Euler-Lagrange Equation

(d/dt)(dL/dq*)-(dL/dq)=Q
(The fully generalized version of Lagrange’s equation for a single degree of freedom. One such equation can be written for each degree of freedom a system has.)

In words, the time rate of change of the partial derivative of the Lagrangian with respect to generalized velocity minus the partial derivative of the Lagrangian with respect to generalized coordinate is equal to the generalized force.

The first term in the equation is the time rate of change of generalized momentum, which is another name for the partial derivative of the Lagrangian with respect to generalized velocity. Evaluating the following derivative corresponding to a moving particle in a potential U,


partial L/partial q*
hence that partial derivative is called the generalized momentum of the system. In this single particle system, the first term is thus mq**.

The second term, the negative partial derivative of the Lagrangian with respect to a generalized coordinate, is a conservative force acting on the system; the system’s kinetic energy does not depend on generalized position, so the term becomes the negative partial derivative of the potential energy with respect to the generalized coordinate, which in Cartesian coordinates is equal to the force acting on a system from the potential U.
The final term Q is called the generalized force, and the term is only used if non-conservative forces exist in the system.

 In conservative conditions, the rate of change of generalized momenta plus the negative partial derivative of potential energy with respect to generalized position acting on a system sum to zero, and so the Euler-Lagrange equation is as follows:
d/dt(partialL/partialq*)=partialL/partialq
Introduction - Core Terminology and Concepts - Principle of Least Action - Example of Lagrangian Mechanics

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