@Space_Dust_art@lardygew it's something like
1. a symmetry argument to prove you only need the even powers
2. the x^2 term simplifies down to the 1/2(kx^2) and L^2/2I terms cause i don't care about ellipse oddities here
3. the 1/2,1/4,c, and d are arbitrary and don't affect final geometry reasoning
@Space_Dust_art@lardygew i'm a MechE primarily so it just reminded me of a buckling beam problem which is how I approached it
like the idea is that we really don't care that much about the coeffs, just the final signage which dictates what sort of shape it becomes
@Space_Dust_art@lardygew it's something like
1. a symmetry argument to prove you only need the even powers
2. the x^2 term simplifies down to the 1/2(kx^2) and L^2/2I terms cause i don't care about ellipse oddities here
3. the 1/2,1/4,c, and d are arbitrary and don't affect final geometry reasoning
@lardygew an intuitive explanation would be:
an ice-skater is doing a spin on the ice while hugging their chest and feels unstable, so they throw their arms out. they now spin slower and are more stable, and it takes work to bring their arms back in, so they remain stretched outwards.