Lagrangian Mechanics
Overview
Principle of Least Action
Lagrangian mechanics is based upon the principle of least action, which states that the path that a classical system
takes is the path that minimizes the action, S.
The action is defined as
{% S = \int dt L(\dot{q},q,t) %}
Here, L is a function the coordinates and the first derivatives, {% (q_i,\dot{q_i}) %}. It is caluculated as the kinetic energy
minus the potential energy.
{% L = K - U %}
Euler Lagrange Equations
The Euler Lagrange equations are derived from the principle of least action. They are set of
differential equations that solve for the path specified by the principle of least action.
{% \delta L = \frac{\partial{L}}{\partial{q_i}} \delta q_i + \frac{\partial{L}}{\partial{\dot{q_i}}} \delta \dot{q_i} %}
{% \delta S = 0 = \int dt [\frac{\partial{L}}{\partial{q_i}} \delta q_i + \frac{\partial{L}}{\partial{\dot{q_i}}} \delta \dot{q_i}] %}
Then by integrating by parts
{% \delta S = \int dt [\frac{\partial{L}}{\partial{q_i}} \delta q_i - \frac{d}{dt} (\frac{\partial{L}}{\partial{\dot{q_i}}}) \delta q_i] + \delta q_i \frac{\partial{L}}{\partial{\dot{q_i}}} |_a^b %}
The last term is zero by assumption.
{% \int dt \delta q_i [\frac{\partial{L}}{\partial{q_i}} - \frac{d}{dt} (\frac{\partial{L}}{\partial{\dot{q_i}}})] = 0 %}
From which we get the Euler Lagrange equations
{% \frac{\partial{L}}{\partial{q_i}} - \frac{d}{dt} (\frac{\partial{L}}{\partial{\dot{q_i}}}) = 0 %}
