This is a comprehensive treatment of the time-integral variational “principles” of mechanics for systems subject to general nonlinear and possibly nonholonomic velocity constraints (i.e., equations of the form f(t, q, q˙) = 0, where t = time and q/q˙ = Lagrangean coordinates/velocities), in general nonlinear nonholonomic coordinates. The discussion is based on the Maurer-Appell-Chetaev-Hamel definition of virtual displacements and subsequent formulation of the corresponding nonlinear transitivity (or transpositional) equations. Also, a detailed analysis of the latter supplies a hitherto missing clear geometrical interpretation of the well-known discrepancies between the equations of motion obtained by formal application of the calculus of variations (mathematics) and those obtained from the principle of d’Alembert-Lagrange (mechanics); i.e., admissible adjacent paths (mathematics) are locally nonvirtual; and adjacent paths built from locally virtual displacements (mechanics) are not admissible. (These discrepancies, although revealed about a century ago, for systems under Pfaffian constraints (Hertz (1894), Ho¨lder (1896), Hamel (1904), Maurer (1905), and others) seem to be relatively unknown and/or misunderstood among today’s engineers.) The discussion includes all relevant nonlinear nonholonomic variational principles, in both unconstrained and constrained forms of their integrands, and the corresponding nonlinear nonholonomic equations of motion. Such time-integral formulations are useful both conceptually and computationally (e.g., multibody dynamics).

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