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Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard Euler-Lagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For the reduction of a Lagrangian system on a level set of a conserved quantity, a key object is the Routhian, which is the Lagrangian minus the mechanical connection paired with the fixed value of the momentum map. For unconstrained systems, we use a velocity shifted Lagrangian, which plays the role of the Routhian in the constrained theory. Hamilton’s variational principle for the Euler-Lagrange equations breaks up into two sets of equations that represent a set of Euler-Lagrange equations with gyroscopic forcing that can be written in terms of the curvature of the connection for horizontal variations, and into the Euler-Poincar´e equations for the vertical variations. This new set of equations is what we call the reduced Euler-Lagrange equations, and it includes the Euler-Poincaré and the Hamel equations as special cases. We illustrate this methodology for a rigid body with internal rotors and for a particle moving in a magnetic field.

The integrable structure of the three-wave equations is discussed in the setting of geometric mechanics. Lie-Poisson structures with quadratic Hamiltonian are associated with the three-wave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to be compatible. The analogy between this system and the rigid-body or Euler equations is discussed. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. We show that using piecewise continuous controls, the transfer of energy among three 1 waves can be controlled. The so called quasi-phase-matching control strategy, which is used in a host of nonlinear optical devices to convert laser light from one frequency to another, is described in this context. Finally, we discuss the connection between piecewise constant controls and billiards.

In the paper, we develop an idea of interconnection of Dirac structures and their associated Lagrange- Dirac dynamical systems. First, we briefly review the Lagrange-Dirac dynamical systems (namely, implicit Lagrangian systems) associated to induced Dirac structures. Second, we describe an idea of interconnection of Dirac structures; namely, we show how two distinct Lagrange-Dirac systems can be interconnected through a Dirac structure on the product of configuration spaces. Third, we also show the variational structure of the interconnected Lagrange-Dirac dynamical system in the context of the Hamilton-Pontryagin-d’Alembert principle. Finally, we demonstrate our theory by an example of mass-spring mechanical systems.

Following the spectacular results of the Cassini mission, NASA and ESA plan to return to Titan. For missions such as this to the giant planets and their moons, the primary challenge for trajectory designers is to minimize ΔV requirements while simultaneously ensuring a reasonable time of flight. Employing a combination of invariant manifolds in the planar circular restricted three-body problem and multiple resonant gravity assists allows for the design of trajectories with a very low ΔV. However, these trajectories typically exhibit long flight times. In this study, desired resonances are targeted that, at any single node, minimize the time of flight. The resulting time of flight for a trajectory created using this methodology is compared to that of a trajectory utilizing the maximum single point decrease in semi-major axis. Then, using this framework, the effect of the Jacobi constant on the trajectory’s total ΔV and time of flight is explored. The total trajectory ΔV is shown to vary over the range of Jacobi constants tested due to the interaction between the ΔV required for capture at Titan and the resonances encompassed by the targeted invariant manifold exit region. Over the range of Jacobi constants tested, the total ΔV varies by 28 m/s while the time of flight varies by 3.2 months between the minimum and maximum cases. The lowest Jacobi constant tested results in a 23-month trajectory and a total ΔV of 626 m/s, including a controlled insertion into a 1000 km circular orbit about Titan.

We show how to \concatenate" variational principles over different bases into one over a single base, thereby providing a unifed Lagrangian treatment of interacting systems. As an example we study a Klein Gordon feld interacting with a mesically charged particle. We employ our method to give a novel group-theoretic derivation of the kinetic stress-energy-momentum tensor density corresponding to the particle.

With an environment comparable to that of primordial Earth, a surface strewn with liquid hydrocarbon lakes, and an atmosphere denser than that of any other moon in the solar system, Saturn's largest moon Titan is a treasure trove of potential scientific discovery and is the target of a proposed NASA mission scheduled for launch in roughly one decade. A chief consideration associated with the design of any such mission is the constraint imposed by fuel limitations that accompany the spacecraft's journey between celestial bodies. In this study, we explore the use of patched three-body models in conjunction with a discrete mechanical optimization algorithm for the design of a fuel-efficient Saturnian moon tour focusing on Titan. In contrast to the use of traditional models for trajectory design such as the patched conic approximation, we exploit subtleties of the three-body problem, a classic problem from celestial mechanics that asks for the motion of three masses in space under mutual gravitational interaction, in order to slash fuel costs. In the process, we demonstrate the aptitude of the DMOC (Discrete Mechanics and Optimal Control) optimization algorithm in handling celestial mechanical trajectory optimization problems.

