1,591 research outputs found
Variational approach to second-order impulsive dynamic equations on time scales
The aim of this paper is to employ variational techniques and critical point
theory to prove some conditions for the existence of solutions to nonlinear
impulsive dynamic equation with homogeneous Dirichlet boundary conditions. Also
we will be interested in the solutions of the impulsive nonlinear problem with
linear derivative dependence satisfying an impulsive condition.Comment: 17 page
Three Lectures: Nemd, Spam, and Shockwaves
We discuss three related subjects well suited to graduate research. The
first, Nonequilibrium molecular dynamics or "NEMD", makes possible the
simulation of atomistic systems driven by external fields, subject to dynamic
constraints, and thermostated so as to yield stationary nonequilibrium states.
The second subject, Smooth Particle Applied Mechanics or "SPAM", provides a
particle method, resembling molecular dynamics, but designed to solve continuum
problems. The numerical work is simplified because the SPAM particles obey
ordinary, rather than partial, differential equations. The interpolation method
used with SPAM is a powerful interpretive tool converting point particle
variables to twice-differentiable field variables. This interpolation method is
vital to the study and understanding of the third research topic we discuss,
strong shockwaves in dense fluids. Such shockwaves exhibit stationary
far-from-equilibrium states obtained with purely reversible Hamiltonian
mechanics. The SPAM interpolation method, applied to this molecular dynamics
problem, clearly demonstrates both the tensor character of kinetic temperature
and the time-delayed response of stress and heat flux to the strain rate and
temperature gradients. The dynamic Lyapunov instability of the shockwave
problem can be analyzed in a variety of ways, both with and without symmetry in
time. These three subjects suggest many topics suitable for graduate research
in nonlinear nonequilibrium problems.Comment: 40 pages, with 21 figures, as presented at the Granada Seminar on the
Foundations of Nonequilibrium Statistical Physics, 13-17 September, as three
lecture
On Minimum-time Paths of Bounded Curvature with Position-dependent Constraints
We consider the problem of a particle traveling from an initial configuration
to a final configuration (given by a point in the plane along with a prescribed
velocity vector) in minimum time with non-homogeneous velocity and with
constraints on the minimum turning radius of the particle over multiple regions
of the state space. Necessary conditions for optimality of these paths are
derived to characterize the nature of optimal paths, both when the particle is
inside a region and when it crosses boundaries between neighboring regions.
These conditions are used to characterize families of optimal and nonoptimal
paths. Among the optimality conditions, we derive a "refraction" law at the
boundary of the regions that generalizes the so-called Snell's law of
refraction in optics to the case of paths with bounded curvature. Tools
employed to deduce our results include recent principles of optimality for
hybrid systems. The results are validated numerically.Comment: Expanded version of paper in Automatic
Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3)
Optimal control problems are formulated and efficient computational
procedures are proposed for combined orbital and rotational maneuvers of a
rigid body in three dimensions. The rigid body is assumed to act under the
influence of forces and moments that arise from a potential and from control
forces and moments. The key features of this paper are its use of computational
procedures that are guaranteed to preserve the geometry of the optimal
solutions. The theoretical basis for the computational procedures is
summarized, and examples of optimal spacecraft maneuvers are presented.Comment: IEEE Conference on Decision and Control, 2006. 6 pages, 19 figure
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