A reappraisal of Lagrangians with non-quadratic velocity dependence and branched Hamiltonians

Abstract

Time and again, non-conventional forms of Lagrangians with non-quadratic velocity dependence have found attention in the literature. For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics including specifically the types of the Liénard class; for another, very often the problem of their quantization opens up multiple branches of the corresponding Hamiltonians, ending up with the presence of singularities in the associated eigenfunctions. In this article, we furnish a brief review of the classical theory of such Lagrangians and the associated branched Hamiltonians, starting with the example of Liénard-type systems. We then take up other cases where the Lagrangians depend upon the velocity with powers greater than two while still having a tractable mathematical structure, while also describing the associated branched Hamiltonians for such systems. For various examples, we emphasize upon the emergence of the notion of momentum-dependent mass in the theory of branched Hamiltonians.

v1: Preliminary version, comments are welcome; v2: Published in Symmetry

References in corpus (13)

• The physics of exceptional points
• Recent progress on the description of relativistic spin: vector model of spinning particle and rotating body with gravimagnetic moment in General Relativity
• Spectral singularities in PT-symmetric periodic finite-gap systems
• Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator
• A review on contact Hamiltonian and Lagrangian systems
• Non-divergent representation of non-Hermitian operator near the exceptional point with application to a quantum Lorentz gas
• Chaotic Maps, Hamiltonian Flows, and Holographic Methods
• Logistic Map Potentials
• Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians
• Exceptional point in a coupled Swanson system
• Generalized Uncertainty principle and momentum-dependent effective mass Schrodinger equation
• PT-symmetric model with an interplay between kinematical and dynamical non-localities
• Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach

Cited by in corpus (4)

• Generalized Liénard systems and isochronous connections
• Quasi-harmonic spectra from branched Hamiltonians
• Quantum Einsteinian Cubic Cosmology
• An alternative interpretation of the Grioli gyroscope suspension points