D'Alembert principle in the velocity space (Q1969933)

The authors consider the Newton's dynamical equations \(F_i+ R_i=m_i \ddot r_i\), where the configuration \(r_i=r_i(q_1,\dots,q_n,t)\) depends on generalized coordinates, and principal and constraining forces \(F_i\), \(R_i\) are functions of variables \(r_i\), \(\dot r_i,t\). By using the virtual displacement \(\delta\dot r_i\) in velocity space, the authors derive the d'Alembert principle \(\Sigma(\dot Q_s-{d \over dt}{\partial S\over \partial \ddot q_s}+{\partial S\over\partial\dot q_s})\delta \dot q_s=0\) in the velocity space, where \(\dot Q_s=\Sigma (\dot F_i+\dot R_i)\partial \dot r_i/ \partial \dot q_s\), and \(S={1\over 2}\Sigma m_i\ddot r_i^2\) is the accelerated energy (i.e. the dynamical energy in the velocity space). The results can be generalized in the case of nonholonomic constraints \(f_k(q_1, \dots, q_n,\dot q_1,\dots, \dot q_n,t)=0\).