We found equilibrium points by setting the derivative of the potential energy to zero:
We then determined if these points were stable or unstable by looking at the second derivative of the potential energy:
By setting
If we consider the typical form of a differential equation,
We can see that we are seeking the points where the differential equation is zero,
This approach is a powerful way to understand the behavior of a system. And we can do so geometrically!
Let
We can integrate this with
Find
Instead find the equilibrium points (
Sketch the differential equation
Consider now the differential equation
That yields the following solution ():
The SHO is a linear system. It's boring. It's predictable. It's stable. But it can help us understand nonlinear 2nd order ODEs and thus more complex systems.
Consider the physical pendulum. The equation of motion is
Or more simply:
In the case of small angles,