Realization of nonintegrable constraints in classical mechanics. (English. Russian original) Zbl 0579.70014

Sov. Phys., Dokl. 28, 735-737 (1983); translation from Dokl. Akad. Nauk SSSR 272, 550-554 (1983).
A multidimensional mechanical system with the Lagrangian \[ L_{\epsilon}=1/2(A(q)\dot q\cdot \dot q)+(\alpha \epsilon /2)(a(q)\dot q)^ 2+V(q) \] is considered. The system is acted upon by the potential forces with the force function V(q) and by the dissipative function \(F_{\epsilon}=(-\beta \epsilon /2)(a(q)\dot q)^ 2\). The equation of motion in Lagrange’s form \[ (1)\quad d/dt(\partial L\epsilon /\partial \dot q)-\partial L\epsilon /\partial q=\partial F\epsilon /\partial q. \] The values of \(\alpha >0\) and \(\beta\geq 0\) are fixed and \(\epsilon\to \infty\). The problem is to describe the limiting solutions of (1). It is shown that for all t the limiting motion q(t) satisfies the ”constraint” equation (2) \(a(q)\dot q=0\). The limiting case of large \(\mu =\beta /\alpha\) is studied. For every fixed \(\mu\) we have the equations of motion of a mechanical system with the Lagrangian \(L_ 0\) and the constraint (2). The family of intrinsically noncontradictory mathematical models represents the synthesis of conventional non-holonomic dynamics and vacconomic dynamics.
Reviewer: A.B.Borisov


70F25 Nonholonomic systems related to the dynamics of a system of particles
70H03 Lagrange’s equations