Canonical Hamiltonian System Research Articles

Quantum Symmetries in the Maxwell–Chern–Simons Theory Coupled to Scalar Fields

The Maxwell-Chern-Simons gauge theory coupled to a complex scalar field is quantized in the Becchi-Rouet-Stora-Tyutin (BRST) path integral formalism. On the basis of the symmetries of a constrained canonical (Hamiltonian) system, we get the quantal conserved angular momentum of the system under the global symmetry transformation. It is shown that fractional spin also appears at the quantum level. The canonical Ward identities for this system are derived under local gauge transformation.

Integrable vs. nonintegrable geodesic soliton behavior

We study confined solutions of certain evolutionary partial differential equations (PDE) in 1+1 space–time. The PDE we study are Lie–Poisson Hamiltonian systems for quadratic Hamiltonians defined on the dual of the Lie algebra of vector fields on the real line. These systems are also Euler–Poincaré equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric is different from the L 2 norm of the velocity. These PDE possess a finite-dimensional invariant manifold of particle-like (measure-valued) solutions we call “pulsons”. We solve the particle dynamics of the two-pulson interaction analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pulson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. The results for head-on antisymmetric collisions of pulsons tend to be singularity formation. Numerical simulations of these PDE show that their evolution by geodesic dynamics for confined (or compact) initial conditions in various nonintegrable cases possesses the same type of multi-soliton behavior (elastic collisions, asymptotic sorting by pulse height) as the corresponding integrable cases do. We conjecture this behavior occurs because the integrable two-pulson interactions dominate the dynamics on the invariant pulson manifold, and this dynamics dominates the PDE initial value problem for most choices of confined pulses and initial conditions of finite extent.

Canonical Variables for Multiphase Solutions of the KP Equation

The KP equation has a large family of quasiperiodic multiphase solutions. These solutions can be expressed in terms of Riemann‐theta functions. In this paper, a finite‐dimensional canonical Hamiltonian system depending on a finite number of parameters is given for the description of each such solution. The Hamiltonian systems are completely integrable in the sense of Liouville. In effect, this provides a solution of the initial‐value problem for the theta‐function solutions. Some consequences of this approach are discussed.

On the Hamiltonian approach: Applications to geophysical flows

Abstract. This paper presents developments of the Harniltonian Approach to problems of fluid dynamics, and also considers some specific applications of the general method to hydrodynamical models. Nonlinear gauge transformations are found to result in a reduction to a minimum number of degrees of freedom, i.e. the number of pairs of canonically conjugated variables used in a given hydrodynamical system. It is shown that any conservative hydrodynamic model with additional fields which are in involution may be always reduced to the canonical Hamiltonian system with three degrees of freedom only. These gauge transformations are associated with the law of helicity conservation. Constraints imposed on the corresponding Clebsch representation are determined for some particular cases, such as, for example. when fluid motions develop in the absence of helicity. For a long time the process of the introduction of canonical variables into hydrodynamics has remained more of an intuitive foresight than a logical finding. The special attention is allocated to the problem of the elaboration of the corresponding regular procedure. The Harniltonian Approach is applied to geophysical models including incompressible (3D and 2D) fluid motion models in curvilinear and lagrangian coordinates. The problems of the canonical description of the Rossby waves on a rotating sphere and of the evolution of a system consisting of N singular vortices are investigated.

Elastic rods, rigid bodies, quaternions and the last quadrature

The Kirchhoff kinetic analogy relates the governing equations for the statics of elastic rods and the dynamics of rigid bodies. We discuss the analogy in light of several different Hamiltonian formulations, including a non–canonical description of rod equilibria. We focus on the last three quadratures that are required to reconstruct the rod centreline from the frame variables, which form the complete configuration space in the rigid body interpretation. In particular, we demonstrate that if the frame evolution is formulated as a canonical Hamiltonian system involving quaternions (or Euler parameters), then the last quadratures can all be computed explicitly in terms of algebraic relations involving invariants (or integrals) of the evolution, independent of whether or not the entire system is completely integrable.

On Hamiltonian Balanced Dynamics and the Slowest Invariant Manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.

An unconstrained Hamiltonian formulation for incompressible fluid flow

The equations governing the time evolution of an ideal fluid in material coordinates are expressed as an unconstrained canonical Hamiltonian system. The incompressibility of the flow is consequent upon certain first integrals of the motion. The variable conjugate to the configuration field is not the usual linear momentum, but is instead a quantity that is related to linear momentum through an auxiliary scalar field whose time derivative is the pressure. The definition of the Hamiltonian involves a minimization with respect to this auxiliary field. The method of derivation may be generally applied to obtain unconstrained Hamiltonian descriptions of Lagrangian field equations subject to pointwise constraints.

Quantum vortex waves in superfluid helium: A Hamiltonian approach

In this paper we construct a canonical Hamiltonian system of equations governing the classical motion of a vortex filament in an incompressible fluid in a three-dimensional configuration. Subsequently we quantize the system, derive the frequency spectrum of energy and compute some thermodynamical parameters of the quantum excitations, such as specific heat and the mean-square displacement of the zero-point motion.

Vorticity equation of two-dimensional hydrodynamics of an incompressible fluid as canonical Hamiltonian system

The canonical Hamiltonian structure for a planar vortex flow of incompressible fluid is shown. New transformations determining canonical variables for Rossby waves are found. A general method of group analysis of local Poisson brackets is presented and used for those purposes.