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Definition/Summary
The Lagrangian is a function that summarizes equations of motion. It appears in the action, a quantity whose extremum (minimum or maximum) yields the classical equations of motion by use of the Euler-Lagrange equation. In quantum mechanics, the action, and thus the Lagrangian, appears in the path integral, and the eikonal or geometric-optics limit of the path integral is finding the extremum of the action.
Though originally developed for Newtonian-mechanics particle systems, Lagrangians have been used for relativistic particle systems, classical fields, and quantum-mechanical systems. Lagrangians are often more convenient to work with than the original equations of motion, because one can easily change variables in them to make a problem more easily tractable.
Lagrangians are also easier for testing for symmetries, since unlike equations of motion, they are unchanged by symmetries.
Equations
Action principle: the action I for time t and system coordinate q(t) is the integral of the Lagrangian L:
[itex]I = \int L(q(t), \dot q(t), t) dt[/itex]
The extremum of the action yields the Euler-Lagrange equation, which gives:
[itex]\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q(t)}\right) - \frac{\partial L}{\partial q(t)} = 0[/itex]
with appropriate terms for any higher derivatives which may be present.
It is easily generalized to multiple independent variables [itex]x_i[/itex] and multiple dependent variables [itex]q_a(x)[/itex]:
[itex]\sum_i \frac{\partial}{\partial x_i}\left(\frac{\partial L}{\partial (\partial q_a(x) / \partial x_i)}\right) - \frac{\partial L}{\partial q_a(x)} = 0[/itex]
Extended explanation
Let us start with a Newtonian equation of motion for a particle with position q and mass m moving in a potential V(q,t):
[itex]m \frac{d^2 q}{dt^2} = - \frac{\partial V}{\partial q}[/itex]
It can easily be derived from this Lagrangian with the Euler-Lagrange equations:
[itex]L = T - V[/itex]
where the kinetic energy has its familiar Newtonian value:
[itex]T = \frac12 m \left( \frac{dq}{dt} \right)^2[/itex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
The Lagrangian is a function that summarizes equations of motion. It appears in the action, a quantity whose extremum (minimum or maximum) yields the classical equations of motion by use of the Euler-Lagrange equation. In quantum mechanics, the action, and thus the Lagrangian, appears in the path integral, and the eikonal or geometric-optics limit of the path integral is finding the extremum of the action.
Though originally developed for Newtonian-mechanics particle systems, Lagrangians have been used for relativistic particle systems, classical fields, and quantum-mechanical systems. Lagrangians are often more convenient to work with than the original equations of motion, because one can easily change variables in them to make a problem more easily tractable.
Lagrangians are also easier for testing for symmetries, since unlike equations of motion, they are unchanged by symmetries.
Equations
Action principle: the action I for time t and system coordinate q(t) is the integral of the Lagrangian L:
[itex]I = \int L(q(t), \dot q(t), t) dt[/itex]
The extremum of the action yields the Euler-Lagrange equation, which gives:
[itex]\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q(t)}\right) - \frac{\partial L}{\partial q(t)} = 0[/itex]
with appropriate terms for any higher derivatives which may be present.
It is easily generalized to multiple independent variables [itex]x_i[/itex] and multiple dependent variables [itex]q_a(x)[/itex]:
[itex]\sum_i \frac{\partial}{\partial x_i}\left(\frac{\partial L}{\partial (\partial q_a(x) / \partial x_i)}\right) - \frac{\partial L}{\partial q_a(x)} = 0[/itex]
Extended explanation
Let us start with a Newtonian equation of motion for a particle with position q and mass m moving in a potential V(q,t):
[itex]m \frac{d^2 q}{dt^2} = - \frac{\partial V}{\partial q}[/itex]
It can easily be derived from this Lagrangian with the Euler-Lagrange equations:
[itex]L = T - V[/itex]
where the kinetic energy has its familiar Newtonian value:
[itex]T = \frac12 m \left( \frac{dq}{dt} \right)^2[/itex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!