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Lagrange
(‖ lagrɑ̃ʒ, ləˈgrɒndʒ)
The name of Joseph Louis Lagrange (1736–1813), the Italian-born mathematician who worked in Prussia and France, used attrib. and in the possessive to designate various concepts introduced by him or arising out of his work, as Lagrange('s) equation, each of a set of equations of motion in classical dynamics relating the total kinetic energy T of a system to a set of generalized co-ordinates qr and forces Qr, and to the time t, and having the form d(∂T/∂q̇r)/dt - ∂T/∂qr = Qr. (In many contexts interchangeable with Lagrangian.)
1858Rep. Brit. Assoc. Adv. Sci. 1857 i. 12 The force function U is independent of the differential coefficients η′,..and, consequently, of the variables ω,.., hence, writing H = T - U, the equations take the form dη/dt = dH/dω, dω/dt = -dH/dη,..which correspond to the condensed form obtained by writing T - V = R in Lagrange's equations. 1902Encycl. Brit. XXVII. 568/1 Hence the typical Lagrange's equation may be now written in the form d(∂T/∂q̇r)/dt - ∂T/∂qr = -∂V/∂qr, or, again, ṗr = -∂(V - T)/∂qr. Ibid., A classical example of the application of Lagrange's equations is to the motion of a top. 1942Synge & Griffith Princ. Mech. xv. 453 Two features of Lagrange's equations should be emphasized. First, there is no unique set of generalized coordinates; however we choose them, the equations of motion always have the form (15.215). Secondly, since only working forces contribute to δW, reactions of constraint are automatically eliminated. [Note] Except where forces of friction do work. 1958Condon & Odishaw Handbk. Physics v. ii. 18/1 If the total number of systems is N, σnj = N σnjEj = E... Using the method of Lagrange multipliers, introduce multipliers β and λ and find the set of nj's which make δ[log P - λ(N - σnj) + β(E - σnjEj)] = 0. 1962J. Riordan Stochastic Service Syst. iv. 66, g(y)e-g(y) = y. The solution of this, obtained by Lagrange expansion, is g(y) = {Summ}n = 1 (nn - 1yn)/n! 1967M. G. Smith Introd. Theory Partial Differential Equations i. 3 Comparing (1.2.8) and (1.2.9) we have the Lagrange equations ṗk = ∂L/∂qk and pk = ∂L/∂q̇k. |
