Symmetries and dynamics in constrained systems

Hamiltonian Dynamics of a spaceship in Alcubierre and Gödel metrics: Recursion operators and underlying master symmetries

We study the Hamiltonian dynamics of a spaceship in the background of Alcubierre and Gödel metrics. We derive the Hamiltonian vector fields governing the system evolution, construct and discuss related recursion operators generating the constants of motion. Besides, we characterize relevant master symmetries.

Hamiltonian analysis for topological and Yang-Mills theories expressed as a constrained BF-like theory

The Hamiltonian analysis for the Euler and Second-Chern classes is performed. We show that, in spite of the fact that the Second-Chern and Euler invariants give rise to the same equations of motion, their corresponding symplectic structures on the phase space are different, therefore, one can expect different quantum formulations. In addition, the symmetries of actions written as a BF-like theory that lead to Yang-Mills equations of motion are studied. A close relationship with the results obtained in previous works for the Second-Chern and Euler classes is found.

Dynamical Symmetries of Dirac Hamiltonian

Several dynamical symmetries of the Dirac Hamiltonian are reviewed in a systematic manner and the conditions under which such symmetries hold. These include relativistic spin and orbital angular momentum symmetries, SO(4)\times SU_σ(2) symmetry for the Dirac Hydrogen atom, SU(3)\times SU_σ(2) symmetry for the relativistic simple harmonic oscillator. The energy spectrum in each case is calculated from group-theoretic considerations.

Symmetry, singularities and integrability in complex dynamics III: approximate symmetries and invariants

The different natures of approximate symmetries and their corresponding first integrals/invariants are delineated in the contexts of both Lie symmetries of ordinary differential equations and Noether symmetries of the Action Integral. Particular note is taken of the effect of taking higher orders of the perturbation parameter. Approximate symmetries of approximate first integrals/invariants and the problems of calculating them using the Lie method are considered.

Normal forms and gauge symmetries of local dynamics

A systematic procedure is proposed for deriving all the gauge symmetries of the general, not necessarily variational, equations of motion. For the variational equations, this procedure reduces to the Dirac-Bergmann algorithm for the constrained Hamiltonian systems with certain extension: it remains applicable beyond the scope of Dirac's conjecture. Even though no pairing exists between the constraints and the gauge symmetry generators in general non-variational dynamics, certain counterparts still can be identified of the first- and second-class constraints without appealing to any Poisson str…

Heisenberg dynamics for non self-adjoint Hamiltonians: symmetries and derivations

In some recent literature the role of non self-adjoint Hamiltonians, $H\neq H^\dagger$, is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schrödinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, *-derivations and integrals of motion is discussed.

Abstract Dynamical Systems: Remarks on Symmetries and Reduction

We review how an algebraic formulation for the dynamics of a physical system allows to describe a reduction procedure for both classical and quantum evolutions.

Hidden Symmetries of Dynamics in Classical and Quantum Physics

This article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description of physical systems as varied as non-relativistic, relativistic, with or without gravity, classical or quantum, and are related to the existence of conserved quantities of the dynamics and integrability. In recent years their study has grown intensively, due to the discovery of non-trivial examples that apply to different types of theories and different numb…

Symmetries and Dynamics of Discrete Systems

We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program constructs and investigates phase portraits of discrete dynamical systems modulo groups of their symmetries, searches dynamical systems possessing specific properties, e.g., reversibility, computes microcanonical partition functions and searches phase transitions in mesoscopic systems. Some computational results and observations are presented. In particular, we exp…

定向球模型非平衡动力学中的动力学对称性

Dynamical symmetries in the non-equilibrium dynamics of the directed spherical model

本文研究淬火定向球模型的非平衡弛豫动力学，发现新的弹道及各向异性弹道标度区域，并用meta-Schrödinger不变性和meta-共形不变性刻画其动力学对称性。