Nonholonomic Hamilton-Jacobi equation and integrability
Tomoki Ohsawa Anthony M. Bloch
Journal of Geometric Mechanics 2009, 1(4): 461-481 doi: 10.3934/jgm.2009.1.461
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 Léon, and Martín 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.
keywords: Hamilton-Jacobi equation Nonholonomic mechanics Integrable systems.
Dynamics of generalized Euler tops with constraints
Dmitry V. Zenkov Anthony M. Bloch
Conference Publications 2001, 2001(Special): 398-405 doi: 10.3934/proc.2001.2001.398
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Controlled Lagrangians and stabilization of discrete mechanical systems
Anthony M. Bloch Melvin Leok Jerrold E. Marsden Dmitry V. Zenkov
Discrete & Continuous Dynamical Systems - S 2010, 3(1): 19-36 doi: 10.3934/dcdss.2010.3.19
Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria of discrete mechanical systems with symmetry and equilibria of discrete mechanical systems with broken symmetry. Unexpected phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic shaping procedure. New terms in the controlled shape equation that are necessary for potential shaping in the discrete setting are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline, and the application of the theory to the construction of digital feedback controllers is also discussed.
keywords: digital control. discrete mechanics Matching
Embedded geodesic problems and optimal control for matrix Lie groups
Anthony M. Bloch Peter E. Crouch Nikolaj Nordkvist Amit K. Sanyal
Journal of Geometric Mechanics 2011, 3(2): 197-223 doi: 10.3934/jgm.2011.3.197
This paper is devoted to a detailed analysis of the geodesic problem on matrix Lie groups, with left invariant metric, by examining representations of embeddings of geodesic flows in suitable vector spaces. We show how these representations generate extremals for optimal control problems. In particular we discuss in detail the symmetric representation of the so-called $n$-dimensional rigid body equation and its relation to the more classical Euler description. We detail invariant manifolds of these flows on which we are able to define a strict notion of equivalence between representations, and identify naturally induced symplectic structures.
keywords: generalized rigid body mechanics. Optimal control geodesics
Variational Integrators for Hamiltonizable Nonholonomic Systems
Oscar E. Fernandez Anthony M. Bloch P. J. Olver
Journal of Geometric Mechanics 2012, 4(2): 137-163 doi: 10.3934/jgm.2012.4.137
We report on new applications of the Poincaré and Sundman time-transformations to the simulation of nonholonomic systems. These transformations are here applied to nonholonomic mechanical systems known to be Hamiltonizable (briefly, nonholonomic systems whose constrained mechanics are Hamiltonian after a suitable time reparameterization). We show how such an application permits the usage of variational integrators for these non-variational mechanical systems. Examples are given and numerical results are compared to the standard nonholonomic integrator results.
keywords: Nonholonomic systems Hamiltonization variational integrators.
Neighboring extremal optimal control for mechanical systems on Riemannian manifolds
Anthony M. Bloch Rohit Gupta Ilya V. Kolmanovsky
Journal of Geometric Mechanics 2016, 8(3): 257-272 doi: 10.3934/jgm.2016007
In this paper, we extend neighboring extremal optimal control, which is well established for optimal control problems defined on a Euclidean space (see, e.g., [8]) to the setting of Riemannian manifolds. We further specialize the results to the case of Lie groups. An example along with simulation results is presented.
keywords: optimal control Lie groups neighboring extremal optimal control mechanical systems Riemannian manifolds.
Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems
Anthony M. Bloch Peter E. Crouch Nikolaj Nordkvist
Journal of Geometric Mechanics 2013, 5(1): 1-38 doi: 10.3934/jgm.2013.5.1
In this paper we define ``embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on $n$-spheres.
keywords: continuous dynamics mechanical systems Optimal control integrators. discrete dynamics global analysis

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