2011/09/30 by Melvin Leok, Tomoki Ohsawa, Diana Sosa · 25 citations
Engineering · Mathematics · Physics and Astronomy · #Classical mechanics #Computer science #Control and Dynamics of Mobile Robots #Control and Stability of Dynamical Systems #Degenerate energy levels #Dynamics and Control of Mechanical Systems #Hamiltonian (control theory) #Hamiltonian mechanics #Hamilton–Jacobi equation #Holonomic #Lagrangian #Lagrangian system #Mathematical optimization #Mathematical physics #Mathematics #Mobile robot #Nonholonomic system #Physics #Quantum mechanics #Robot #math-ph #math.MP #math.SG #msc:37J60 #msc:70F25 #msc:70H20 #msc:70H45
paper · pdf · doi:10.1063/1.4736733
published in Journal of Mathematical Physics 53(7) (American Institute of Physics) · 44 pages, 3 figures
openalex publication_date 2012/07/01 · arxiv created 2012/09/12 · arxiv updated 2012/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton–Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac–Hamilton–Jacobi equation reduces to a variant of the nonholonomic Hamilton–Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac–Hamilton–Jacobi equation can be used to exactly integrate the equations of motion.