equilibrium (null) solution is unstable in the sense that (non-linear) motions, starting in its neighbourhood, cease to exist after finite time. The same phenomenon has been shown by Knops, Levine and Payne(19) (see also Hills and Knops(14)) to hold for a general class of materials which includes as a special case one possessing the potential energy considered in our example.

QUARK LAGRANGIAN DIAGONALIZATION VERSUS NON-DIAGONAL KINETIC TERMS

Modern Physics Letters A ◽

10.1142/s0217732309030084 ◽

2009 ◽

Vol 24 (04) ◽

pp. 273-275 ◽

Cited By ~ 4

Author(s):

Q. DURET ◽

B. MACHET ◽

M. I. VYSOTSKY

Keyword(s):

Standard Model ◽

Quadratic Part ◽

The Standard Model

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.

Normalisation holomorphe de structures de Poisson

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385709000480 ◽

2009 ◽

Vol 30 (4) ◽

pp. 1165-1199

Author(s):

PHILIPP LOHRMANN

Keyword(s):

Singular Point ◽

Normal Form ◽

Lie Algebra ◽

Poisson Structure ◽

Linear Part ◽

The Other ◽

Diophantine Condition ◽

Quadratic Part ◽

Other Hand ◽

The One

AbstractWe 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.

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

Journal of Applied Mechanics ◽

10.1115/1.2789028 ◽

1998 ◽

Vol 65 (1) ◽

pp. 209-217 ◽

Cited By ~ 1

Author(s):

E. A. Butcher ◽

S. C. Sinha

Keyword(s):

Perturbation Theory ◽

Classical Method ◽

Time Dependent ◽

Fast Time ◽

Quadratic Part ◽

Canonical Perturbation ◽

Internal Excitation ◽

Quadratic Hamiltonian ◽

Unperturbed Part ◽

Time Periodic

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.

Hyperbolic Roussarie fields with degenerate quadratic part

Russian Mathematical Surveys ◽

10.1070/rm9893 ◽

2021 ◽

Vol 76 (2) ◽

Author(s):

Natalia Gennadievna Pavlova ◽

Alexey Olegovich Remizov

Keyword(s):

Quadratic Part

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500411300056x ◽

2013 ◽

Vol 155 (3) ◽

pp. 529-556 ◽

Cited By ~ 3

Author(s):

MAHIR HADŽIĆ ◽

GERHARD REIN

Keyword(s):

Steady State ◽

Linear Stability ◽

Particle Distribution ◽

Particle Energy ◽

Steady States ◽

Einstein Equations ◽

Spherically Symmetric ◽

Quadratic Part ◽

Non Linear ◽

The Stability

AbstractThe 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.