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Abstract: This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether’s theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization

In this paper we construct homogeneous star products of Weyl type on every cotangent bundle T ∗ Q by means of the Fedosov procedure using a symplectic torsion-free connection on T ∗ Q homogeneous of degree zero with respect to the Liouville vector field. By a fibrewise equivalence transformation we construct a homogeneous Fedosov star product of standard ordered type equivalent to the homogeneous Fedosov star product of Weyl type. Representations for both star product algebras by differential operators on functions on Q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex-valued functions on T ∗ Q polynomial in the momenta (where an arbitrary fixed torsion-free connection ∇0 on Q is used). Motivated by the flat case T ∗ R n another homogeneous star product of Weyl type corresponding to the Weyl ordering prescription is constructed. The example of the cotangent bundle of an arbitrary Lie group is explicitly computed and the star product given by Gutt is rederived in our approach.

We show that, given a certain transversality condition, the set of relative equilibria E near pe ∈ E of a Hamiltonian system with symmetry is locally Whitney-stratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentum-generator pairs (µ, ξ) of the relative equilibria. The dimension of the stratum of the conjugacy class (K) is dimG+2dim Z(K) −dim K, where Z(K) is the center of K, and transverse to this stratum E is locally diffeomorphic to the commuting pairs of the Lie algebra of K/Z(K). The stratum E(K) is a symplectic submanifold of P near pe ∈ E if and only if pe is nondegenerate and K is a maximal torus of G. We also show that there is a dense subset of G-invariant Hamiltonians on P for which all the relative equilibria are transversal. Thus, generically, the types of singularities that can be found in the set of relative equilibria of a Hamiltonian system with symmetry are those types found amongst the singularities at zero of the sets of commuting pairs of certain Lie subalgebras of the symmetry group.

Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T ∗ Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T ∗ Q/G, decomposed as a Whitney sum bundle, T ∗ (Q/G) � �g ∗ over Q/G. The splitting arises naturally from a choice of connection on the G-principal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found.

We give explicit differential equations for the dynamics near relative equilibria of Hamiltonian systems. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that is inherited from the symplectic structure and symmetry properties of the Hamiltonian system is analysed and the effects of time reversing symmetries are included. The results will be applicable to the stability and bifurcation theories of relative equilibria of Hamiltonian systems.

There has recently been considerable interest in sending a spacecraft to orbit Europa, the smallest of the four Galilean moons of Jupiter. The trajectory design involved in effecting a capture by Europa presents formidable challenges to traditional conic analysis since the regimes of motion involved depend heavily on three-body dynamics. New three-body perspectives are required to design successful and efficient missions which take full advantage of the natural dynamics. Not only does a three-body approach provide low-fuel trajectories, but it also increases the flexibility and versatility of missions. We apply this approach to design a new mission concept wherein a spacecraft “leap-frogs ” between moons, orbiting each for a desired duration in a temporary capture orbit. We call this concept the “Petit Grand Tour.” For this application, we apply dynamical systems techniques developed in a previous paper to design a Europa capture orbit. We show how it is possible, using a gravitional boost from Ganymede, to go from a jovicentric orbit beyond the orbit of Ganymede to a ballistic capture orbit around Europa. The main new technical result is the employment of dynamical channels in the phase space — tubes in the energy surface which naturally link the vicinity of Ganymede to the vicinity of Europa.

Some interesting aspects of motion and control for systems such as those found in biological and robotic locomotion, attitude control of spacecraft and underwater vehicles, and steering of cars and trailers, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can move forward or rotate in place. When the amplitude of the motion increases, the resulting net displacements normally increase as well. These observations lead to the general idea that when certain variables in a system move 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. Geometric tools that have been useful to describe this phenomenon are \connections", mathematical objects that are extensively used in general relativity and other parts of theoretical physics. The theory of connections, which isnow part of the general subject of geometric mechanics, has also been helpful in the study of the stability or instability ofa system and in its bifurcations under parameter variations. This approach, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in the attitude dynamics of spacecraft and

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle [Mar84, Mar85] and Guillemin and Sternberg [GS84] for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model it can be used the so called Chu map [Chu75] instead, which exists for any canonical action, unlike the momentum map. Hamilton’s equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.

We prove a convergent strategy for a group of mobile sensors to generate contour plots, i.e., to automatically detect and track level curves of a scalar field in the plane. The group can consist of as few as four mobile sensors, where each sensor can take only a single measurement at a time. The shape of the formation of mobile sensors is determined to minimize the least mean square error in the estimates of the scalar field and its gradient. The algorithm to generate a contour plot is based on feedback control laws for each sensor platform. The control laws serve two purposes: to guarantee that the center of the formation moves along one level curve at unit speed; and to stabilize the shape of the formation. We prove that both goals can be achieved asymptotically. We show simulation results that illustrate the performance of the control laws in noisy environments. 1.