Another way in which Newton equations are often rewritten
is due to Hamilton:

where

It is easy to see that these equations are equivalent to the Newton's
equation too. In order to see that first observe that

Hence the total energy of a material point can be rewritten as:

Now you can easily see that:

so the equation

simply states that . The second Hamilton equation states that:

so it is equivalent to the Newton equation.

Hamilton equations are often expressed in terms of generalized
coordinates too and written in the following way:

where