A universal instability of many-dimensional oscillator systems

Publisher's description: The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule.

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Lectures on Mechanics

Time-reversal symmetry in dynamical systems: a survey

Reduction in the category of Poisson manifolds is defined and some basic properties are derived.
The context is chosen to include the usual theorems on reduction of symplectic manifolds,
as well as results such as the Dirac bracket and the reduction to the Lie-Poisson bracket.

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Reduction of Poisson manifolds

/pdf/homoclinic-orbits-in-reversible-systems-and-their-123fy30vuj.pdf

Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics

On averaging, reduction, and symmetry in hamiltonian systems

/pdf/reduction-of-the-semisimple-1-1-resonance-51qh4733ta.pdf

Reduction of the semisimple 1:1 resonance

This paper surveys some recent techniques and results for Hamiltonian systems in the context of the Henon-Heiles Hamiltonian. The paper attempts to give some reasonable interpretation of various numerical (computer) results alongside recent mathematical techniques and results presented as they apply to this Hamiltonian. In particular, various computed periodic orbits are identified with geometrically constructed periodic orbits and rigorous conclusions concerning the stability status of the orbit as the energy of the system changes are indicated whenever possible. The paper concludes with a discussion of some related Hamiltonians and suggested computer experiments.

David L. Rod

Papers

On averaging, reduction, and symmetry in hamiltonian systems

Reduction of the semisimple 1:1 resonance

A survey of the Hénon-Heiles Hamiltonian with applications to related examples

Pathology in dynamical systems. III. Analytic Hamiltonians

Isolated unstable periodic orbits