The Lagrange–Charpit Theory of the Hamilton–Jacobi Problem

The Lagrange–Charpit theory is a geometric method of determining a complete integral by means of a constant of the motion of a vector field defined on a phase space associated to a nonlinear PDE of first order. In this article, we establish this theory on the symplectic structure of the cotangent bundle $$T^{*}Q$$ T ∗ Q of the configuration manifold Q. In particular, we use it to calculate explicitly isotropic submanifolds associated with a Hamilton–Jacobi equation.

The Information Geometry of Sensor Configuration

In problems of parameter estimation from sensor data, the Fisher information provides a measure of the performance of the sensor; effectively, in an infinitesimal sense, how much information about the parameters can be obtained from the measurements. From the geometric viewpoint, it is a Riemannian metric on the manifold of parameters of the observed system. In this paper, we consider the case of parameterized sensors and answer the question, “How best to reconfigure a sensor (vary the parameters of the sensor) to optimize the information collected?” A change in the sensor parameters results in a corresponding change to the metric. We show that the change in information due to reconfiguration exactly corresponds to the natural metric on the infinite-dimensional space of Riemannian metrics on the parameter manifold, restricted to finite-dimensional sub-manifold determined by the sensor parameters. The distance measure on this configuration manifold is shown to provide optimal, dynamic sensor reconfiguration based on an information criterion. Geodesics on the configuration manifold are shown to optimize the information gain but only if the change is made at a certain rate. An example of configuring two bearings-only sensors to optimally locate a target is developed in detail to illustrate the mathematical machinery, with Fast Marching methods employed to efficiently calculate the geodesics and illustrate the practicality of using this approach.

The geometry of equations of motion: particles in equivalent universes

The equations of motion for the simplest non-holonomically constrained system of particles are formulated using six methods: Newton–Euler, Lagrange, Maggi, Gibbs–Appell, Kane, and Boltzmann–Hamel. The challenging tasks of exploring and explaining the relationships and equivalences between these formulations is accomplished by constructing a single representative particle for the system of particles. The single particle is constrained to move on a configuration manifold. The explicit construction of sets of tangent vectors to the manifold and their relation to the forces acting on the single particle are used to provide several helpful geometric interpretations of the relationships between the formulations. These interpretations can also be extended to help understand the relationships between different formulations of the equations of motion for more complex systems, including systems of rigid bodies and particles.

Towards optimal path computation in a simplicial complex

Computing optimal path in a configuration space is fundamental to solving motion planning problems in robotics and autonomy. Graph-based search algorithms have been widely used to that end, but they suffer from drawbacks. We present an algorithm for computing the shortest path through a metric simplicial complex that can be used to construct a piece-wise linear discrete model of the configuration manifold. In particular, given an undirected metric graph, G, which is constructed as a discrete representation of an underlying configuration manifold (a larger “continuous” space typically of dimension greater than one), we consider the Rips complex, [Formula: see text], associated with it. Such a complex, and hence shortest paths in it, represent the underlying metric space more closely than what the graph does. Our algorithm requires only a local connectivity-based description of an abstract graph, [Formula: see text], and a cost/length function, [Formula: see text], as inputs. No global information such as an embedding or a global coordinate chart is required. The local nature of the proposed algorithm makes it suitable for configuration spaces of arbitrary topology, geometry, and dimension. We not only develop the search algorithm for computing shortest distances, but we also present a path reconstruction algorithm for explicitly computing the shortest paths through the simplicial complex. The complexity of the presented algorithm is comparable with that of Dijkstra’s search, but, as the results presented in this paper demonstrate, the shortest paths obtained using the proposed algorithm represent the geodesic paths in the original metric space significantly more closely.

A geometric Hamilton–Jacobi theory on a Nambu–Jacobi manifold

In this paper, we propose a geometric Hamilton–Jacobi (HJ) theory on a Nambu–Jacobi (NJ) manifold. The advantage of a geometric HJ theory is that if a Hamiltonian vector field [Formula: see text] can be projected into a configuration manifold by means of a one-form [Formula: see text], then the integral curves of the projected vector field [Formula: see text] can be transformed into integral curves of the vector field [Formula: see text] provided that [Formula: see text] is a solution of the HJ equation. This procedure allows us to reduce the dynamics to a lower-dimensional manifold in which we integrate the motion. On the other hand, the interest of a NJ structure resides in its role in the description of dynamics in terms of several Hamiltonian functions. It appears in fluid dynamics, for instance. Here, we derive an explicit expression for a geometric HJ equation on a NJ manifold and apply it to the third-order Riccati differential equation as an example.

A Boundary Layer Approach to Multibody Systems Involving Single Frictional Impacts

A systematic theoretical approach is presented, revealing dynamics of a class of multibody systems. Specifically, the motion is restricted by a set of bilateral constraints, acting simultaneously with a unilateral constraint, representing a frictional impact. The analysis is carried out within the framework of Analytical Dynamics and uses some concepts of differential geometry, which provides a foundation for applying Newton's second law. This permits a successful and illuminating description of the dynamics. Starting from the unilateral constraint, a boundary is defined, providing a subspace of allowable motions within the original configuration manifold. Then, the emphasis is focused on a thin boundary layer. In addition to the usual restrictions imposed on the tangent space, the bilateral constraints cause a correction of the direction where the main impulse occurs. When friction effects are negligible, the dominant action occurs along this direction and is described by a single nonlinear ordinary differential equation (ODE), independent of the number of the original generalized coordinates. The presence of friction increases this to a system of three ODEs, capturing the essential dynamics in an appropriate subspace, arising by bringing the image of the friction cone from the physical to the configuration space. Moreover, it is shown that the classical Darboux–Keller approach corresponds to a special case of the new method. Finally, the theoretical results are complemented by a selected set of numerical results for three examples.

The Hamiltonian & Phase Space

This chapter discusses the Hamiltonian and phase space. Hamilton’s equations can be derived in several ways; this chapter follows two pathways to arrive at the same result, thus giving insight into the motivation for forming these equations. The importance of deriving the same result in several ways is that it shows that, in physics, there are often several mathematical avenues to go down and that approaching a problem with, say, the calculus of variations can be entirely as valid as using a differential equation approach. The chapter extends the arenas of classical mechanics to include the cotangent bundle momentum phase space in addition to the tangent bundle and configuration manifold, and discusses conjugate momentum. It also introduces the Hamiltonian as the Legendre transform of the Lagrangian and compares it to the Jacobi energy function.

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.