D'Alembert-Lagrange principle

One of the basic, most general, differential variational principles of classical mechanics, expressing necessary and sufficient conditions for the correspondence of the real motion of a system of material points, subjected by ideal constraints, to the applied active forces. In the d'Alembert–Lagrange principle the positions of the system in its real motion are compared with infinitely close positions permitted by the constraints at the given moment of time.

According to the d'Alembert–Lagrange principle, during a real motion of a system the sum of the elementary works performed by the given active forces and by the forces of inertia for all the possible displacements is equal to or less than zero,

$$\sum_\nu(F_\nu-m_\nu w_\nu)\delta r_\nu\leq0,\label{*}\tag{*}$$

at any moment of time. The equality is valid for the possible reversible displacements, the symbol $\leq$ is valid for the possible irreversible displacements $\delta r_\nu$; $F_\nu$ are the given applied forces, and $m_\nu$ and $w_\nu$ are, respectively, the masses and the accelerations of the points. Equation \eqref{*} is the general equation of the dynamics of systems with ideal constraints; it comprises all the equations and laws of motion, so that one can say that all dynamics is reduced to the single general formula \eqref{*}.

The principle was established by J.L. Lagrange [1] by generalization of the principle of virtual displacements (cf. Virtual displacements, principle of) with the aid of the d'Alembert principle. For systems subject to bilateral constraints Lagrange based himself on formula \eqref{*} to deduce the general properties and laws of motion of bodies, as well as the equations of motion, which he applied to solve a number of problems in dynamics including the problems of motions of non-compressible, compressible and elastic liquids, thus combining "dynamics and hydrodynamics as branches of the same principle and as conclusions drawn from a single general formula" .

References

[1]	J.L. Lagrange, "Mécanique analytique" , 1–2 , Paris (1788) ((Also: Oeuvres, Vol. 11.))

Comments

The d'Alembert–Lagrange principle is quite close to the variational principle stating that the evolution path of a mechanical system subject to (holonomic) constraints constitutes an extremal for the action integral, cf. [a2], §21.

References

[a1]	H. Goldstein, "Classical mechanics" , Addison-Wesley (1962)
[a2]	V.I. Arnol'd, "Méthodes mathématiques de la mécanique classique" , MIR (1976) (Translated from Russian)