Lagrange Equations

Newton equation (4.1), which is a second order differential equation, can be easily reduced to two first order differential equations:

  $$\begin{eqnarray*}\frac{\textrm{d}\boldsymbol{r}}{\textrm{d} t} &=& \boldsymbol{v... ...ol{v}}{\textrm{d} t} &=& -\frac{q}{m} \boldsymbol{\nabla} V\\ \end{eqnarray*}$$

There are various interesting ways, in which these equations can be rewritten.

Consider the following scalar function

$$L = \frac{m v^2}{2} - qV$$ (4.6)

Note: avoid a confusion with the angular momentum L. Traditionally capital ``L'' is used for both, but function L defined by equation (4.6) is a different thing altogether. It is called a Lagrangian and is a function of 6 variables, x, y, z, v_x, v_y, and v_z. Using this function equations (4.6) can be rewritten in the following form:

 

$$\frac{\textrm{d}}{\textrm{d}t} \frac{\partial L}{\partial v_x} - \frac{\partial L}{\partial x}$$ = 0

$$\frac{\textrm{d}}{\textrm{d}t} \frac{\partial L}{\partial v_y} - \frac{\partial L}{\partial y}$$ = 0

$$\frac{\textrm{d}}{\textrm{d}t} \frac{\partial L}{\partial v_z} - \frac{\partial L}{\partial z}$$ = 0

It is easy to see that this is indeed the case:

$$\frac{\textrm{d}}{\textrm{d}t} \frac{\partial L}{\partial v_... ...}t} m 2 \frac{v_x}{2} = m \frac{\textrm{d}}{\textrm{d}t} v_x$$

So the first term of the Lagrange equation, $\frac{\textrm{d}}{\textrm{d}t}\frac{\partial L}{\partial v_x}$ is simply m a_x. The second term evaluates to:

$$- \frac{\partial L}{\partial x} = - \frac{\partial }{\partial x} \left(-q V\right) = q \nabla_x V$$

Consequently the equation reduces to:

$$m a_x + q \nabla_x V = 0$$

or

$$m a_x = - q\nabla_x V$$

and similarly for y and z components.

An amazing thing about these equations is not that they are equivalent to the Newton equation of motion, but that they look the same also in non-Cartesian coordinates. Mathematicians and physicists often write them in the following form:

$$\frac{\textrm{d}}{\textrm{d}t} \frac{\partial L}{\partial ... ...i} - \frac{\partial L}{\partial q_i} = 0, \quad i = 1, 2, 3$$ (4.7)

where q_i is referred to as a generalized coordinate, and a dot above a symbol, as before, denotes a time derivative.