3,424 research outputs found
Spatial discretization of partial differential equations with integrals
We consider the problem of constructing spatial finite difference
approximations on a fixed, arbitrary grid, which have analogues of any number
of integrals of the partial differential equation and of some of its
symmetries. A basis for the space of of such difference operators is
constructed; most cases of interest involve a single such basis element. (The
``Arakawa'' Jacobian is such an element.) We show how the topology of the grid
affects the complexity of the operators.Comment: 24 pages, LaTeX sourc
Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci
In this paper we continue our study of bifurcations of solutions of
boundary-value problems for symplectic maps arising as Hamiltonian
diffeomorphisms. These have been shown to be connected to catastrophe theory
via generating functions and ordinary and reversal phase space symmetries have
been considered. Here we present a convenient, coordinate free framework to
analyse separated Lagrangian boundary value problems which include classical
Dirichlet, Neumann and Robin boundary value problems. The framework is then
used to {prove the existence of obstructions arising from} conformal symplectic
symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary
value problems. Under non-degeneracy conditions, a group action by conformal
symplectic symmetries has the effect that the flow map cannot degenerate in a
direction which is tangential to the action. This imposes restrictions on which
singularities can occur in boundary value problems. Our results generalise
classical results about conjugate loci on Riemannian manifolds to a large class
of Hamiltonian boundary value problems with, for example, scaling symmetries
A note on the motion of surfaces
We study the motion of surfaces in an intrinsic formulation in which the
surface is described by its metric and curvature tensors. The evolution
equations for the six quantities contained in these tensors are reduced in
number in two cases: (i) for arbitrary surfaces, we use principal coordinates
to obtain two equations for the two principal curvatures, highlighting the
similarity with the equations of motion of a plane curve; and (ii) for surfaces
with spatially constant negative curvature, we use parameterization by
Tchebyshev nets to reduce to a single evolution equation. We also obtain
necessary and sufficient conditions for a surface to maintain spatially
constant negative curvature as it moves. One choice for the surface's normal
motion leads to the modified-Korteweg de Vries equation,the appearance of which
is explained by connections to the AKNS hierarchy and the motion of space
curves.Comment: 10 pages, compile with AMSTEX. Two figures available from the author
A minimal-variable symplectic integrator on spheres
We construct a symplectic, globally defined, minimal-coordinate, equivariant
integrator on products of 2-spheres. Examples of corresponding Hamiltonian
systems, called spin systems, include the reduced free rigid body, the motion
of point vortices on a sphere, and the classical Heisenberg spin chain, a
spatial discretisation of the Landau-Lifschitz equation. The existence of such
an integrator is remarkable, as the sphere is neither a vector space, nor a
cotangent bundle, has no global coordinate chart, and its symplectic form is
not even exact. Moreover, the formulation of the integrator is very simple, and
resembles the geodesic midpoint method, although the latter is not symplectic
Integrable four-dimensional symplectic maps of standard type
We search for rational, four-dimensional maps of standard type (x_{n+1} -
2x_n + x_{n-1} = eps f(x,eps)) possessing one or two polynomial integrals.
There are no non-trivial maps corresponding to cubic oscillators, but we find a
four-parameter family of such maps corresponding to quartic oscillators. This
seems to be the only such example.Comment: 5 pages, compile with plain TEX. No figures. To appear in Physics
Letters
Symplectic integrators for spin systems
We present a symplectic integrator, based on the canonical midpoint rule, for
classical spin systems in which each spin is a unit vector in .
Unlike splitting methods, it is defined for all Hamiltonians, and is
-equivariant. It is a rare example of a generating function for
symplectic maps of a noncanonical phase space. It yields an integrable
discretization of the reduced motion of a free rigid body
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