Reduction theory is concerned with mechanical systems with symmetries. It constructs a lower dimensional reduced space in which associated conservation laws are taken out and symmetries are \factored out" and studies the relation between the dynamics of the given system with the dynamics on the reduced space. This subject is important in many areas, such as stability of relative equilibria, geometric phases and integrable systems.

A mission design technique that uses invariant manifold techniques together with the optimal control algorithm DMOC produces locally optimal, low ΔV trajectories. Previously, invariant manifolds of the planar circular restricted three body problem (PCR3BP) have been used to design trajectories with relatively small ΔV . Using local optimal control methods, specifically DMOC, it is possible to reduce the ΔV even further. This method is tested on a trajectory which begins in Earth orbit and ends in ballistic capture at the Moon. DMOC produces locally optimal trajectories with much smaller total ΔV applied in a distributed way along the trajectory. Additionally, DMOC allows for variable flight times, leading to smaller ΔV necessary for lunar orbit insertion. Results from different Earth to Moon missions are presented in table form to show how the DMOC results fit in with actual missions and different trajectory types. The ΔV of the DMOC results are, on average, 23%-25% better than the ΔV of trajectories produced using a Hohmann transfer.

Some interesting aspects of motion and control, such as those found in biological and robotic locomotion and attitude control of spacecraft, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can rotate or move forward. This observation leads to the general idea that when one variable in a system moves in a periodic fashion, motion of the Whole object can result. This property can be used for control purposes; the position and attitude Of a satellite, for example, are often controlled by periodic motions of parts of the satellite, such as spinning rotors. One of the geometric tools that has been used to describe this phenomenon is that of connections, a notion that is used extensively in general relativity and other parts of theoretical physics. This tool, part of the general subject Of geometric mechanics, has been helpful in the study of both the stability and instability of a system and system bifurcations, that is, changes in the nature of the system dynamics, as some parameter changes. Geometric mechanics, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in attitude dynamics. Theory is also being developed for systems with rolling constraints such as those found in a simple rolling wheel. This paper explains how some of these tools of geometric mechanics are used in the study of motion control and locomotion generation.

The first part of this paper considers the problem of solving an equation of the form F(x,y) = 0, for y = φ(x) as a function of x, where F : X x Y → Z is a smooth nonlinear mapping between Banach spaces. The focus is on the case in which the mapping F is degenerate at some point (x^*; y^*) with respect to y, i.e., when F' _y (x^*; y^*), the derivative of F with respect to y, is not invertible and, hence, the classical Implicit Function Theorem is not applicable. We present pth-order generalizations of the Implicit Function Theorem for this case. The second part of the paper uses these pth-order implicit function theorems to derive sufficient conditions for the existence of a solution of degenerate nonlinear boundary-value problems for second-order ordinary differential equations in cases close to resonance. The last part of the paper presents a modified perturbation method for solving degenerate second-order boundary value problems with a small parameter. The results of this paper are based on the constructions of p-regularity theory, whose basic concepts and main results are given in the paper Factor-analysis of nonlinear mappings: p- regularity theory by Tret'yakov and Marsden (Communications on Pure and Applied Analysis, 2 (2003), 425-445).

The equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange-d’Alembert principle. This paper derives a structure-preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two-link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem.

This paper presents a methodology for generating locally optimal control policies for mechanical systems that undergo collisions at point contacts. Principles of nonsmooth mechanics for rigid bodies are used in both continuous and discrete time, and provide impact models for a variety of collision behaviors. The discrete Euler-Lagrange (DEL) equations that follow from the discrete time analyses yield variational integration schemes for the dierent impact models. These DEL equations play a pivotal role in the method of Discrete Mechanics and Optimal Control (DMOC), which generates locally optimal control policies as the solution to equality constrained nonlinear optimization problems. The DMOC method is demonstrated on a 4-link planar walking robot model, generating locally optimal periodic walking gaits.

In this paper, we will develop two different representations for induced Dirac structures and their associated IPCL systems; namely, (1) a standard representation with using Lagrange multipliers; and (2) a representation without using Lagrange multipliers. Those representations are consistent with those developed by Courant and Weinstein. Specifically, the second representation without using Lagrange multipliers may be crucial in formulation of constrained mechanical systems since it systematically enables one to eliminate unnecessary constraint forces. In mechanics, it is known that the elimination of constraint forces can be done by the orthogonal complement method or the null space method, although the link with Dirac structures has not been clarified. The present paper fills this gap to show that the orthogonal complement method can be incorporated into the context of Dirac structures and the associated IPCL systems and we will further show the link with the topological method in electrical network theory using the so-called fundamental cut-set and loop matrices. In the paper, we shall illustrate out ideas by an example of L-C circuits.

