Pré-Publication, Document De Travail Année : 2005

Les équations de mécanique classique en formes laconiques

Résumé

1. Various versions of a new general form (proposed in [B,C]) of dynamical equations in holonomic and nonholonomic mechanics are discussed. The most unusual aspect in our form is the following: the calculation of almost all terms in dynamical equations is performed with the aid of Poisson brackets before the canonical momenta will acquire their usual meaning (formal Poisson brackets). The form is concise and easy to memorize. It is also convenient for symbolic computation. It presents a universal approach to replace or to obtain the usual equations for non-holonomic systems given by Chaplygin, Maggi, Voronets, Boltzmann, Hamel, Przeborski, Johnsen. 2. A new approach to mechanical systems with servo-constraints is proposed. The idea is to insert a non-classical energy of accelerations in the Gauss principle of least constraint. 3. For systems with servo-constraints, a new form of equations also on formal Poisson brackets is given. 4.A similar form of equations is discussed for a large class of dynamical systems including the vakonomic systems of V.V.Kozlov. 5. In [A], a pseudo-canonical structure (Poisson bracket which not satisfy the Poisson identity) was defined directly on the tangent bundle or a sub-bundle for a Riemannian manifold. Сonsider a second-order equation, which has degree two in the velocities, is reversible and has a non-degenerate quadratic integral (energy). The equation takes the form of Newton's law written in a Riemannian connexion with torsion for the metric defined by the integral. A non-holonomic system appear if a tangent sub-bundle is invariant for the flow. It possesses supplementary linear and quadratic integrals under precise geometric conditions. In the present paper, for reversible system defined only on the sub-bundle, sufficient conditions are now given to represent a classical non-holonomic system. If the system has two degrees of freedom, these conditions are equivalent to the non-integrability of the equations of constraints. 6. Chaplygin's theorem on the multiplier is generalized. 7. A new phenomenon of isoenergic Hamiltonian for non-holonomic systems of 3/2 degrees of freedom is presented. MAIN AUTHOR'S ARTICLES: [A] Geometric formalism of classical dynamics. Canonical reformulations of fundamental theorems for configurations with torsion (in particular, nonholonomic). (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1986, no. 3, 74-81. [B] Universal characteristic function and moving-frame method in the dynamics of systems with nonintegrable constraints. Mosc. Univ. Mech. Bull. 44, No.2, 1-10 (1989); translation from Vestn. Mosk. Univ., Ser. I. 1989, No.2, 60-66. [C] Equations of classical mechanics in new form. (Russian) Vestn. Mosk. Univ., Ser. I 2003, No.3, 67-76. 87

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hal-00015612 , version 1 (09-12-2005)

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Yaroslav Vsevolodovich Tatarinov. Les équations de mécanique classique en formes laconiques. 2005. ⟨hal-00015612⟩

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