719 research outputs found
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
Higher Order Variational Integrators: a polynomial approach
We reconsider the variational derivation of symplectic partitioned
Runge-Kutta schemes. Such type of variational integrators are of great
importance since they integrate mechanical systems with high order accuracy
while preserving the structural properties of these systems, like the
symplectic form, the evolution of the momentum maps or the energy behaviour.
Also they are easily applicable to optimal control problems based on mechanical
systems as proposed in Ober-Bl\"obaum et al. [2011].
Following the same approach, we develop a family of variational integrators
to which we refer as symplectic Galerkin schemes in contrast to symplectic
partitioned Runge-Kutta. These two families of integrators are, in principle
and by construction, different one from the other. Furthermore, the symplectic
Galerkin family can as easily be applied in optimal control problems, for which
Campos et al. [2012b] is a particular case.Comment: 12 pages, 1 table, 23rd Congress on Differential Equations and
Applications, CEDYA 201
High order variational integrators in the optimal control of mechanical systems
In recent years, much effort in designing numerical methods for the
simulation and optimization of mechanical systems has been put into schemes
which are structure preserving. One particular class are variational
integrators which are momentum preserving and symplectic. In this article, we
develop two high order variational integrators which distinguish themselves in
the dimension of the underling space of approximation and we investigate their
application to finite-dimensional optimal control problems posed with
mechanical systems. The convergence of state and control variables of the
approximated problem is shown. Furthermore, by analyzing the adjoint systems of
the optimal control problem and its discretized counterpart, we prove that, for
these particular integrators, dualization and discretization commute.Comment: 25 pages, 9 figures, 1 table, submitted to DCDS-
Discrete mechanics and variational integrators
This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented
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