Journal of Geometry and Physics 49 (2004) 385–417 On classical mechanical systems with non-linear constraints Gláucio Terra a,∗ , Marcelo H. Kobayashi b,1 a Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, 05508-090 São Paulo, SP, Brazil b Departamento de Engenharia Mecˆ anica, Instituto Superior Técnico, Secção de Mec ˆ anica Aeroespacial, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal Received 15 April 2003 Abstract In the present work, we analyze classical mechanical systems with non-linear constraints in the velocities. We prove that the d’Alembert–Chetaev trajectories of a constrained mechanical system satisfy both Gauss’ principle of least constraint and Hölder’s principle. In the case of a free mechanics, they also satisfy Hertz’s principle of least curvature if the constraint manifold is a cone. We show that the Gibbs–Maggi–Appell (GMA) vector field (i.e. the second-order vector field which defines the d’Alembert–Chetaev trajectories) conserves energy for any potential energy if, and only if, the constraint is homogeneous (i.e. if the Liouville vector field is tangent to the constraint manifold). We introduce the Jacobi–Carathéodory metric tensor and prove Jacobi–Carathéodory’s theorem assuming that the constraint manifold is a cone. Finally, we present a version of Liouville’s theorem on the conservation of volume for the flow of the GMA vector field. © 2003 Published by Elsevier B.V. MSC: 70A05; 70G45; 53C80 JGP SC: Geometrical mechanics; Chemical mechanics Keywords: Constrained mechanical systems; Non-linear constraints; Non-holonomic mechanics 1. Introduction The aim of this paper is to develop a geometric formulation of the dynamics of non-linearly constrained mechanical systems based on Newton’s law. ∗ Corresponding author. E-mail addresses: glaucio@ime.usp.br (G. Terra), marcelo@popsrv.ist.utl.pt (M.H. Kobayashi). 1 Present address: Department of Mechanical Engineering, University of Hawaii-Manoa, 2540 Dole Street – Holmes Hall 302, Honolulu HI 96822. 0393-0440/$ – see front matter © 2003 Published by Elsevier B.V. doi:10.1016/j.geomphys.2003.08.005