4,207 research outputs found
Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms
This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a
discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamiltonās principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noetherās theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators
Numerical integration of variational equations
We present and compare different numerical schemes for the integration of the
variational equations of autonomous Hamiltonian systems whose kinetic energy is
quadratic in the generalized momenta and whose potential is a function of the
generalized positions. We apply these techniques to Hamiltonian systems of
various degrees of freedom, and investigate their efficiency in accurately
reproducing well-known properties of chaos indicators like the Lyapunov
Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs).
We find that the best numerical performance is exhibited by the
\textit{`tangent map (TM) method'}, a scheme based on symplectic integration
techniques which proves to be optimal in speed and accuracy. According to this
method, a symplectic integrator is used to approximate the solution of the
Hamilton's equations of motion by the repeated action of a symplectic map ,
while the corresponding tangent map , is used for the integration of the
variational equations. A simple and systematic technique to construct is
also presented.Comment: 27 pages, 11 figures, to appear in Phys. Rev.
R-adaptive multisymplectic and variational integrators
Moving mesh methods (also called r-adaptive methods) are space-adaptive
strategies used for the numerical simulation of time-dependent partial
differential equations. These methods keep the total number of mesh points
fixed during the simulation, but redistribute them over time to follow the
areas where a higher mesh point density is required. There are a very limited
number of moving mesh methods designed for solving field-theoretic partial
differential equations, and the numerical analysis of the resulting schemes is
challenging. In this paper we present two ways to construct r-adaptive
variational and multisymplectic integrators for (1+1)-dimensional Lagrangian
field theories. The first method uses a variational discretization of the
physical equations and the mesh equations are then coupled in a way typical of
the existing r-adaptive schemes. The second method treats the mesh points as
pseudo-particles and incorporates their dynamics directly into the variational
principle. A user-specified adaptation strategy is then enforced through
Lagrange multipliers as a constraint on the dynamics of both the physical field
and the mesh points. We discuss the advantages and limitations of our methods.
Numerical results for the Sine-Gordon equation are also presented.Comment: 65 pages, 13 figure
A variational problem on Stiefel manifolds
In their paper on discrete analogues of some classical systems such as the
rigid body and the geodesic flow on an ellipsoid, Moser and Veselov introduced
their analysis in the general context of flows on Stiefel manifolds. We
consider here a general class of continuous time, quadratic cost, optimal
control problems on Stiefel manifolds, which in the extreme dimensions again
yield these classical physical geodesic flows. We have already shown that this
optimal control setting gives a new symmetric representation of the rigid body
flow and in this paper we extend this representation to the geodesic flow on
the ellipsoid and the more general Stiefel manifold case. The metric we choose
on the Stiefel manifolds is the same as that used in the symmetric
representation of the rigid body flow and that used by Moser and Veselov. In
the extreme cases of the ellipsoid and the rigid body, the geodesic flows are
known to be integrable. We obtain the extremal flows using both variational and
optimal control approaches and elucidate the structure of the flows on general
Stiefel manifolds.Comment: 30 page
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