## 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

### MSC:

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