This paper describes the place of the book by Guckenheimer and Holmes (Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, Berlin, 1983) in the research and literature on nonlinear dynamics.

In this paper, we apply Dirac structures and the associated theory of implicit Lagrangian systems to electric networks. We show how a Dirac structure on the flux linkage phase can be induced from a KCL (Kirchhoff Current Law) constraint distribution on a configuration charge space in analogy with mechanics. In this context, a notion of implicit port-controlled Lagrangian systems is developed. As a specific illustrative example, it is demonstrated that a one-dimensional L-C transmission line can be formulated in the context of implicit port-controlled Lagrangian systems, where the transmission line may be regarded as an interconnected system of a chain of constituent primitive modules, each of which is given by an L-C circuit with external ports.

Nonholonomic systems are, roughly speaking, mechanical systems with constraints on their velocity that are not derivable from position constraints. They arise, for instance, in mechanical systems that have rolling contact (for example, the rolling of wheels without slipping) or certain kinds of sliding contact (such as the sliding of skates). They are a remarkable generalization of classical Lagrangian and Hamiltonian systems in which one allows position constraints only. There are some fascinating differences between nonholonomic systems and classical Hamiltonian or Lagrangian systems. Among other things: nonholonomic systems are nonvariational—they arise from the Lagrange-d’Alembert principle and not from Hamilton’s principle; while energy is preserved for nonholonomic systems, momentum is not always preserved for systems with symmetry (i.e., there is nontrivial dynamics associated with the nonholonomic generalization of Noether’s theorem); nonholonomic systems are almost Poisson but not Poisson (i.e., there is a bracket that together with the energy on the phase space defines the motion, but the bracket generally does not satisfy the Jacobi identity); and finally, unlike the Hamiltonian setting, volume may not be preserved in the phase space, leading to interesting asymptotic stability in some cases, despite energy conservation. The purpose of this article is to engage the reader’s interest by highlighting some of these differences along with some current research in the area. There has been some confusion in the literature for quite some time over issues such as the variational character of nonholonomic systems, so it is appropriate that we begin with a brief review of the history of the subject.

The purpose of this paper is to describe the general setting for the application of techniques from geometric mechanics and dynamical systems to the problem of asteroid pairs. It also gives some preliminary results on transport calculations and the associated problem of calculating binary asteroid escape rates. The dynamics of an asteroid pair, consisting of two irregularly shaped asteroids interacting through their gravitational potential is an example of a full body problem or FBP in which two or more extended bodies interact. One of the interesting features of the binary asteroid problem is that there is coupling between their translational and rotational degrees of freedom. General FBPs have a wide range of other interesting aspects as well, including the 6-DOF guidance, control, and dynamics of vehicles, the dynamics of interacting or ionizing molecules, the evolution of small body, planetary, or stellar systems, and almost any other problem where distributed bodies interact with each other or with an external field. This paper focuses on the specific case of asteroid pairs using techniques that are generally applicable to many other FBPs. This particular full 2-body problem (F2BP) concerns the dynamical evolution of two rigid bodies mutually interacting via a gravitational field. Motivation comes from planetary science, where these interactions play a key role in the evolution of asteroid rotation states and binary asteroid systems. The techniques that are applied to this problem fall into two main categories. The first is the use of geometric mechanics to obtain a description of the reduced phase space, which opens the door to a number of powerful techniques such as the energy-momentum method for determining the stability of equilibria and the use of variational integrators for greater accuracy in simulation. Secondly, techniques from computational dynamical systems are used to determine phase space structures important for transport phenomena and dynamical evolution.

The purpose of this paper is to survey some recent advances in variational integrators for both finite dimensional mechanical systems as well as continuum mechanics. These advances include the general development of discrete mechanics, applications to dissipative systems, collisions, spacetime integration algorithms, AVI’s (Asynchronous Variational Integrators), as well as reduction for discrete mechanical systems. To keep the article within the set limits, we will only treat each topic briefly and will not attempt to develop any particular topic in any depth. We hope, nonetheless, that this paper serves as a useful guide to the literature as well as to future directions and open problems in the subject.