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74
Multisymplectic geometry, variational integrators, and nonlinear PDEs
 Comm. Math. Phys
, 1998
"... Abstract: This paper presents a geometricvariational 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 v ..."
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Cited by 85 (21 self)
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Abstract: This paper presents a geometricvariational 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 infinitedimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sineGordon equation, including computational results and a comparison with other discretization
Homogeneous Fedosov star products on cotangent bundles I: Weyl and . . .

, 1997
"... 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 torsionfree connection on T ∗ Q homogeneous of degree zero with respect to the Liouville vector field. By a fibrewise equivalence transformation we ..."
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Cited by 59 (10 self)
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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 torsionfree 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 complexvalued functions on T ∗ Q polynomial in the momenta (where an arbitrary fixed torsionfree 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.
Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics
, 2000
"... This paper applies dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted threebody problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the des ..."
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Cited by 45 (19 self)
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This paper applies dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted threebody problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits, one around the libration point L1 and the other around L2, withthe two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the “interior ” and “exterior”
Relative equilibria of Hamiltonian systems with symmetry: linearization, smoothness
, 1995
"... 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 Whitneystratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentumgenerat ..."
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Cited by 42 (9 self)
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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 Whitneystratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentumgenerator 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 Ginvariant 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.
Symmetry and Bifurcations of Momentum Mappings
 Comm. Math. Phys
"... Abstract. The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is ..."
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Cited by 33 (15 self)
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Abstract. The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the YangMills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface. 1.
Generating contour plots using multiple sensor platforms
 in Proc. of 2005 IEEE Symposium on Swarm Intelligence
, 2005
"... 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 sha ..."
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Cited by 31 (10 self)
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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.
The Orbit Bundle Picture of Cotangent Bundle Reduction
, 2000
"... 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 realiza ..."
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Cited by 21 (15 self)
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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 Gprincipal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found.
Hamiltonian Systems Near Relative Equilibria
 J. Dierential Equations
, 1999
"... 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 ..."
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Cited by 16 (6 self)
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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.
Constructing a low energy transfer between Jovian moons
 Contemporary Mathematics
, 2002
"... 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 ..."
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Cited by 16 (8 self)
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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 threebody dynamics. New threebody perspectives are required to design successful and efficient missions which take full advantage of the natural dynamics. Not only does a threebody approach provide lowfuel trajectories, but it also increases the flexibility and versatility of missions. We apply this approach to design a new mission concept wherein a spacecraft “leapfrogs ” 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.