2,348 research outputs found
Poisson integrators
An overview of Hamiltonian systems with noncanonical Poisson structures is
given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are
presented. Numerical integrators using generating functions, Hamiltonian
splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems
and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems
and the discrete gradient methods are also presented.Comment: 30 page
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
Discrete Hamiltonian Variational Integrators
We consider the continuous and discrete-time Hamilton's variational principle
on phase space, and characterize the exact discrete Hamiltonian which provides
an exact correspondence between discrete and continuous Hamiltonian mechanics.
The variational characterization of the exact discrete Hamiltonian naturally
leads to a class of generalized Galerkin Hamiltonian variational integrators,
which include the symplectic partitioned Runge-Kutta methods. We also
characterize the group invariance properties of discrete Hamiltonians which
lead to a discrete Noether's theorem.Comment: 23 page
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