With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.

* Lagrange polynomials

In numerical analysis, Lagrange polynomials are used for polynomial interpolation.

Lagrange interpolation is susceptible to Runge's phenomenon, and the fact that changing the interpolation points requires recalculating the entire interpolant can make Newton polynomials easier to use.

of Lagrange basis polynomials

As can be seen in the following derivation the weights are derived from the Lagrange basis polynomials.

Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm.

where are the Lagrange basis polynomials associated with the points.

The Lebesgue constant can be expressed in terms of the Lagrange basis polynomials:

The Lagrangian points (; also Lagrange points, L-points, or libration points ) are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be part of a constant-shape pattern with two larger objects ( such as a satellite with respect to the Earth and Moon ).

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

Lagrange proved that for any natural number n that is not a perfect square there are x and y > 0 that satisfy Pell's equation.

These leading and trailing points are called the and Lagrange points.

Most inner moons of planets have synchronous rotation, so their synchronous orbits are, in practice, limited to their leading and trailing ( and ) Lagrange points, as well as the and Lagrange points, assuming they do not fall within the body of the moon.

Supporting this theory, extrasolar planets have been discovered in Lagrange points of each other, and are expected to collide, after co-orbiting for millions of years.

However, these linear orbits are not as stable as, for example, the equilateral Lagrange orbits and.

In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler – Lagrange equations and Hamilton's equations.

The Euler – Lagrange equations of motion for the functional E are then given in local coordinates by

Lagrange also reformulated the principles of classical mechanics, emphasizing energy more than force and developing a method to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic.

If is a maximum of for the original constrained problem, then there exists such that is a stationary point for the Lagrange function ( stationary points are those points where the partial derivatives of are zero, i. e. ).

are called Lagrange Multipliers and this optimization method itself is called The Method of Lagrange Multipliers.

In optimal control theory, the Lagrange multipliers are interpreted as costate variables, and Lagrange multipliers are reformulated as the minimization of the Hamiltonian, in Pontryagin's minimum principle.

The ITN makes particular use of Lagrange points as locations where trajectories through space are redirected using little or no energy.

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when and Take to obtain We have

Around 1770, Joseph Louis Lagrange began the groundwork that unified the many different tricks that had been used up to that point to solve equations, relating them to the theory of groups of permutations, in the form of Lagrange resolvents.

* Conceptual introduction ( plus a brief discussion of Lagrange multipliers in the calculus of variations as used in physics )

The STScI is currently developing similar processes for JWST, although the operational details will be very different due to its different instrumentation and spacecraft constraints, and its location at the Sun-Earth L2 Lagrange point (~ 1. 5 million km from Earth ) rather than the low Earth orbit (~ 565 km ) used by HST.

* Lagrange multiplier, a scalar variable used in mathematics to solve an optimisation problem for a given constraint.

Although generalized to triple integrals by Lagrange in 1773, and used by Legendre, Laplace, Gauss, and first generalized to n variables by Mikhail Ostrogradski in 1836, it resisted a fully rigorous formal proof for a surprisingly long time, and was first satisfactorily resolved 125 years later, by Élie Cartan in a series of papers beginning in the mid-1890s (; ).

Although Lagrange only sought to describe classical mechanics, the action principle that is used to derive the Lagrange equation was later recognized to be applicable to quantum mechanics as well.

The identity was used by Lagrange to prove his four square theorem.

** Konami VRC7, used in one Famicom game Lagrange Point.

Jumps are normally made to and from points far above a solar system's ecliptic, usually where the gravitational influence in the system is most stable ; however, so-called " pirate points " exist where local gravitational pull is stable enough to used ; though quicker, using such points is also more dangerous due the random appearance of so-called " Lagrange points ".

the additional parameter is a Lagrange multiplier used in the minimization of the kriging error to honor the unbiasedness condition.

When the probability is derived from the Gibbs measure, as it would be for any Markovian process, then can also be understood to be a Lagrange multiplier ; Lagrange multipliers are used to enforce constraints, such as holding the expectation value of some quantity constant.

Whereas optimization methods are nearly as old as calculus, dating back to Isaac Newton, Leonhard Euler, Daniel Bernoulli, and Joseph Louis Lagrange, who used them to solve problems such as the shape of the catenary curve, numerical optimization reached prominence in the digital age.

Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.

In spaceflight, especially for NASA's Constellation Program, the term sortie has been used for a flight of the Orion spacecraft beyond the confluence of low-Earth orbit, such as a flight to the Moon or to the Sun-Earth L < sub > 2 </ sub > Lagrange Point.

Lagrange first used the method in 1766.

* Konami's VRC7 soundchip is a YM2413 derivative, used exclusively on the NES game Lagrange Point.

Lagrange Point has the distinction of being the only game ever released with Konami's VRC7 sound generator Integrated Circuit, which allowed for a drastic improvement in the quality of the music and sound effects used in the game.

* Games used in: Lagrange Point, Tiny Toon Adventures 2

This advanced audio was used only in the Famicom game Lagrange Point ; while the Japanese version of Tiny Toon Adventures 2 also used the VRC7, it did not make use of the extended audio.