On the energy conservation theorem for the system with rheonomous constraints

Veljko A. Vujičić

Theoretical and applied mechanics, Tome 6 (1980) no. 1, p. 133 .

Résumé

It is general accepted that the law of energy conseryation for mechanical system with rheonomous constraints is not valid. The aim of this paper is to proove that it is possible to evaluate the energy integral (2.21) of iheonomous system when the potential force exists. Motion of the system with $n$ degrees of freedom is described by $n+1$ equations (1.4). The equation (3.6) has been added to the system of second order Lagrange differential equations (1.16) that correspond to the coordinate-time $q^0=t$. Two equations (4.1) hawe been added to the Hamilton equation. This set of equations finally leads to the law of kinetic energy variation of rheonomous system for the real motion in the form (1.7) and also to the energy integral in form (1.12) or (2.20). By equations (2.14) and (2.15) the way to find out the energy conservation theorem is shown, using $n+1$ differential equations of motion that corresponds to the time instant $t$. The energy integrals obtained (2.20) were compared with Yacoby first energy integral. It was showu that these integrals are obtained for the particular conditions as the first integrals of $n$ Lagrange differential equations of second order while the energy integral obtained is the integral of the system of $n+1$ differential equations.