50 research outputs found
Integrability of a conducting elastic rod in a magnetic field
We consider the equilibrium equations for a conducting elastic rod placed in
a uniform magnetic field, motivated by the problem of electrodynamic space
tethers. When expressed in body coordinates the equations are found to sit in a
hierarchy of non-canonical Hamiltonian systems involving an increasing number
of vector fields. These systems, which include the classical Euler and
Kirchhoff rods, are shown to be completely integrable in the case of a
transversely isotropic rod; they are in fact generated by a Lax pair. For the
magnetic rod this gives a physical interpretation to a previously proposed
abstract nine-dimensional integrable system. We use the conserved quantities to
reduce the equations to a four-dimensional canonical Hamiltonian system,
allowing the geometry of the phase space to be investigated through Poincar\'e
sections. In the special case where the force in the rod is aligned with the
magnetic field the system turns out to be superintegrable, meaning that the
phase space breaks down completely into periodic orbits, corresponding to
straight twisted rods.Comment: 19 pages, 1 figur
Phase space structures governing reaction dynamics in rotating molecules
Recently the phase space structures governing reaction dynamics in
Hamiltonian systems have been identified and algorithms for their explicit
construction have been developed. These phase space structures are induced by
saddle type equilibrium points which are characteristic for reaction type
dynamics. Their construction is based on a Poincar{\'e}-Birkhoff normal form.
Using tools from the geometric theory of Hamiltonian systems and their
reduction we show in this paper how the construction of these phase space
structures can be generalized to the case of the relative equilibria of a
rotational symmetry reduced -body system. As rotations almost always play an
important role in the reaction dynamics of molecules the approach presented in
this paper is of great relevance for applications.Comment: 28 pages, 7 figures, pdflate
Perturbations of Superintegrable Systems
A superintegrable system has more integrals of motion than degrees d of freedom. The quasi-periodic motions then spin around tori of dimension n < d. Already under integrable perturbations almost all n-tori will break up; in the non-degenerate case the resulting d-tori have n fast and d −n slow frequencies. Such d-parameter families of d-tori do survive Hamiltonian perturbations as Cantor families of d-tori. A perturbation of a superintegrable system that admits a better approximation by a non-degenerate integrable perturbation of the superintegrable system is said to remove the degeneracy. In the minimal case d = n+1 this can be achieved by means of averaging, but the more integrals of motion the superintegrable system admits the more difficult becomes the perturbation analysis