Stability of equilibrium of nonholonomic rheonomic systems

Mila Kažić

Theoretical and applied mechanics, Tome 8 (1982) no. 1, p. 41 .

Résumé

In this article, the motion of a system of $N$ material points acted upon by $K$ holonomic and $L$ non-holonomic connections, where both are rheonomic, is replaced by an equivalent problem - the motion of a representative point in the expanded phase space $V_{2n+2}$ with coordinates: \[ q^0=t, q^1,\dots,q^n;\quad p_0=lpha_{0\mu}\dot q^{\bar\mu} p_1,\dots,p_n \] In relation to the given space, the first dif. equations of motion, and then diff. equations of the disturbed state of equilibrium. The stability of the equilibrium state is tested using Lyapunov's direct method. However, the Lyapunov function $V$ is assumed in the form of the sum of two functions $T$ and $W$ (as is done in the papers that follow in the cited literature). $T$ is the kinetic energy, which is shown to be positive definite with respect to the generalized impulses $p_0,p_1,\dots,p_n$; and we look for the function $W$ as a function only of the generalized coordinates $q_0,q^1,\dots,q^n$ and we require it to have the properties of a Lyapunov function. In this way, the problem is, at least in principle, simplified - the required function $W$ depends on twice as few variables as the function $V$.