Deriving generalized variational principles in general mechanics by using Lagrangian multiplier method (Q1974182)

The ancient method of integration by parts (due to Lagrange) is formally applied to derive Euler-Lagrange system for variational integral \(\int(T(q,\dot q,t)-V(q,t))dt\) subjected to nonholonomic constraints \(f_i(q,\dot q,t)=0\). The derivatives \(\dot q=(dq^1/dt, \dots, dq^m/ dt)\) may be regarded as new variable functions. Then \(T(q,\dot q,t)= T(q,v,t)\) is inserted into the integral, the constraints are completed by equation \(\dot q-v=0\), the resulting Euler-Lagrange system is appropriately modified, and two kinds of variables are employed to formulate a generalized variational principle.