Tomoki Ohsawa

Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization

Abstract We develop Hamilton–Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton–Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton–Jacobi equation associated with the Hamiltonized system also solve the nonholonomic Hamilton–Jacobi equation associated with the original Chaplygin system. The results are illustrated through several examples.

Nonholonomic Hamilton-Jacobi equation and integrability

We discuss an extension of the Hamilton-Jacobi theory to nonholonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. We give an intrinsic proof of a nonholonomic analogue of the Hamilton-Jacobi theorem. Our intrinsic proof clarifies the difference from the conventional Hamilton-Jacobi theory for unconstrained systems. The proof also helps us identify a geometric meaning of the conditions on the solutions of the Hamilton-Jacobi equation that arise from nonholonomic constraints. The major advantage of our result is that it provides us with a method of integrating the equations of motion just as the unconstrained Hamilton-Jacobi theory does. In particular, we build on the work by Iglesias-Ponte, de Leon, and Martin de Diego [15] so that the conventional method of separation of variables applies to some nonholonomic mechanical systems. We also show a way to apply our result to systems to which separation of variables does not apply.

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 Jacobis 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.

Discrete Hamilton-Jacobi theory and discrete optimal control

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. The correspondence between discrete and continuous Hamiltonian mechanics naturally gives rise to a discrete analogue of Jacobis solution to the Hamilton-Jacobi equation. We prove discrete analogues of Jacobis solution to the Hamilton-Jacobi equation and of the geometric Hamilton-Jacobi theorem. These results are readily applied to the discrete optimal control setting, and some well-known results in discrete optimal control theory, such as the Bellman equation, follow immediately. We also apply the theory to discrete linear Hamiltonian systems, and show that the discrete Riccati equation follows as a special case.

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.

Symmetry Reduction of Optimal Control Systems and Principal Connections

This paper explores the role of symmetries and reduction in nonlinear control and optimal control systems. The focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry. In particular, we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups, as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting. Our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems. Affine and kinematic optimal control systems are of particular interest: We explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems.

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.

The Siegel Upper Half Space is a Marsden–Weinstein Quotient: Symplectic Reduction and Gaussian Wave Packets

AbstractWe show that the Siegel upper half space

Discrete Dirac Structures and Implicit Discrete Lagrangian and Hamiltonian Systems

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