Integrating the equation of motion for a one dimensional problem.
Posted by peeterjoot on January 2, 2010
While linear approximations, such as the small angle approximation for the pendum, are often used to understand the dynamics of non-linear systems, exact solutions may be possible in some cases. Walk through the setup for such an exact solution.
The equation to consider solutions of has the form
We have an unpleasant mix of space and time derivatives, preventing any sort of antidifferentiation. Assuming constant mass , and employing the chain rule a refactoring in terms of velocities is possible.
The one dimensional Newton’s law (2.1) now takes the form
This can now be antidifferentiated for
Taking roots and rearranging produces an implicit differential form in terms of time
One can concievably integrate this and invert to solve for position as a function of time, but substitution of a more specific potential is required to go further.
TODO: doing stuff with this.
EDIT: This was a stupid way to do this. It is nothing more than rearranging the Hamiltonian for .