2015/11/30 by Songhao Li, Ari Stern, Xiang Tang
Mathematics · Physics and Astronomy · #Adjoint representation of a Lie algebra #Advanced Topics in Algebra #Algebra over a field #Analytical mechanics #Fibered knot #Geometry #Homogeneous space #Homotopy and Cohomology in Algebraic Topology #Lagrangian #Lagrangian mechanics #Lagrangian system #Lie algebra #Lie algebroid #Lie group #Lie theory #Mathematics #Nonlinear Waves and Solitons #Physics #Pure mathematics #Reduction (mathematics) #math-ph #math.DG #math.DS #math.MP #math.SG
paper · pdf · doi:10.3842/sigma.2017.019
published as SIGMA 13 (2017), 019, 26 pages
arxiv created 2017/03/22 · openalex publication_date 2017/03/22 · arxiv updated 2017/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. Along the way, we also develop some new results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free formulation of the equations of motion. Finally, we extend the foregoing to include fibered and Lie algebroid generalizations of the Hamilton-Pontryagin principle of Yoshimura and Marsden, along with the associated reduction theory.