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

Note: avoid a confusion with the angular momentum

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

So the first term of the Lagrange equation, is simply

Consequently the equation reduces to:

or

and similarly for

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:

where