2007/12/14 by Joris Vankerschaver, J. Vankerschaver, David Martı́n de Diego +1
Engineering · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Control and Dynamics of Mobile Robots #math-ph #math.MP #msc:37K05 #msc:53D20 #msc:70F25 #msc:70S05
paper · pdf · doi:10.1088/1751-8113/41/3/035401
published as J.Phys.A41:035401,2008 · 18 pages
arxiv created 2007/12/14 · openalex publication_date 2008/01/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic symmetries are then introduced as symmetry generators whose virtual work is zero along the constraint submanifold, and we show that for every such symmetry, there exists a so-called momentum equation, describing the evolution of the associated component of the momentum map. Keeping up with the underlying geometric philosophy, a small modification of the derivation of the momentum lemma allows us to also treat generalized nonholonomic symmetries, which are vector fields along a projection. Such symmetries arise for example in practical examples of nonholonomic field theories such as the Cosserat rod, for which we recover both energy conservation (a previously known result) and a modified conservation law associated with spatial translations.