DAlembert-Lagrange Principle

one of the primary principles of mechanics that offers a general method for solving problems of dynamics and statics. Named after the French scientists J. D’AIembert and J. Lagrange, it combines the virtual work principle and d’Alembert’s principle. If the inertial force J, is combined with the active forces F₁, which are acting on the points of a mechanical system, then according to the d’Alembert-Lagrange principle, when the mechanical system moves with ideal constraints at each moment in time, the sum of the elementary work of the active forces δA₁^a and the elementary work of the inertial forces δA₁^u on any virtual displacement of the system is equal to zero. Mathematically, the d’Alembert-Lagrange principle is expressed by an equality, which is also called the general equation of mechanics:

∑(δA_i^a + δA_i^u) = ∑(F_i cos α _i + J_i cos β_i)δs_i = 0

Where δs_i is the magnitude of virtual displacements of the points of the system, α _i and β_i are the angles between the directions of the corresponding forces and virtual displacements, and the inertial forces J_i = −m_iw_i in which m_i is the mass of the points of the system and vf,, their acceleration. The advantage of the d’Alembert-Lagrange principle is that it makes it possible to study the motion of a system without introducing unknown constraints into the equations.