Extended Lagrangian formalism and main general principles of mechanics

The main topic of this paper is the formulation of the main general principles of mechanics for the rheonomic systems with the nonstationary constraints f μ [ r → ν , φ a ( t ) ] = 0 in the extended Lagrangian formalism, founded by the author himself [Mušicki, Dj., 2004. Eur. J. Mech. A Solids 23, 975]. The starting point of this formulation is the introduction of the new quantities, suggested by these constraints, which change according to the law τ a = φ a ( t ) . In transition to the generalized coordinates, which here always refer to some moving frame of reference, it is demonstrated that these new quantities determine the position of this reference frame with respect to an immobile one. They are taken as the additional, independent generalized coordinates, whose dependence on time is given a priori. 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 dʹAlembert–Lagrangeʹs 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 Hamiltonʹs principle, the Lagrangeʹs principle of the least action and the associated Jacobiʹs one are formulated. With the aid of the corresponding generalized Hölder–Vossʹs relation, a correlation between these integral principles is established. Finally, the obtained results are illustrated by a simple, but characteristic example.