The Energy-Momentum Method

This paper develops the energy momentum methodJor studying stability and bifurcation of Lagrangian and Hamiltonian systems with symmetry. The method was specifically designed to deal with the stability of rotating structures. The relation with the energy-Casimir method is given and the energy-momentum method is shown to be more general. Stability of rigid body motion is given 10 illustrate the method. Some discussion of its applicability to general rotating systems and block diagonalization is also given

Stress tensors, Riemannian metrics and the alternative descriptions in elasticity

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Symplectic-energy-momentum preserving variational integrators

The purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving. To do this, a space–time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy. Criteria for the solvability of the time steps and some numerical examples are given

Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments

We consider Hamiltonian closures of the Vlasov equation using the phase-space moments of the distribution function. We provide some conditions on the closures imposed by the Jacobi identity. We completely solve some families of examples. As a result, we show that imposing that the resulting reduced system preserves the Hamiltonian character of the parent model shapes its phase space by creating a set of Casimir invariants as a direct consequence of the Jacobi identity

The Dynamical History of Chariklo and its Rings

Chariklo is the only small Solar system body confirmed to have rings. Given the instability of its orbit, the presence of rings is surprising, and their origin remains poorly understood. In this work, we study the dynamical history of the Chariklo system by integrating almost 36,000 Chariklo clones backwards in time for one Gyr under the influence of the Sun and the four giant planets. By recording all close encounters between the clones and planets, we investigate the likelihood that Chariklo's rings could have survived since its capture to the Centaur population. Our results reveal that Chariklo's orbit occupies a region of stable chaos, resulting in its orbit being marginally more stable than those of the other Centaurs. Despite this, we find that it was most likely captured to the Centaur population within the last 20 Myr, and that its orbital evolution has been continually punctuated by regular close encounters with the giant planets. The great majority (> 99%) of those encounters within one Hill radius of the planet have only a small effect on the rings. We conclude that close encounters with giant planets have not had a significant effect on the ring structure. Encounters within the Roche limit of the giant planets are rare, making ring creation through tidal disruption unlikely

Variational Multisymplectic Formulations of Nonsmooth Continuum Mechanics

This paper develops the foundations of the multisymplectic formulation of nonsmooth continuum mechanics. It may be regarded as a PDE generalization of previous techniques that developed a variational approach to collision problems. These methods have already proved of value in computational mechanics, particularly in the development of asynchronous integrators and efficient collision methods. The present formulation also includes solid-fluid interactions and material interfaces and, in addition, lays the groundwork for a treatment of shocks

Optimal trajectory generation in ocean flows

In this paper it is shown that Lagrangian Coherent Structures (LCS) are useful in determining near optimal trajectories for autonomous underwater gliders in a dynamic ocean environment. This opens the opportunity for optimal path planning of autonomous underwater vehicles by studying the global flow geometry via dynamical systems methods. Optimal glider paths were computed for a 2-dimensional kinematic model of an end-point glider problem. Numerical solutions to the optimal control problem were obtained using Nonlinear Trajectory Generation (NTG) software. The resulting solution is compared to corresponding results on LCS obtained using the Direct Lyapunov Exponent method. The velocity data used for these computations was obtained from measurements taken in August, 2000, by HF-Radar stations located around Monterey Bay, CA

Lagrangian coherent structures in n-dimensional systems

Numerical simulations and experimental observations reveal that unsteady fluid systems can be divided into regions of qualitatively different dynamics. The key to understanding transport and stirring is to identify the dynamic boundaries between these almost-invariant regions. Recently, ridges in finite-time Lyapunov exponent fields have been used to define such hyperbolic, almost material, Lagrangian coherent structures in two-dimensional systems. The objective of this paper is to develop and apply a similar theory in higher dimensional spaces. While the separatrix nature of these structures is their most important property, a necessary condition is their almost material nature. This property is addressed in this paper. These results are applied to a model of Rayleigh-Bénard convection based on a three-dimensional extension of the model of Solomon and Gollub

Stability Analysis of a Rigid Body with Attached Geometrically Nonlinear Rod by the Energy-Momentum Method

This paper applies the energy-momentum method to the problem of nonlinear stability of relative equilibria of a rigid body with attached flexible appendage in a uniformly rotating state. The appendage is modeled as a geometrically exact rod which allows for finite bending, shearing and twist in three dimensions. Application of the energy-momentum method to this example depends crucially on a special choice of variables in terms of which the second variation block diagonalizes into blocks associated with rigid body modes and internal vibration modes respectively. The analysis yields a nonlinear stability result which states that relative equilibria are nonlinearly stable provided that; (i) the angular velocity is bounded above by the square root of the minimum eigenvalue of an associated linear operator and, (ii) the whole assemblage is rotating about the minimum axis of inertia

Magnetic fields and differential rotation on the pre-main sequence

Maps of magnetic field topologies of rapidly rotating stars obtained over the last decade or so have provided unique insight into the operation of stellar dynamos. However, for solar-type stars many of the targets imaged to date have been lower-mass zero-age main sequence stars. We present magnetic maps and differential rotation measurements of two-higher mass pre-main sequence stars HD 106506 (~10 Myrs) and HD 141943 (~15 Myrs). These stars should evolve into mid/late F-stars with predicted high differential rotation and little magnetic activity. We investigate what effect the extended convection zones of these pre-main sequence stars has on their differential rotation and magnetic topologies. ©2009 American Institute of Physic