Hernán Cendra

Lagrangian reduction by stages

This booklet studies the geometry of the reduction of Lagrangian systems with symmetry in a way that allows the reduction process to be repeated; that is, it develops a context for Lagrangian reduction by stages. The Lagrangian reduction procedure focuses on the geometry of variational structures and how to reduce them to quotient spaces under group actions. This philosophy is well known for the classical cases, such as Routh reduction for systems with cyclic variables (where the symmetry group is Abelian) and Euler{Poincare reduction (for the case in which the conguration space is a Lie group) as well as Euler-Poincare reduction for semidirect products. The context established for this theory is a Lagrangian analogue of the bundle picture on the Hamiltonian side. In this picture, we develop a category that includes, as a special case, the realization of the quotient of a tangent bundle as the Whitney sum of the tangent of the quotient bundle with the associated adjoint bundle. The elements of this new category, called the Lagrange{Poincare category, have enough geometric structure so that the category is stable under the procedure of Lagrangian reduction. Thus, reduction may be repeated, giving the desired context for reduction by stages. Our category may be viewed as a Lagrangian analog of the category of Poisson manifolds in Hamiltonian theory. We also give an intrinsic and geometric way of writing the reduced equations, called the Lagrange{Poincare equations, using covariant derivatives and connections. In addition, the context includes the interpretation of cocycles as curvatures of connections and is general enough to encompass interesting situations involving both semidirect products and central extensions. Examples are given to illustrate the general theory. In classical Routh reduction one usually sets the conserved quantities conjugate to the cyclic variables equal to a constant. In our development, we do not require the imposition of this constraint. For the general theory along these lines, we refer to the complementary work of Marsden, Ratiu and Scheurle [2000], which studies the Lagrange-Routh equations.

Geometric Mechanics, Lagrangian Reduction, and Nonholonomic Systems

This paper outlines some features of general reduction theory as well as the geometry of nonholonomic mechanical systems. In addition to this survey nature, there are some new results. Our previous work on the geometric theory of Lagrangian reduction provides a convenient context that is herein generalized to nonholonomic systems with symmetry. This provides an intrinsic geometric setting for many of the results that were previously understood primarily in coordinates. This solidification and extension of the basic theory should have several interesting consequences, some of which are spelled out in the final section of the paper. Two important references for this work are Cendra, Marsden and Ratiu [2000], hereafter denoted CMR and Bloch, Krishnaprasad, Marsden and Murray [1996], hereafter denoted BKMM.

The Maxwell–Vlasov equations in Euler–Poincaré form

Lows well-known action principle for the Maxwell–Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of Eulerian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we first transform this action principle into purely Eulerian variables. Hamiltons principle for the Eulerian description of Lows action principle then casts the Maxwell–Vlasov equations into Euler–Poincare form for right invariant motion on the diffeomorphism group of position-velocity phase space, [openface R]6. Legendre transforming the Eulerian form of Lows action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Euler–Poincare equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie–Poisson bracket on the dual of a semidirect product Lie algebra. Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell–Vlasov Poisson structure is known, whose ingredients are the Lie–Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born–Infeld brackets for the Maxwell field. We discuss the relationship between these two Hamiltonian formulations. We also discuss the general Kelvin–Noether theorem for Euler–Poincare equations and its meaning in the plasma context.

Lin constraints, Clebsch potentials and variational principles

The Poisson bracket formulation of fluid, plasma and rigid body type systems has undergone considerable recent development using techniques of symmetry group reduction. The relationship between this approach and that using Lin constraints and Clebsch potentials is established. The connection is made in the setting of abstract Clebsch variables as well as that of variational principles on reduced spaces. Variational principles for both the Clebsch and reduced form (such as fluids in spatial representation) are derived from the standard variational principle of Hamilton in material (Lagrangian) representation using reduction theory.

Generalized nonholonomic mechanics, servomechanisms and related brackets

It is well known that nonholonomic systems obeying D’Alembert’s principle are described on the Hamiltonian side, after using the Legendre transformation, by the so-called almost-Poisson brackets. In this paper we define the Lagrangian and Hamiltonian sides of a class of generalized nonholonomic systems (GNHS), obeying a generalized version of D’Alembert’s principle, such as rubber wheels (like some simplified models of pneumatic tires) and certain servomechanisms (like the controlled inverted pendulum), and show that corresponding equations of motion can also be described in terms of a bracket. We present essentially all possible brackets in terms of which the mentioned equations can be written down, which include the brackets that appear in the literature, and point out those (if any) that are naturally related to each system. In particular, we show there always exists a Leibniz bracket related to a GNHS, and conversely, that every Leibniz system is a GNHS. The control of the inverted pendulum on a cart ...

Variational principles on principal fiber bundles: A geometry theory of Clebsch potentials and Lin constraints

The geometric theory of Lin constraints and variational principles in terms of Clebsch variables proposed recently by Cendra and Marsden [1987] will be generalized to include those systems defined not only on configuration spaces which are products of Lie groups and vector spaces but on configuration spaces which are principal bundles with structural group G. This generalization includes, for example, fluids with free boundaries, Yang-Mills fields, and it will be very useful, as it will be shown later, to illustrate some aspects of the theory of particles moving in a Yang-Mills field in both its variational and Hamiltonian aspects.

Geometric mechanics and the dynamics of asteroid pairs

This paper studies, using the technique of Lagrangian reduction, the geometric mechanics of a pair of asteroids in orbit about each other under mutual gravitational attraction.

Lagrangian systems with higher order constraints

A class of mechanical systems subject to higher order constraints (i.e., constraints involving higher order derivatives of the position of the system) are studied. We call them higher order constrained systems (HOCSs). They include simplified models of elastic rolling bodies, and also the so-called generalized nonholonomic systems (GNHSs), whose constraints only involve the velocities of the system (i.e., first order derivatives in the position of the system). One of the features of this kind of systems is that D’Alembert’s principle (or its nonlinear higher order generalization, the Chetaev’s principle) is not necessarily satisfied. We present here, as another interesting example of HOCS, systems subjected to friction forces, showing that those forces can be encoded in a second order kinematic constraint. The main aim of the paper is to show that every HOCS is equivalent to a GNHS with linear constraints, in a canonical way. That is to say, systems with higher order constraints can be described in terms ...

Distance of a complex coefficient stable polynomial from the boundary of the stability set

In this paper, the distance in the 1 ≤ 2p ≤ ∞ norm from a complex coefficient polynomial to the border of its Hurwitz region is analyzed. Simplified expressions for 2p=1, 2, ∞ are also obtained.

Degenerate bifurcations and catastrophe sets via frequency analysis

We present some bifurcation conditions using the well-known stability analysis of feedback systems. A general ordinary differential equation system is formulated in two parts: one that considers the linear part and the other that includes the memoryless nonlinear part, in a similar way as the describing function. The bifurcation conditions are obtained using the results of the generalized Nyquist stability criterion (GNSC) with some explicit formulae derived from some properties of the complex variable We analyse simultaneously both static and dynamic (Hopf) bifurcations and their degeneracies in a rich example, a continuous stirred-tank reactor (CSTR), in which two consecutive, irreversible, first-order reactions A→B→C occur