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Lagrangian, a. and n. Math.
(‖ lagrɑ̃ʒɪən, ləˈgrɒndʒɪən)
Also lagrangian, Lagrangean.
[f. prec. + -ian, -an.]
A. adj. Of or pertaining to the work of J. L. Lagrange (see prec.); of the kind introduced by Lagrange or associated with his work; spec. applied to (a) Lagrange('s) equation; (b) the difference between the kinetic energy and the potential energy of a system expressed as a function of generalized co-ordinates, their time derivatives, and time. (In many contexts interchangeable with Lagrange('s).)
1858A. Cayley in Rep. Brit. Assoc. Adv. Sci. 1857 i. 2 The above-mentioned form is par excellence the Lagrangian form of the equations of motion, and the one which has given rise to almost all the ulterior developments of the theory. Ibid. 15 When there is no force function..the forms corresponding to the untransformed forms in T and U are as follows, viz. the Lagrangian form is dq/dt = q′, d(dT/dq′)/dt - dT/dq = Q, and the Hamiltonian form is dq/dt = dT/dp, dp/dt = - dT/dq + Q. 1870Rep. Brit. Assoc. Adv. Sci. 1869 ii. 10 Equations of motion in the Lagrangean form. 1882Lagrangian method [see Eulerian a.]. 1904E. T. Whittaker Treat. Analytical Dynamics ii. 38 If we introduce a new function L of the variables q1, q2,{ddd}qn, q̇1,{ddd}q̇n, t, defined by the equation L = T - V, then Lagrange's equations can be written d(∂L/∂q̇ir)/dt - ∂L/∂qr = 0 (r = 1, 2,{ddd}n). The function L is called the Kinetic Potential, or Lagrangian function; this single function completely specifies, so far as dynamical investigations are concerned, a holonomic system for which the forces are conservative. 1908J. H. Jeans Math. Theory Electr. & Magn. xvi. 484 The Lagrangian equation corresponding to the coordinate r is found to be..d(∂T/∂ṙ)/dt - ∂(T - w)/∂r = R. Ibid. 486 Let Q be the charge on the positive plate at any instant, and let this be taken as a Lagrangian coordinate. 1949Math. Tables & Other Aids to Computation III. 466 Tables of 4-point and 6-point Lagrangean interpolation coefficients. 1954D. ter Haar Elem. Statistical Mech. 445 The values of the xi for which f is [an] extremum while equations (MA4.01) are satisfied can be determined from the equations ∂f / ∂xi + {Summ}pj = 1λi ∂gj / ∂xi = 0, i = 1,{ddd}, n. The xi are now functions of the λj, but these quantities, which are called the undetermined multipliers or the Lagrangian multipliers, can be eliminated by substituting for the xi into equations (MA4.01) and solving for the λj. 1957L. Fox Numerical Solution Two-Point Boundary Probl. ii. 31 Lagrangian [interpolation] formulae..may be defined as formulae in which all significant differences have been replaced by pivotal values. 1964Oceanogr. & Marine Biol. II. 14 The velocity along a particular trajectory varied more slowly than one would expect from the variability between one trajectory and another, suggesting that the Lagrangian scale of motion was noticeably larger than the Eulerian scale.
B. n. The Lagrangian function (see (b) above).
1938R. C. Tolman Princ. Statistical Mech. ii. 22 For more complicated mechanical systems, or even non⁓mechanical ones, suitable choices of the Lagrangian may be possible which will make Hamilton's principle still applicable. 1961J. Thewlis et al. Encycl. Dict. Physics II. 99/2 This function, called the Lagrangian of the given mechanical system, contains everything for the unique determination of the motion of the system, provided that we know its initial position and velocity. 1964E. A. Power Introd. Quantum Electrodynamics ix. 129 Then for the interaction with the electromagnetic field the substitution p → p - (e/c)A, iℏ∂/∂t → iℏ∂/∂t - eϕ is made in the Lagrangian of the free electron according to the principle of minimal electromagnetic coupling. 1973Physics Bull. Mar. 183/3 Nonlinear lagrangians.
Add:[A.] b. Astron. Lagrangian point: each of the five points in the orbital plane of two bodies at which a third body of negligible mass will remain stationary relative to them.
1962Struve & Zebergs Astron. 20th Cent. ii. 46 Particles of very small mass..which tend to linger near the Lagrangian point that lies about 70,000 mi. beyond the apex of the earth's shadow. 1975Physics Bull. Nov. 485/2 Gravity vanishes at all Lagrangian points. 1985Sci. Amer. Sept. 43/1 An excellent position for a satellite equipped to measure the small-scale distribution of velocities over the surface of the sun would be an orbit around the inner Lagrangian point, a point of gravitational equilibrium between the earth and the sun. |