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81
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 56 (22 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”
Asynchronous Variational Integrators
 ARCH. RATIONAL MECH. ANAL.
, 2003
"... We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation t ..."
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Cited by 44 (10 self)
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We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system. A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.
Variational time integrators
 Int. J. Numer. Methods Eng
"... The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in de ..."
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Cited by 37 (9 self)
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The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in detail. We present selected numerical examples which demonstrate the excellent accuracy, conservation and convergence characteristics of AVIs. In these tests, AVIs are found to result in substantial speedups, at equal accuracy, relative to explicit Newmark. A mathematical proof of convergence of the AVIs is also presented in this paper. Finally, we develop the subject of horizontal variations and configurational forces in discrete dynamics. This theory leads to exact pathindependent characterizations of the configurational forces acting on discrete systems. Notable examples are the configurational forces acting on material nodes in a finite element discretisation; and the Jintegral at the tip of a crack in
Dimensional Model Reduction in Nonlinear Finite Element Dynamics of Solids and Structures
 International Journal for Numerical Methods in Engineering
"... this paper given as wallclock measurements on an 225 MHz SGI Octane workstation with R10K processor, 128 MB of memory, and 32 KB instruction and data caches, and 1 MB secondary cache. Copyright c ..."
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Cited by 30 (5 self)
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this paper given as wallclock measurements on an 225 MHz SGI Octane workstation with R10K processor, 128 MB of memory, and 32 KB instruction and data caches, and 1 MB secondary cache. Copyright c
Geometric, Variational Integrators for Computer Animation
 EUROGRAPHICS/ACM SIGGRAPH SYMPOSIUM ON COMPUTER ANIMATION, M.P. CANI, J. O’BRIEN (EDITORS)
, 2006
"... We present a generalpurpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physicsbased animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically pre ..."
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Cited by 26 (3 self)
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We present a generalpurpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physicsbased animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically preserves important invariants, such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite simply; finally, our simple methodology allows for the design of highorder accurate schemes if needed. Two key properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the implementation of the method. These properties are achieved using a discrete form of a general variational principle called the PontryaginHamilton principle, expressing time integration in a geometric manner. We demonstrate the applicability of our integrators to the simulation of nonlinear elasticity with implementation details.
Geometric mechanics, Lagrangian reduction and nonholonomic systems
 in Mathematics Unlimited2001 and Beyond
, 2001
"... This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has ..."
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Cited by 24 (5 self)
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This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in mechanics ranges from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many
Nonsmooth Lagrangian mechanics and variational collision integrators
 SIAM Journal on Applied Dynamical Systems
, 2003
"... Abstract. Variational techniques are used to analyze the problem of rigidbody dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noetherstyle momentum ..."
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Cited by 24 (8 self)
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Abstract. Variational techniques are used to analyze the problem of rigidbody dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noetherstyle momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplecticmomentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the longtime stable energy behavior typical of variational methods is demonstrated.
Reduction theory and the LagrangeRouth Equations
 J. Math. Phys
, 2000
"... Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as e ..."
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Cited by 21 (8 self)
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Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, stability theory, integrable systems, as well as geometric phases.
Hamilton–Pontryagin integrators on Lie groups
, 2007
"... Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical ..."
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Cited by 21 (6 self)
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Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical