Some basic properties of infinite dimensional Hamiltonian systems

We consider some fundamental properties of infinite dimensional Hamiltonian systems, both linear and nonlinear. For exemple, in the case of linear systems, we prove a symplectic version of the teorem of M. Stone. In the general case we establish conservation of energy and the moment function for system with symmetry. (The moment function was introduced by B. Kostant and J .M. Souriau). For infinite dimensional systems these conservation laws are more delicate than those for finite dimensional systems because we are dealing with partial as opposed to ordinary differential equations

Is SGR 1900+14 a Magnetar?

We present RXTE observations of the soft gamma--ray repeater SGR 1900+14 taken September 4-18, 1996, nearly 2 years before the 1998 active period of the source. The pulsar period (P) of 5.1558199 +/- 0.0000029 s and period derivative (Pdot) of (6.0 +/- 1.0) X 10^-11 s/s measured during the 2-week observation are consistent with the mean Pdot of (6.126 +/- 0.006) X 10^-11 s/s over the time up to the commencement of the active period. This Pdot is less than half that of (12.77 +/- 0.01) X 10^-11 s/s observed during and after the active period. If magnetic dipole radiation were the primary cause of the pulsar spindown, the implied pulsar magnetic field would exceed the critical field of 4.4 X 10^13 G by more than an order of magnitude, and such field estimates for this and other SGRs have been offered as evidence that the SGRs are magnetars, in which the neutron star magnetic energy exceeds the rotational energy. The observed doubling of Pdot, however, would suggest that the pulsar magnetic field energy increased by more than 100% as the source entered an active phase, which seems very hard to reconcile with models in which the SGR bursts are powered by the release of magnetic energy. Because of this, we suggest that the spindown of SGR pulsars is not driven by magnetic dipole radiation, but by some other process, most likely a relativistic wind. The Pdot, therefore, does not provide a measure of the pulsar magnetic field strength, nor evidence for a magnetar.Comment: 14 pages, aasms4 latex, figures 1 & 2 changed, accepted by ApJ letter

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

Magnetic Field Limits on SGRs

We measure the period and spin-down rate for SGR 1900+14 during the quiescient period two years before the recent interval of renewed burst activity. We find that the spin-down rate doubled during the burst activity which is inconsistent with both mangetic dipole driven spin down and a magnetic field energy source for the bursts. We also show that SGRs 1900+14 and 1806-20 have braking indices of $\sim$1 which indicate that the spin-down is due to wind torques and not magnetic dipole radiation. We further show that a combination of dipole radiation, and wind luminosity, coupled with estimated ages and present spin parameters, imply that the magnetic fields of SGRs 1900+14 and 1806-20 are less than the critical field of 4$\times10^{13}$ G and that the efficiency for conversion of wind luminosity to x-ray luminosity is <2%.Comment: 5 pages, 2 figures submitted to 5th Huntsville GRB Symposium proceeding

Hamiltonian systems with symmetry, coadjoint orbits and plasma physics

The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the Lie-Poisson structure on the dual of a Lie algebra. These results are applied to plasma physics. We show in three steps how the Maxwell-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. First, the Poisson-Vlasov equations are shown to be in Hamiltonian form relative to the Lie-Poisson bracket on the dual of the (nite dimensional) Lie algebra of innitesimal canonical transformations. Then we write Maxwell's equations in Hamiltonian form using the canonical symplectic structure on the phase space of the electromagnetic elds, regarded as a gauge theory. In the last step we couple these two systems via the reduction procedure for interacting systems. We also show that two other standard models in plasma physics, ideal MHD and two- uid electrodynamics, can be written in Hamiltonian form using similar group theoretic techniques

Cometary Astrometry

Modern techniques for making cometary astrometric observations, reducing these observations, using accurate reference star catalogs, and computing precise orbits and ephemerides are discussed in detail and recommendations and suggestions are given in each area

Generalized poisson brackets and nonlinear Liapunov stability application to reduces mhd

A method is presented for obtaining Liapunov functionals (LF) and proving nonlinear stability. The method uses the generalized Poisson bracket (GPB) formulation of Hamiltonian dynamics. As an illustration, certain stationary solutions of ideal reduced MHD (RMHD) are shown to be nonlinearly stable. This includes Grad-Shafranov and Alfven solutions