4,940 research outputs found
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 correction, as well as the double
spherical pendulum. The 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
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
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
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
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 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
A variational principle for actions on symmetric symplectic spaces
We present a definition of generating functions of canonical relations, which
are real functions on symmetric symplectic spaces, discussing some conditions
for the presence of caustics. We show how the actions compose by a neat
geometrical formula and are connected to the hamiltonians via a geometrically
simple variational principle which determines the classical trajectories,
discussing the temporal evolution of such ``extended hamiltonians'' in terms of
Hamilton-Jacobi-type equations. Simplest spaces are treated explicitly.Comment: 28 pages. Edited english translation of first author's PhD thesis
(2000
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
On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets
The role of projectors associated with Poisson brackets of constrained
Hamiltonian systems is analyzed. Projectors act in two instances in a bracket:
in the explicit dependence on the variables and in the computation of the
functional derivatives. The role of these projectors is investigated by using
Dirac's theory of constrained Hamiltonian systems. Results are illustrated by
three examples taken from plasma physics: magnetohydrodynamics, the
Vlasov-Maxwell system, and the linear two-species Vlasov system with
quasineutrality
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.
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