Extended Lagrangian formalism and main general principles of mechanics

In so extended Lagrangian formalism the main differential and integral principles of mechanics are formulated, in the form where the influence of the nonstationary constraints is expressed explicitly. So, starting from the work of the ideal forces of constraints along arbitrary virtual displacements of the particles, the corresponding d'Alembert-Lagrange's principle is formulated, and from it an extended system of the Lagrangian equations is obtained. By transition to the integral principles via the corresponding central Lagrangian equation, the general Hamilton's principle, the Lagrange's principle of the least action and the associated Jacobi's one are formulated. With the aid of the corresponding generalized Hölder-Voss's relation, a correlation between these integral principles is established. Finally, the obtained results are illustrated by a simple, but characteristic example.