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One form of the Lagrangian equation is as follows:
One form of the Lagrangian equation is as follows:
<math>{\partial L }/ {\partial \dot q} - {\partial L} /{\partial q} = Q </math>
<math>\frac{\partial}{\partial t}({\partial L }/ {\partial \dot q}) - {\partial L} /{\partial q} = Q </math>
Where:
Where:
L, the Lagrangian function, is defined as: L = T - V, where T is the total kinetic energy of the system and V is the total potential energy of the system
L, the Lagrangian function, is defined as L = T - V, where T is the total kinetic energy of the system and V is the total potential energy of the system, q is the generalized coordinate, and Q is the generalized force. <math>{\partial L }/ {\partial \dot q}</math> is known as the generalized momentum.
q is the generalized coordinate
Q is the generalized force
===Example===
Consider a mass m attached a spring with spring constant k. The system has a single degree of freedom: x, the displacement of the mass from its equilibrium position. Then,
<math>T=\frac{1}{2}m\dot x^2</math>
<math>V=\frac{1}{2}kx^2</math>
Thus,
<math>
{\partial L }/ {\partial \dot x} = m\dot x
</math>
<math>
{\partial L}/{\partial x} = -kx
</math>
(Note that the generalized momentum is the same as the "normal" Newtonian momentum of mass times velocity in this problem.) <math>Q=0</math> for this simple problem, and so the equation of motion is
<math> m\ddot x = -kx </math>
which is the same as the result one arrives at by just considering the forces acting on the mass in Newtonian mechanics.
[[category:physics]]
[[category:physics]]