Lagrangian Reduction by Stages

By H. Cendra, Jerrold E. Marsden, Tudor S. Ratiu. 2001

259 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/cc48aaa98768176b9d95e60105bc9053f431c76a

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Citations in the corpus

The Orbit Bundle Picture of Cotangent BundleReduction (2000, methodology): “This approach was motivated by Cendra and Marsden [1987] and Cendra, Ibort, and Marsden [1987]. The work of Bloch, Krishnaprasad, Marsden, and Ratiu [1996] generalized the Euler–Poincaré variational structure to general Lie groups and Cendra, Marsden, and Ratiu [2000] carried out a Lagrangian reduct”
Structure-preserving model reduction for mechanical systems (2003, methodology): “This variational view for obtaining structure preservation also occurs in the now very popular method of Lagrangian reduction of Marsden and Scheurle (see [4] for a current account and literature).”
Averaging and control of nonlinear systems (2003, methodology): “[29, 30] emphasizes the principal bundle part by explicitly deriving the equations of motion from a reduced perspective, avoiding the “connection within a connection.”