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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:

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

Consider the following scalar function

 (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:

 = 0 = 0 = 0

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

So the first term of the Lagrange equation, is simply m ax. The second term evaluates to:

Consequently the equation reduces to:

or

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:

 (4.7)

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

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