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A constructive method of establishing the validity of the theory of systems with non-retaining constraints. (English. Russian original) Zbl 0711.70021

J. Appl. Math. Mech. 52, No. 6, 691-699 (1988); translation from Prikl. Mat. Mekh. 52, No. 6, 883-894 (1988).
Summary: The formal axiomatic approach to establishing the validity of the theory of constrained systems has obvious disadvantages: the source of the initial axioms (such as the Befreiungsprinzip and the conditions for constraints to be ideal) remains unclear. A constructive method is proposed for establishing the validity of the main principles of the dynamics of unilaterally constrained systems (including systems with collisions). The idea of the method is related to the analysis of physical methods for realising constraints (stiff systems, anisotropic viscosity, and apparent additional masses). This approach yields simple equations of motion, suitable for the entire time interval and more accurately incorporating the actual dynamics. Several problems of the mechanics of oscillatory systems with collisions are solved by the method. In particular, conditions are determined for the stability of periodic oscillatory modes and a study is made of the evolution of motion with inealistic collisions when the coefficient of restitution is close to unity. Total integrability is established and a qualitative analysis is presented of the problem of parabolic billiards in a uniform force field.

MSC:

70H03 Lagrange’s equations
70F20 Holonomic systems related to the dynamics of a system of particles
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70F35 Collision of rigid or pseudo-rigid bodies
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