Lagrangianadj.n.

/ləˈɡrɒnʒɪən/

/ləˈɡrɒndʒɪən/

Forms: Also lagrangian, Lagrangean.

Etymology: < Lagrange n. + -ian suffix, -an suffix.

Mathematics.

A. adj.

Of or pertaining to the work of J. L. Lagrange (see Lagrange n.); of the kind introduced by Lagrange or associated with his work; spec. applied to (a) Lagrange n.; (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).)

ΚΠ

1858 A. Cayley in Rep. Brit. Assoc. Advancem. 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.

1858 A. Cayley in Rep. Brit. Assoc. Advancem. Sci. 1857 i. 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.

1870 Rep. Brit. Assoc. Advancem. Sci. 1869 ii. 10 Equations of motion in the Lagrangean form.

1882 G. M. Minchin Uniplanar Kinematics 139 In the first case what we have done for the fixed-space point P we imagine to be done for all fixed-space points; and in the second case we imagine our record to be similarly kept for every individual fluid particle..They are..often called the Eulerian and the Lagrangian methods, respectively.

1904 E. T. Whittaker Treat. Analyt. Dynamics ii. 38 If we introduce a new function L of the variables q₁, q₂,…q_n, q̇₁,…q̇_n, t, defined by the equation L = T − V, then Lagrange's equations can be written d(∂L/∂q̇_ir)/dt − ∂L/∂q_r = 0 (r = 1, 2,…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.

1908 J. 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.

1908 J. H. Jeans Math. Theory Electr. & Magn. xvi. 486 Let Q be the charge on the positive plate at any instant, and let this be taken as a Lagrangian coordinate.

1949 Math. Tables & Other Aids Computation 3 466 Tables of 4-point and 6-point Lagrangean interpolation coefficients.

1954 D. ter Haar Elem. Statistical Mech. 445 The values of the x_i for which f is [an] extremum while equations (MA4.01) are satisfied can be determined from the equations ^∂f/ _{∂x_i} + ∑pj = 1λ_i^∂gj/ _{∂x_i} = 0, i = 1,…, n. The x_i are now functions of the λ_j, but these quantities, which are called the undetermined multipliers or the Lagrangian multipliers, can be eliminated by substituting for the x_i into equations (MA4.01) and solving for the λ_j.

1957 L. 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.

1964 Oceanogr. & Marine Biol. 2 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).

ΘΚΠ

the world > matter > physics > mechanics > dynamics > [noun] > specific function

H1835

Hamiltonian1926

Lagrangian1938

1938 R. 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.

1961 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.

1964 E. 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.

1973 Physics Bull. Mar. 183/3 Nonlinear lagrangians.

Draft additions 1997

Lagrangian point n. Astronomy 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.

ΘΚΠ

the world > the universe > heavenly body > movement of heavenly bodies > [noun] > movement in orbit > orbit > parts of orbit

aphelion1656

perihelion1666

vector1704

elongation1708

orb1733

periaster1851

orbitale1901

Lagrangian point1962

1962 Struve & 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.

1975 Physics Bull. Nov. 485/2 Gravity vanishes at all Lagrangian points.

1985 Sci. 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.

This entry has not yet been fully updated (first published 1976; most recently modified version published online December 2020).

单词	lagrangian
释义	Lagrangianadj.n. /ləˈɡrɒnʒɪən//ləˈɡrɒndʒɪən/ Forms: Also lagrangian, Lagrangean. Etymology: < Lagrange n. + -ian suffix, -an suffix. Mathematics. A. adj. Of or pertaining to the work of J. L. Lagrange (see Lagrange n.); of the kind introduced by Lagrange or associated with his work; spec. applied to (a) Lagrange n.; (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).) ΚΠ 1858 A. Cayley in Rep. Brit. Assoc. Advancem. 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. 1858 A. Cayley in Rep. Brit. Assoc. Advancem. Sci. 1857 i. 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. 1870 Rep. Brit. Assoc. Advancem. Sci. 1869 ii. 10 Equations of motion in the Lagrangean form. 1882 G. M. Minchin Uniplanar Kinematics 139 In the first case what we have done for the fixed-space point P we imagine to be done for all fixed-space points; and in the second case we imagine our record to be similarly kept for every individual fluid particle..They are..often called the Eulerian and the Lagrangian methods, respectively. 1904 E. T. Whittaker Treat. Analyt. Dynamics ii. 38 If we introduce a new function L of the variables q₁, q₂,…q_n, q̇₁,…q̇_n, t, defined by the equation L = T − V, then Lagrange's equations can be written d(∂L/∂q̇_ir)/dt − ∂L/∂q_r = 0 (r = 1, 2,…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. 1908 J. 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. 1908 J. H. Jeans Math. Theory Electr. & Magn. xvi. 486 Let Q be the charge on the positive plate at any instant, and let this be taken as a Lagrangian coordinate. 1949 Math. Tables & Other Aids Computation 3 466 Tables of 4-point and 6-point Lagrangean interpolation coefficients. 1954 D. ter Haar Elem. Statistical Mech. 445 The values of the x_i for which f is [an] extremum while equations (MA4.01) are satisfied can be determined from the equations ^∂f/ _{∂x_i} + ∑pj = 1λ_i^∂gj/ _{∂x_i} = 0, i = 1,…, n. The x_i are now functions of the λ_j, but these quantities, which are called the undetermined multipliers or the Lagrangian multipliers, can be eliminated by substituting for the x_i into equations (MA4.01) and solving for the λ_j. 1957 L. 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. 1964 Oceanogr. & Marine Biol. 2 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). ΘΚΠ the world > matter > physics > mechanics > dynamics > [noun] > specific function H1835 Hamiltonian1926 Lagrangian1938 1938 R. 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. 1961 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. 1964 E. 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. 1973 Physics Bull. Mar. 183/3 Nonlinear lagrangians. Draft additions 1997 Lagrangian point n. Astronomy 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. ΘΚΠ the world > the universe > heavenly body > movement of heavenly bodies > [noun] > movement in orbit > orbit > parts of orbit aphelion1656 perihelion1666 vector1704 elongation1708 orb1733 periaster1851 orbitale1901 Lagrangian point1962 1962 Struve & 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. 1975 Physics Bull. Nov. 485/2 Gravity vanishes at all Lagrangian points. 1985 Sci. 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. This entry has not yet been fully updated (first published 1976; most recently modified version published online December 2020). < adj.n.1858
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