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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

 \begin{displaymath}
L = \frac{m v^2}{2} - qV
\end{displaymath} (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, vx, vy, and vz. Using this function equations (4.6) can be rewritten in the following form:
 
$\displaystyle \frac{\textrm{d}}{\textrm{d}t}
\frac{\partial L}{\partial v_x}
- \frac{\partial L}{\partial x}$ = 0  
$\displaystyle \frac{\textrm{d}}{\textrm{d}t}
\frac{\partial L}{\partial v_y}
- \frac{\partial L}{\partial y}$ = 0  
$\displaystyle \frac{\textrm{d}}{\textrm{d}t}
\frac{\partial L}{\partial v_z}
- \frac{\partial L}{\partial z}$ = 0  

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

\begin{displaymath}\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
\end{displaymath}

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

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

Consequently the equation reduces to:

\begin{displaymath}m a_x + q \nabla_x V = 0
\end{displaymath}

or

\begin{displaymath}m a_x = - q\nabla_x V
\end{displaymath}

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:

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

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


next up previous index
Next: Hamilton Equations Up: Kinematics and Dynamics of Previous: Conservation of Energy and
Zdzislaw Meglicki
2001-02-26