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 $J_2$ correction, as well as the double spherical pendulum. The $J_2$ 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

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

Resonant Geometric Phases for Soliton Equations

The goal of the present paper is to introduce a multidimensional generalization of asymptotic reduction given in a paper by Alber and Marsden [1992], to use this to obtain a new class of solutions that we call resonant solitons, and to study the corresponding geometric phases. The term "resonant solitons" is used because those solutions correspond to a spectrum with multiple points, and they also represent a dividing solution between two different types of solitons. In this sense, these new solutions are degenerate and, as such, will be considered as singular points in the moduli space of solitons

Stabilization of Relative Equilibria II

In this paper, we obtain feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We use a notion of stability ‘modulo the group action’ developed by Patrick [1992]. We deal with both internal instability and with instability of the rigid motion. The methodology is that of potential shaping, but the system is allowed to be internally underactuated, i.e., have fewer internal actuators than the dimension of the shape space

Complex geometric asymptotics for nonlinear systems on complex varieties

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Stabilization of relative equilibria

This paper discusses the problem of obtaining feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We show how to stabilize an internally unstable relative equilibrium using internal actuators. The methodology is that of potential shaping, but the system is allowed to be underactuated, i.e., have fewer actuators than the dimension of the shape space. The theory is illustrated with the problem of stabilization of the cowboy relative equilibrium of the double spherical pendulum

Control and stabilization of systems with homoclinic orbits

In this paper we consider the control of two physical systems, the near wall region of a turbulent boundary layer and the rigid body, using techniques from the theory of nonlinear dynamical systems. Both these systems have saddle points linked by heteroclinic orbits. In the fluid system we show how the structure of the phase space can be used to keep the system near an (unstable) saddle. For the rigid body system we discuss passage along the orbit as a possible control manouver, and show how the Energy-Casimir method can be used to analyze stabilization of the system about the saddles

Controllability for Distributed Bilinear Systems

This paper studies controllability of systems of the form ${{dw} / {dt}} = \mathcal {A}w + p(t)\mathcal {B}w$ where $\mathcal{A}$ is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators $e^{\mathcal{A}t}$ on a Banach space $X$, $\mathcal{B}:X \to X$ is a $C^1$ map, and $p \in L^1 ([0,T];\mathbb{R})$ is a control. The paper (i) gives conditions for elements of $X$ to be accessible from a given initial state $w_0$ and (ii) shows that controllability to a full neighborhood in $X$ of $w_0$ is impossible for $\dim X = \infty$. Examples of hyperbolic partial differential equations are provided

Controllability and stabilizability of distributed bilinear systems: Recent results and open problems

This paper describes recent results for controlling and stabilizing control systems of the form ú(t) = Au(t) + p(t) B(u(t)) where A is the infinitesimal generator C∞ semigroup on a Banach space X, B' map from X into X, and p(t) is a real valued control. Application to a vibrating beam problem is given for illusstration of the theory