PlanetPhysics/Lagrange's Equations

There are certain general principles or theorems in mechanics, such as Lagrange's equations, Hamilton's principle, the principle of least work, and Gauss' principle of least constraint, which afford general solutions of certain types of problems. Such general principles have therefore the advantage over ordinary methods in that once having found the general solution, any particular problem may be solved by merely routine processes.

The general form of Lagrange's equation for the generalized coordinates $$q_i$$ is given as

$$ Q_i = \frac{d}{dt} \left ( \frac{ \partial T}{\partial \dot{q_i}} \right ) - \frac{\partial T}{\partial q_i} $$

where $$T$$ is the kinetic energy and $$Q_i$$ are the generalized forces which is related to the system forces through

$$ Q_i = f_j \frac{\partial x_j}{\partial q_i} $$

The more common form, used when the forces for the dynamical system can be found from a scalar potential function $$V$$, is

$$ \frac{d}{dt} \left ( \frac{\partial L}{\partial \dot{q_i}} \right ) - \frac{\partial L}{\partial q_i} = 0 $$

where $$L$$, the Lagrangian function (or, simply, Lagrangian), is the difference between the kinetic and potential energy $$L = T - V$$