Symplectic Formalism Research Articles

The principle of virtual work and a symplectic reduction of non-local continuum mechanics

We study a symplectic formulation of continuum mechanics on the Fréchet manifold of (smooth) embeddings J : M → N. From a virtual work principle we derive a generalized Hamiltonian dynamics, which allows to include the description non-hyperelastic media as well as a non-linear and non-local material behaviour. Dividing out the (rigid) translation group, the deformation gradient and the stress tensor appear—via Marsden-Weinstein reduction—as natural geometric objects. The balance law on the reduced phase space becomes a weak equation for exact differential forms, which relates the dynamics of the deformation gradient to the stress. This equation is subjected to an interesting gauge freedom and yields the classical continuum dynamics as well as boundary conditions for the stress.

On the symplectic formalism for general relativity

A covariant formalism for the hamiltonian formulation of general relativity in arbitrary dimensions is presented. Specifically, a presymplectic form on the solution space for the vacuum equations in n dimensions is given. The basic variables are taken to be a soldering form and a torsion-free connection on an SO ( p , q )-bundle over the space-time manifold M . It is shown how the present formalism is related to the standard ADM-formalism.

A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition

In a previous paper I laid the foundations of a covariant Hamiltonian framework for the calculus of variations in general. The purpose of the present work is to demonstrate, in the context of classical field theory, how this covariant Hamiltonian formalism may be space + time decomposed. It turns out that the resulting “instantaneous” Hamiltonian formalism is an infinite- dimensional version of Ostrogradskiǐ's theory and leads to the standard symplectic formulation of the initial value problem. The salient features of the analysis are: (i) the instantaneous Hamiltonian formalism does not depend upon the choice of Lepagean equivalent; (ii) the space + time decomposition can be performed either before or after the covariant Legendre transformation has been carried out, with equivalent results; (iii) the instantaneous Hamiltonian can be recovered in natural way from the multisymplectic structure inherent in the theory; and (iv) the space + time split symplectic structure lives on the space of Cauchy data for the evolution equations, as opposed to the space of solutions thereof.

Geometrical formulation of bond graph dynamics with application to mechanisms

A differential-geometric formulation of the dynamics associated with a bond graph model of energy-conserving physical systems is presented. Based on the generalized bond graph notation and the symplectic gyrator element, the causal procedure of constructing the bond graph dynamics is shown to fit with the symplectic formalism of classical mechanics. A simple mechanical example illustrates the geometric-differential formulation of its dynamics and the question of its order with respect to reduction procedures. For bond graphs containing dependent energy-storage fields, the multiple causality assignments are interpreted in differential-geometric terms as a set of coordinate charts of the constrained state-space. Finally, the bond graph dynamics of models of mechanisms is formulated, where the energy variables, the momenta, belong now to a certain cotangent bundle over the inertial positions associated with the bodies of the mechanism.

A differential algebraic approach to mechanics with perfect nonholonomic constraints

Differential algebra permits the generalization of several features of symplectic formalism to mechanics with perfect nonholonomic constraints.

A symplectic formalism for quantum scattering on the hyperboloid

The author studies the quantum scattering of geodesic curves on the hyperboloid via established techniques in geometric quantisation. The wave operators are shown to be the dual Radon transform, and the S matrix thus calculated is the same as the one from Coulomb scattering. The procedure used can be extended to treat any non-compact symmetric spaces since they involve only geometric and group theoretic arguments.

Symplectic Landau-Ginzburg fixed points and the localization problem

The symplectic σ-model formulation of the localization problem suggests that a fixed point of Sp( n + + n −)/Sp( n − symmetry describes the localization transition. We find such a fixed point in a 4 - ϵ expansion of a Landau-Ginzburg theory of symplectic symmetry. However, it is generally unstable, indicating a breakdown of universality, the σ-model, or smoothness in critical behavior as a function of spatial dimension.

Symplectic formulation of relativistic quantum mechanics

It is shown that the covariant harmonic oscillator formalism in the light-cone coordinate system discussed in previous papers is a realization of the symplectic group. It is shown in particular that the Lorentz transformation of the wave function along a given direction corresponds to a one-parameter subgroup of Sp(4). The diagonal form in the light-cone coordinate system is discussed in detail. The oscillator formalism is known to represent the Poincaré group for relativistic extended hadrons, while serving as a simple calculational device for basic high-energy hadronic phenomena. Likewise, the symplectic formulation given in the present paper may serve as the basic spacetime/momentum-energy symmetry for a relativistic quantum mechanics of bound-state quarks.