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 motion of a continuum body is expressed by the mapping function ( Figure 2 ),

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.

In the generalized Lagrangian mechanics, the generalized coordinates q ( or generalized momenta p ) replace the ordinary position ( or momentum ).

called the Lagrangian density, depending on φ, its derivative and the position.

The Lagrangian is a function of the position now and the position a little later ( or, equivalently for infinitesimal time separations, it is a function of the position and velocity ).

For spacecraft in a halo orbit around a Lagrangian point stationkeeping is even more fundamental as such an orbit is unstable ; without an active control with thruster burns the smallest deviation in position / velocity would result in the spacecraft leaving the orbit completely.

*, Lagrangian point 1, the most intuitive position for an object to be gravitationally stationary relative to two larger objects ( such as a satellite with respect to the Earth and Moon ).

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.

High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics ; these are abstract spaces, independent of the physical space we live in.

The action of a physical system is the integral over time of a Lagrangian function ( which may or may not be an integral over space of a Lagrangian density function ), from which the system's behavior can be determined by the principle of least action.

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 another example, if a physical process exhibits the same outcomes regardless of place or time ( having the same outcome, say, somewhere in Asia on a Tuesday or in America on a Friday ), then its Lagrangian is symmetric under continuous translations in space and time: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.

It is assumed that the symmetry of the Lagrangian is rotational, i. e., that the Lagrangian does not depend on the absolute orientation of the physical system in space.

The same principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system.

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.

Frieden states, if the effect has an intrinsic information level J, and is observed with information level I, then the physical information is defined to be the difference I − J, which Frieden calls the information Lagrangian.

The solution was to realize that the quantities initially appearing in the theory's formulae ( such as the formula for the Lagrangian ), representing such things as the electron's electric charge and mass, as well as the normalizations of the quantum fields themselves, did not actually correspond to the physical constants measured in the laboratory.

In Lagrangian mechanics, because of Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler – Lagrange equation for the action of the system.

The guiding principle of technicolor is " naturalness ": basic physical phenomena should not require fine-tuning of the parameters in the Lagrangian that describes them.

According to Noether's theorem, if the action ( the integral over time of its Lagrangian ) of a physical system is invariant under rotation, then angular momentum is conserved.

Note that the constant term m²v² in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.

However self interaction never fades away and, besides many other effects, it causes a difference between the Lagrangian mass and the physical mass of the boson.

Over time, new mathematical techniques, notably the Lagrangian, greatly simplified the application of these physical laws to more complex problems.

Instead of fixing the gauge by constraining the gauge field a priori via an auxiliary equation, one adds to the " physical " ( gauge invariant ) Lagrangian a gauge breaking term

For example renormalization in QED modifies the mass of the free field electron to match that of a physical electron ( with an electromagnetic field ), and will in doing so add a term to the free field Lagrangian which must be cancelled by a counterterm in the interaction Lagrangian, that then shows up as a two-line vertex in the Feynman diagrams.

There is a further possibility that the physical vacuum ( ground-state ) does not respect the symmetries implied by the " unbroken " electroweak Lagrangian ( see the article Electroweak interaction for more details ) from which one arrives at the field equations.

In physics, a sigma model is a physical system that is described by a Lagrangian density of the form:

Following are overlapping properties between the Lagrangian and Hamiltonian functions.

One of the most famous examples, of particular interest due to its topological properties, is the O ( 3 ) nonlinear sigma model in 1 + 1 dimensions, with the Lagrangian density

Since Nordström's equation of motion for test particles in an ambient gravitational field also follows from a Lagrangian, this shows that Nordström's second theory can be derived from an action principle and also shows that it obeys other properties we must demand from a self-consistent field theory.