6,399 research outputs found

### Routh reduction and the class of magnetic Lagrangian systems

In this paper, some new aspects related to Routh reduction of Lagrangian
systems with symmetry are discussed. The main result of this paper is the
introduction of a new concept of transformation that is applicable to systems
obtained after Routh reduction of Lagrangian systems with symmetry, so-called
magnetic Lagrangian systems. We use these transformations in order to show
that, under suitable conditions, the reduction with respect to a (full)
semi-direct product group is equivalent to the reduction with respect to an
Abelian normal subgroup. The results in this paper are closely related to the
more general theory of Routh reduction by stages.Comment: 23 page

### Generalized Hamiltonian mechanics

Our purpose is to generalize Hamiltonian mechanics t the case in which the energy function (Hamiltonian), H , is a distribution (generalized function) in the sense of Schwartz. We follow the same general program as in the smooth case. Familiarity with the smooth case is helpful, although we have striven to make the exposition self-contained, starting from calculus on manifold

### Gauge Theory for Finite-Dimensional Dynamical Systems

Gauge theory is a well-established concept in quantum physics,
electrodynamics, and cosmology. This theory has recently proliferated into new
areas, such as mechanics and astrodynamics. In this paper, we discuss a few
applications of gauge theory in finite-dimensional dynamical systems with
implications to numerical integration of differential equations. We distinguish
between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry
is, in essence, re-scaling of the independent variable, while descriptive gauge
symmetry is a Yang-Mills-like transformation of the velocity vector field,
adapted to finite-dimensional systems. We show that a simple gauge
transformation of multiple harmonic oscillators driven by chaotic processes can
render an apparently "disordered" flow into a regular dynamical process, and
that there exists a remarkable connection between gauge transformations and
reduction theory of ordinary differential equations. Throughout the discussion,
we demonstrate the main ideas by considering examples from diverse engineering
and scientific fields, including quantum mechanics, chemistry, rigid-body
dynamics and information theory

### Discrete Routh Reduction

This paper develops the theory of abelian Routh reduction for discrete
mechanical systems and applies it to the variational integration of mechanical
systems with abelian symmetry. The reduction of variational Runge-Kutta
discretizations is considered, as well as the extent to which symmetry
reduction and discretization commute. These reduced methods allow the direct
simulation of dynamical features such as relative equilibria and relative
periodic orbits that can be obscured or difficult to identify in the unreduced
dynamics. The methods are demonstrated for the dynamics of an Earth orbiting
satellite with a non-spherical $J_2$ correction, as well as the double
spherical pendulum. The $J_2$ problem is interesting because in the unreduced
picture, geometric phases inherent in the model and those due to numerical
discretization can be hard to distinguish, but this issue does not appear in
the reduced algorithm, where one can directly observe interesting dynamical
structures in the reduced phase space (the cotangent bundle of shape space), in
which the geometric phases have been removed. The main feature of the double
spherical pendulum example is that it has a nontrivial magnetic term in its
reduced symplectic form. Our method is still efficient as it can directly
handle the essential non-canonical nature of the symplectic structure. In
contrast, a traditional symplectic method for canonical systems could require
repeated coordinate changes if one is evoking Darboux' theorem to transform the
symplectic structure into canonical form, thereby incurring additional
computational cost. Our method allows one to design reduced symplectic
integrators in a natural way, despite the noncanonical nature of the symplectic
structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added,
fixed typo

### Symplectic reduction and topology for applications in classical molecular dynamics

This paper aims to introduce readers with backgrounds in classical molecular dynamics to some ideas in geometric mechanics that may be useful. This is done through some simple but specific examples: (i) the separation of the rotational and internal energies in an arbitrarily floppy N-body system and (ii) the reduction of the phase space accompanying the change from the laboratory coordinate system to the center of mass coordinate system relevant to molecular collision dynamics. For the case of two-body molecular systems constrained to a plane, symplectic reduction is employed to demonstrate explicitly the separation of translational, rotational, and internal energies and the corresponding reductions of the phase space describing the dynamics for Hamiltonian systems with symmetry. Further, by examining the topology of the energy-momentum map, a unified treatment is presented of the reduction results for the description of (i) the classical dynamics of rotating and vibrating diatomic molecules, which correspond to bound trajectories and (ii) the classical dynamics of atomâ€“atom collisions, which correspond to scattering trajectories. This provides a framework for the treatment of the dynamics of larger N-body systems, including the dynamics of larger rotating and vibrating polyatomic molecular systems and the dynamics of moleculeâ€“molecule collisions

### Discrete mechanics and variational integrators

This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Rungeâ€“Kutta schemes are presented

### Resonant Geometric Phases for Soliton Equations

The goal of the present paper is to introduce a multidimensional generalization of asymptotic reduction given in a paper by Alber and Marsden [1992], to use this to obtain a new class of solutions that we call resonant solitons, and to study the corresponding geometric phases. The term "resonant solitons" is used because those solutions correspond to a spectrum with multiple points, and they also represent a dividing solution between two different types of solitons. In this sense, these new solutions are degenerate and, as such, will be considered as singular points in the moduli space of solitons

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