Discrete Routh Reduction

This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry. The reduction of variational Runge-Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical $J_2$ correction, as well as the double spherical pendulum. The $J_2$ problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a nontrivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the noncanonical nature of the symplectic structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added, fixed typo

Nonholonomic constraints in classical field theories

A multisymplectic setting for classical field theories subjected to non-holonomic constraints is presented. The infinite dimensional setting in the space of Cauchy data is also given.Comment: 14 pages; 1 figur

Discrete Variational Optimal Control

This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of variational principles. The key point is to solve the optimal control problem as a variational integrator of a specially constructed higher-dimensional system. The developed framework applies to systems on tangent bundles, Lie groups, underactuated and nonholonomic systems with symmetries, and can approximate either smooth or discontinuous control inputs. The resulting methods inherit the preservation properties of variational integrators and result in numerically robust and easily implementable algorithms. Several theoretical and a practical examples, e.g. the control of an underwater vehicle, will illustrate the application of the proposed approach.Comment: 30 pages, 6 figure

Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot).Comment: 45 page

Generalized Reduction Procedure: Symplectic and Poisson Formalism

We present a generalized reduction procedure which encompasses the one based on the momentum map and the projection method. By using the duality between manifolds and ring of functions defined on them, we have cast our procedure in an algebraic context. In this framework we give a simple example of reduction in the non-commutative setting.Comment: 39 pages, Latex file, Vienna ESI 28 (1993

Geometric aspects of nonholonomic field theories

A geometric model for nonholonomic Lagrangian field theory is studied. The multisymplectic approach to such a theory as well as the corresponding Cauchy formalism are discussed. It is shown that in both formulations, the relevant equations for the constrained system can be recovered by a suitable projection of the equations for the underlying free (i.e. unconstrained) Lagrangian system.Comment: 29 pages; typos remove

Experimental and modeling studies on the synthesis and properties of higher fatty esters of corn starch

This paper describes a systematic study on the synthesis of higher fatty esters of corn starch (starch laurate and starch stéarate) by using the corresponding vinyl esters. The reactions were carried out in DMSO using basic catalysts (Na2HPO4, K2CO3, and Naacetate). The effect of the process variables (vinyl ester to starch ratio, catalyst intake, reaction temperature and type of the catalyst) on the degree of substitution (DS) of the starch laurate and starch stearate esters was determined by performing a total of 54 experiments. The results were adequately modeled using a non-linear multivariable regression model (R2≥0. 96). The basicity of the catalyst and the reaction temperature have the highest impact on the product DS. The thermal and mechanical properties of some representative product samples were determined. High-DS products (DS = 2.26-2.39) are totally amorphous whereas the low-DS ones (DS = 1.45-1.75) are still partially crystalline. The thermal stability of the esterlfied products is higher than that of native starch. Mechanical tests show that the products have a tensile strength (stress at break) between 2.7-3.5 MPa, elongation at break of 3-26%, and modulus of elasticity of 46-113 MPa.

Central extensions, classical non-equivariant maps and residual symmetries

The arising of central extensions is discussed in two contexts. At first classical counterparts of quantum anomalies (deserving being named as "classical anomalies") are associated with a peculiar subclass of the non-equivariant maps. Further, the notion of "residual symmetry" for theories formulated in given non-vanishing EM backgrounds is introduced. It is pointed out that this is a Lie-algebraic, model-independent, concept.Comment: 8 pages, LaTex. Talk given at the International Conference "Renormalization Group and Anomalies in Gravitation and Cosmology", Ouro Preto, Brazil, March 2003. To appear in the Proceeding

Chandra astrometry sets a tight upper limit to the proper motion of SGR 1900+14

The soft gamma-ray repeater (SGR) SGR 1900+14 lies a few arcminutes outside the edge of the shell supernova remnant (SNR) G42.8+0.6. A physical association between the two systems has been proposed - for this and other SGR-SNR pairs - based on the expectation of high space velocities for SGRs in the framework of the magnetar model. The large angular separation between the SGR and the SNR center, coupled with the young age of the system, suggest a test of the association with a proper motion measurement. We used a set of three Chandra/ACIS observations of the field spanning 5 years to perform accurate relative astrometry in order to measure the possible angular displacement of the SGR as a function of time. Our investigation sets a 3-sigma upper limit of 70 mas/yr to the overall proper motion of the SGR. Such a value argues against an association of SGR 1900+14 with G42.8+0.6 and adds further support to the mounting evidence for an origin of the SGR within a nearby, compact cluster of massive stars.Comment: Accepted for publication in The Astrophysical Journal. 4 pages in emulate-apj styl

Geometric gradient-flow dynamics with singular solutions

The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy's Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.Comment: 28 pages, 1 figure, to appear on Physica