Lagrangen.

/ləˈɡrɒnʒ/

/ləˈɡrɒndʒ/

Etymology: < the name of Joseph Louis Lagrange (1736–1813), the Italian-born mathematician who worked in Prussia and France.

Used attributively and in the possessive to designate various concepts introduced by Lagrange or arising out of his work, as Lagrange equation n. (also 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 q_r and forces Q_r, and to the time t, and having the form d(∂T/∂q̇_r)/dt − ∂T/∂q_r = Q_r. (In many contexts interchangeable with Lagrangian.)

ΚΠ

1858 Rep. 27th Meeting Brit. Assoc. Advancem. 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.

1902 Encycl. Brit. XXVII. 568/1 Hence the typical Lagrange's equation may be now written in the form d(∂T/∂q̇_r)/dt − ∂T/∂q_r = −∂V/∂q_r, or, again, ṗ_r = −∂(V − T)/∂q_r.

1902 Encycl. Brit. XXVII. 568/1 A classical example of the application of Lagrange's equations is to the motion of a top.

1942 J. L. Synge & B. A. 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.

1958 E. U. Condon & H. Odishaw Handbk. Physics v. ii. 18/1 If the total number of systems is N, Σn_j = N Σn_jE_j = E... Using the method of Lagrange multipliers, introduce multipliers β and λ and find the set of n_j's which make δ[log P − λ(N − Σn_j) + β(E − Σn_jE_j)] = 0.

1962 J. Riordan Stochastic Service Syst. iv. 66 g(y)e^−g(y) = y. The solution of this, obtained by Lagrange expansion, is g(y) = ∑n = 1 (n^{n − 1}yⁿ)/n!

1967 M. G. Smith Introd. Theory Partial Differential Equations i. 3 Comparing (1.2.8) and (1.2.9) we have the Lagrange equations ṗ_k = ∂L/∂q_k and p_k = ∂L/∂q̇_k.

This entry has not yet been fully updated (first published 1976; most recently modified version published online June 2021).

单词	lagrange
释义	Lagrangen. /ləˈɡrɒnʒ//ləˈɡrɒndʒ/ Etymology: < the name of Joseph Louis Lagrange (1736–1813), the Italian-born mathematician who worked in Prussia and France. Used attributively and in the possessive to designate various concepts introduced by Lagrange or arising out of his work, as Lagrange equation n. (also 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 q_r and forces Q_r, and to the time t, and having the form d(∂T/∂q̇_r)/dt − ∂T/∂q_r = Q_r. (In many contexts interchangeable with Lagrangian.) ΚΠ 1858 Rep. 27th Meeting Brit. Assoc. Advancem. 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. 1902 Encycl. Brit. XXVII. 568/1 Hence the typical Lagrange's equation may be now written in the form d(∂T/∂q̇_r)/dt − ∂T/∂q_r = −∂V/∂q_r, or, again, ṗ_r = −∂(V − T)/∂q_r. 1902 Encycl. Brit. XXVII. 568/1 A classical example of the application of Lagrange's equations is to the motion of a top. 1942 J. L. Synge & B. A. 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. 1958 E. U. Condon & H. Odishaw Handbk. Physics v. ii. 18/1 If the total number of systems is N, Σn_j = N Σn_jE_j = E... Using the method of Lagrange multipliers, introduce multipliers β and λ and find the set of n_j's which make δ[log P − λ(N − Σn_j) + β(E − Σn_jE_j)] = 0. 1962 J. Riordan Stochastic Service Syst. iv. 66 g(y)e^−g(y) = y. The solution of this, obtained by Lagrange expansion, is g(y) = ∑n = 1 (n^{n − 1}yⁿ)/n! 1967 M. G. Smith Introd. Theory Partial Differential Equations i. 3 Comparing (1.2.8) and (1.2.9) we have the Lagrange equations ṗ_k = ∂L/∂q_k and p_k = ∂L/∂q̇_k. This entry has not yet been fully updated (first published 1976; most recently modified version published online June 2021). < n.1858
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