53 research outputs found
Discrete Hamilton-Jacobi Theory
We develop a discrete analogue of Hamilton-Jacobi theory in the framework of
discrete Hamiltonian mechanics. The resulting discrete Hamilton-Jacobi equation
is discrete only in time. We describe a discrete analogue of Jacobi's solution
and also prove a discrete version of the geometric Hamilton-Jacobi theorem. The
theory applied to discrete linear Hamiltonian systems yields the discrete
Riccati equation as a special case of the discrete Hamilton-Jacobi equation. We
also apply the theory to discrete optimal control problems, and recover some
well-known results, such as the Bellman equation (discrete-time HJB equation)
of dynamic programming and its relation to the costate variable in the
Pontryagin maximum principle. This relationship between the discrete
Hamilton-Jacobi equation and Bellman equation is exploited to derive a
generalized form of the Bellman equation that has controls at internal stages.Comment: 26 pages, 2 figure
Controlled Lagrangians and Potential Shaping for Stabilization of Discrete Mechanical Systems
The method of controlled Lagrangians for discrete mechanical systems is
extended to include potential shaping in order to achieve complete state-space
asymptotic stabilization. New terms in the controlled shape equation that are
necessary for matching in the discrete context are introduced. The theory is
illustrated with the problem of stabilization of the cart-pendulum system on an
incline. We also discuss digital and model predictive control.Comment: IEEE Conference on Decision and Control, 2006 6 pages, 4 figure
Controlled Lagrangians and Stabilization of the Discrete Cart-Pendulum System
Matching techniques are developed for discrete
mechanical systems with symmetry. We describe new phenomena
that arise in the controlled Lagrangian approach for mechanical
systems in the discrete context. In particular, one needs
to either make an appropriate selection of momentum levels or
introduce a new parameter into the controlled Lagrangian to
complete the matching procedure. We also discuss digital and
model predictive control
Matching and stabilization of discrete mechanical systems
Controlled Lagrangian and matching techniques are developed for the stabilization of equilibria of discrete mechanical systems
with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in
the discrete context that are not present in the continuous theory. Specifically, a nonconservative force that is necessary for
matching in the discrete setting is introduced. The paper also discusses digital and model predictive controllers
Discrete Routh Reduction
This paper develops the theory of abelian Routh reduction for discrete
mechanical systems and applies it to the variational integration of mechanical
systems with abelian symmetry. The reduction of variational Runge-Kutta
discretizations is considered, as well as the extent to which symmetry
reduction and discretization commute. These reduced methods allow the direct
simulation of dynamical features such as relative equilibria and relative
periodic orbits that can be obscured or difficult to identify in the unreduced
dynamics. The methods are demonstrated for the dynamics of an Earth orbiting
satellite with a non-spherical correction, as well as the double
spherical pendulum. The problem is interesting because in the unreduced
picture, geometric phases inherent in the model and those due to numerical
discretization can be hard to distinguish, but this issue does not appear in
the reduced algorithm, where one can directly observe interesting dynamical
structures in the reduced phase space (the cotangent bundle of shape space), in
which the geometric phases have been removed. The main feature of the double
spherical pendulum example is that it has a nontrivial magnetic term in its
reduced symplectic form. Our method is still efficient as it can directly
handle the essential non-canonical nature of the symplectic structure. In
contrast, a traditional symplectic method for canonical systems could require
repeated coordinate changes if one is evoking Darboux' theorem to transform the
symplectic structure into canonical form, thereby incurring additional
computational cost. Our method allows one to design reduced symplectic
integrators in a natural way, despite the noncanonical nature of the symplectic
structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added,
fixed typo
Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints
We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian)
systems, a generalized formulation of Lagrangian mechanics that can incorporate
degenerate Lagrangians as well as holonomic and nonholonomic constraints. We
refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi
equation. For non-degenerate Lagrangian systems with nonholonomic constraints,
the theory specializes to the recently developed nonholonomic Hamilton-Jacobi
theory. We are particularly interested in applications to a certain class of
degenerate nonholonomic Lagrangian systems with symmetries, which we refer to
as weakly degenerate Chaplygin systems, that arise as simplified models of
nonholonomic mechanical systems; these systems are shown to reduce to
non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian
systems defined with non-closed two-forms. Accordingly, the
Dirac-Hamilton-Jacobi equation reduces to a variant of the nonholonomic
Hamilton-Jacobi equation associated with the reduced system. We illustrate
through a few examples how the Dirac-Hamilton-Jacobi equation can be used to
exactly integrate the equations of motion.Comment: 44 pages, 3 figure
Coupling Non-Gravitational Fields with Simplicial Spacetimes
The inclusion of source terms in discrete gravity is a long-standing problem.
Providing a consistent coupling of source to the lattice in Regge Calculus (RC)
yields a robust unstructured spacetime mesh applicable to both numerical
relativity and quantum gravity. RC provides a particularly insightful approach
to this problem with its purely geometric representation of spacetime. The
simplicial building blocks of RC enable us to represent all matter and fields
in a coordinate-free manner. We provide an interpretation of RC as a discrete
exterior calculus framework into which non-gravitational fields naturally
couple with the simplicial lattice. Using this approach we obtain a consistent
mapping of the continuum action for non-gravitational fields to the Regge
lattice. In this paper we apply this framework to scalar, vector and tensor
fields. In particular we reconstruct the lattice action for (1) the scalar
field, (2) Maxwell field tensor and (3) Dirac particles. The straightforward
application of our discretization techniques to these three fields demonstrates
a universal implementation of coupling source to the lattice in Regge calculus.Comment: 10 pages, no figures, Latex, fixed typos and minor corrections
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