# Lagrangian Mechanics

Lagrangian mechanics is a re-formulation of classical mechanics using Hamilton's Principle of stationary action. Lagrangian mechanics applies to systems whether or not they conserve energy or momentum, and it provides conditions under which energy and/or momentum are conserved. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788.

In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates. The fundamental lemma of the calculus of variations shows that solving the Lagrange equations is equivalent to finding the path for which the action functional is stationary, a quantity that is the integral of the Lagrangian over time.

The use of generalized coordinates may considerably simplify a system's analysis. For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using Newtonian mechanics would require solving for the time-varying constraint force required to keep the bead in the groove. For the same problem using Lagrangian mechanics, one looks at the path of the groove and chooses a set of independent generalized coordinates that completely characterize the possible motion of the bead. This choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the groove on the bead at a given moment.

Read more about Lagrangian Mechanics:  Lagrange Equations of The First Kind, Extensions of Lagrangian Mechanics

### Other articles related to "lagrangian mechanics, lagrangian, mechanics":

Extensions of Lagrangian Mechanics
... denoted by H, is obtained by performing a Legendre transformation on the Lagrangian, which introduces new variables, canonically conjugate to the original variables ... for an alternative formulation of classical mechanics known as Hamiltonian mechanics ... It is a particularly ubiquitous quantity in quantum mechanics (see Hamiltonian (quantum mechanics)) ...
Centrifugal Force - Use of The Term in Lagrangian Mechanics
... See also Lagrangian and Mechanics of planar particle motion Lagrangian mechanics formulates mechanics in terms of generalized coordinates {qk}, which can be as simple as ... The Lagrangian approach to polar coordinates that treats as generalized coordinates, as generalized velocities and as generalized accelerations, is outlined in another ... The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference, but in the case of motion in a central potential the result ...
Centrifical Force - Use of The Term in Lagrangian Mechanics
... See also Lagrangian and Mechanics of planar particle motion Lagrangian mechanics formulates mechanics in terms of generalized coordinates {qk}, which can be as simple as the ... The Lagrangian approach to polar coordinates that treats as generalized coordinates, as generalized velocities and as generalized accelerations, is outlined in another article, and found in many ... The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference, but in the case of motion in a central potential the result is the same as the fictitious centrifugal ...
Mechanics Of Planar Particle Motion - Fictitious Forces in Polar Coordinates - Two Terminologies - Lagrangian Approach
... See also Lagrangian, Lagrangian mechanics, Generalized coordinates, and Euler-Lagrange equations To motivate the introduction of "coordinate" inertial forces by more than a reference to "mathem ... These forces are introduced via the Lagrangian mechanics approach to mechanics based upon describing a system by generalized coordinates usually denoted as {qk} ... In classical mechanics, the Lagrangian is defined as the kinetic energy, of the system minus its potential energy ...
Conserved Quantity - Lagrangian Mechanics
... Suppose a system is defined by the Lagrangian L with generalized coordinates q ... If L has no explicit time dependence (so ), then the energy E defined by is conserved ...

### Famous quotes containing the word mechanics:

It is only the impossible that is possible for God. He has given over the possible to the mechanics of matter and the autonomy of his creatures.
Simone Weil (1909–1943)