Fun with spoons

How do you find the center of mass for any given geometry even if you have no idea on how to solve this analytically?

a) You can try some numerics.
b) You build a 3D replica and try to balance it on your fingertip. Works pretty well for spoons. Take a spoon, try to balance it -> yep, you found the COM.

Some more physics. You balance that spoon and add some small perturbations. Does the spoon hold? Congratulations, you just found a stable minimum.

How to describe this mathematically? -> Taylor series.
Dirty way - do a Taylor series expansion of the equation of motion around your extremum. Try to solve the differential equation with an ansatz combining two exponentials, one with a negative sign in the exponent, one with a positive one and both with different coefficients. The coefficients depend on the initial conditions but the exponents come from the solution of the differential equation.
If the exponent is imaginary you get a linear combination of cosine and sine functions. Congrats! For small perturbations your system will oscillate around the minimum but not leave it. You found a stable (maybe local) minimum.
Your exponent is real. Damn you found an unstable maximum. Try again.