One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description.

In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time.

In the Lagrangian description, the material derivative of is simply the partial derivative with respect to time, and the position vector is held constant as it does not change with time.

Continuity in the Lagrangian description is expressed by the spatial and temporal continuity of the mapping from the reference configuration to the current configuration of the material points.

where the functional form of in the Lagrangian description is not the same as the form of in the Eulerian description.

The vector joining the positions of a particle in the undeformed configuration and deformed configuration is called the displacement vector, in the Lagrangian description, or, in the Eulerian description.

In late 2007, Bagger and Lambert set off renewed interest in M-theory with the discovery of a candidate Lagrangian description of coincident M2-branes, based on a non-associative generalization of Lie Algebra, Nambu 3-algebra or Filippov 3-algebra.

Alternatively, a Lagrangian description of the two-body problem gives a Lagrangian of

Vortex dynamics has retained a characteristic " flavor " deriving from its particle-based ( Lagrangian ) interpretation and from its frequently intuitive, " mechanistic " description of flow phenomena.

This linearisation implies that the Lagrangian description and the Eulerian description are approximately the same as there is little difference in the material and spatial coordinates of a given material point in the continuum.

This mathematical concept is closely related to the description of fluid motion — its kinematics and dynamics — in a Lagrangian frame of reference.

The following Lagrangian contains the complete description of the Brans / Dicke theory:

The vector joining the positions of a particle in the undeformed configuration and deformed configuration is called the displacement vector in the Lagrangian description, or in the Eulerian description, where and are the unit vectors that define the basis of the material ( body-frame ) and spatial ( lab-frame ) coordinate systems, respectively.

Except for special cases like the Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies.

This formalism is analogous to the Lagrangian formalism used in classical mechanics to solve for the motion of a particle under the influence of a field.

Quantum field theory uses this same Lagrangian procedure to determine the equations of motion for quantum fields.

Next, we can substitute this Lagrangian into the Euler-Lagrange equation of motion for a field:

Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation

A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics ( developed in 1788 and 1833, respectively ), it does not apply to systems that cannot be modeled with a Lagrangian alone ( e. g. systems with a Rayleigh dissipation function ).

As an illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion.

As examples, in Lagrangian mechanics the equations of motion are derived by finding stationary points of the action, the time integral of the difference between kinetic and potential energy.

Starting with Lagrangian mechanics, the equations of motion are based on generalized coordinates

If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler – Lagrange equation.

The Lagrangian of a given system is not unique, but solving any equivalent Lagrangians will give the same equations of motion.

Lagrangian mechanics and Noether's theorem together yield a natural formalism for first quantization by including commutators between certain terms of the Lagrangian equations of motion for a physical system.

* If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry.

It thus describes systems where the equations of motion or the Lagrangian obey certain symmetries, but the lowest energy solutions do not exhibit that symmetry.

Lagrangian multiphase models, which are used for dispersed media, are based on solving the Lagrangian equation of motion for the dispersed phase.

is the Euler-Lagrange equation of motion derived from the Lagrangian density,

Since the Lagrangian ( eq ( 1 )) contains second derivatives, the Euler – Lagrange equation of motion for this field is

In the Lagrangian and Hamiltonian formalisms, the constraints of the situation are incorporated into the geometry of the motion, and in doing so the number of coordinates is reduced to only the minimum number needed to define the motion.

For infinitesimal deformations of a continuum body, in which the displacements and the displacement gradients are small compared to unity, i. e., and, it is possible to perform a geometric linearisation of the Lagrangian finite strain tensor, and the Eulerian finite strain tensor.

The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.

Relevant operators are those responsible for perturbations to the free energy, the imaginary time Lagrangian, that will affect the continuum limit, and can be seen at long distances.

* Generalized Lagrangian mean, a method in continuum mechanics to split a flow field into a mean ( average ) part and a wave part

Two moons are known to have small companions at their and Lagrangian points, sixty degrees ahead and behind the body in its orbit.

Recent work by Edward Belbruno and J. Richard Gott of Princeton University suggests that a suitable impacting body could form in a planet's trojan points ( or Lagrangian point ).

In Geometry of Batalin-Vilkovisky quantization, Albert Schwarz has proven that the integral of a function H over a Lagrangian submanifold L depends only on the cohomology class of H and on the homology class of the body of L in the body of the ambient supermanifold.

The name of the video game references Lagrangian points, the five positions in space where a body of negligible mass could be placed which would then maintain its position relative to two existing massive bodies.

In the canonical quantum field theory the S-matrix is represented within the interaction picture by the perturbation series in the powers of the interaction Lagrangian,

where is the interaction Lagrangian and signifies the time-ordered product of operators.

Most of this dust is orbiting the Sun in about the ecliptic plane, with a possible concentration of particles at the Earth – Sun Lagrangian point.

It also turns out that, at least in the case of Sun – Earth-missions, it is actually preferable to place the spacecraft in a large-amplitude () Lissajous orbit, instead of having it sit at the Lagrangian point, because this keeps the spacecraft off the direct Sun – Earth line, thereby reducing the impact of solar interference on Earth – spacecraft communications.

Another interesting and useful property of the collinear Lagrangian points and their associated Lissajous orbits is that they serve as " gateways " to control the chaotic trajectories of the Interplanetary Transport Network.

This same effect is present at the Lagrangian points in the Earth – Moon system, where the analogue of the string is the summed ( or net ) gravitational attraction of the two masses, and the stone is an asteroid or a spacecraft.

At Lagrangian points,,, and, a spacecraft's inertia to move away from the barycenter is balanced by the attraction of gravity toward the barycenter.

In Lagrangian mechanics, a Lagrangian is defined as the difference between the kinetic energy and the potential energy:

In Hamiltonian mechanics, the Lagrangian ( a function of generalized coordinates and their derivatives ) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum.

where the momentum is obtained by differentiating the Lagrangian as above.

As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.

The gauge invariant QCD Lagrangian is

An important theoretical notion concerning the final term of the above Lagrangian is the Wilson loop variable.