Hamiltonian and Linear-Space Structure for Damped Oscillators: II. Critical Points

S. C. Chee; Alec Maassen van den Brink; K. Young

doi:10.1088/0305-4470/37/37/009

Home/Archive/arXiv

arXiv · math-ph/0206027

Hamiltonian and Linear-Space Structure for Damped Oscillators: II. Critical Points

S. C. Chee·Alec Maassen van den Brink·K. Young

View original source ↗Read source document ↗Export metadata ↓

Abstract

The eigenvector expansion developed in the preceding paper for a system of damped linear oscillators is extended to critical points, where eigenvectors merge and the time-evolution operator $H$ assumes a Jordan-block structure. The representation of the bilinear map is obtained in this basis. Perturbations $εΔH$ around an $M$-th order critical point generically lead to eigenvalue shifts $\simε^{1/M}$ dependent on only_one_ matrix element, with the $M$ eigenvalues splitting in equiangular directions in the complex plane. Small denominators near criticality are shown to cancel.

Explore related subjects

math-ph math.DS math.MP math.RA physics.class-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

S. C. Chee, Alec Maassen van den Brink, K. Young. 2004-02-10. Hamiltonian and Linear-Space Structure for Damped Oscillators: II. Critical Points. https://doi.org/10.1088/0305-4470%2F37%2F37%2F009

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Modeling of point charges with distributions, regularizations, and generalized functions

We apply regularization and generalized function techniques to the classical electromagnetic field of a point-charge, following a strategy by the physicist A. Gsponer 2007-2009, thereby extending it and adding rigor.. We show how the Liénard-Wiechert potential emerges essentially from the basic geometry of Minkowski space, namely by action of the d'Alembertian on a generating vector field, which is defined via a spacetime interval between an observer and the point-charge at retarded proper time. Furthermore, for a charged particle in its rest frame, we discuss generalized functions aspects of the electric monopole, magnetic dipole, electron singularity, and self-energy, where infinitely large generalized numbers occur whose concrete representations can be applied in field renormalization.

math-ph↗

arXiv2026

Variational derivation of the homogeneous Boltzmann equation

We introduce a variational formulation of the homogeneous Boltzmann equation, with hard-sphere cross section, which selects the unique energy conserving solution. We prove that this solution arises from the microscopic dynamics, namely Kac's walk, and we establish the propagation in time of entropic chaoticity, under the minimal assumption that the initial distribution is entropically chaotic.

math-ph↗

arXiv2026

Noether symmetries and conservation laws of a class of time-dependent multidimensional nonlinear wave equations

Conservation laws of a class of time-dependent damped nonlinear multidimensional wave equations are derived by Noether's theorem. For arbitrary nonzero damping coefficient and nonlinear interaction term, its infinitesimal variational symmetries span a Euclidean algebra $\euclid(n)$ of space translations and rotations. They produce conservation of linear and angular momentums. For some specific forms of these two terms symmetry algebra is enlarged to a subalgebra of the conformal algebra $\conf(1,n)$ and in this case more interesting conservation laws are found.

math-ph↗