[GistNoesis]

Every time in physics you see quadratic, you should think sphere.

There is some rotation invariance hidden in the velocity physics because you can rotate the velocity vector of an object without having to spend energy (The force you need to apply is perpendicular to the velocity so does no work).

The typical example is you have a ball fall 1m vertically, then have a 90° bend which convert the vertical velocity into horizontal velocity and no vertical velocity, then the ball fall again 1m vertically and have its vertical velocity increased by the same amount as for the first meter. You can then add a 45° degree bend ramp to redirect the ball so that it only has horizontal velocity, and have the ball fall again. For the third bend ramp the incoming velocity will have 2 units horizontal, and 1 unit vertical (I'll let you compute the appropriate angle). A fourth ramp would be 3 units horizontal and 1 unit vertical.

Because we can do this adding velocity in a perpendicular way trick we must then use Pythagoras.

[terminalbraid]

> Every time in physics you see quadratic, you should think sphere.

Not sure how I reconcile that for systems with linear symmetry that don't admit a sphere such as a 1D harmonic oscillator (i.e. a spring). You're confusing the fact that spheres require quadratics but quadratics are not sufficient to admit a sphere.

[GistNoesis]

For 1D harmonic oscillator, the sphere is 2D, and called a circle. It's rotating through time. 1D space + 1D time.

[terminalbraid]

If changing definitions and adding dimensions are fair game, try it for a damped oscillator and tell me where whatever dimension sphere is in there.

If you understood what you were trying to push now you'd recognize that adding in your time dimension to force an argument involving a circle is just a matter of the sign in the 2nd order differential equation. This also falls apart if that sign changes and you get hyperbolic solutions. and I'm sure you'll just say that's just another kind of sphere with negative curvature