Kinetic energy is, strangely, quite a bit like a least squares cost function in an optimization problem. The "dt"s in "dx/dt" hardly matter; it basically represents "dx^2" between the current state and the next.
Kinetic energy is, strangely, quite a bit like a least squares cost function in an optimization problem. The "dt"s in "dx/dt" hardly matter; it basically represents "dx^2" between the current state and the next.
If I follow you, that's not strange. That's exactly how Lagrangian mechanics are formulated (minimizing the action which has exactly the kinetic energy as a term to be minimized against a potential energy term) which rests on well-founded symmetry principles.
Action is linked to spatial symmetry too, and you can find the square there.
Since space is isotropic, a Lagrangian can only depend on a speed vector through its norm. A Lagrangian must also be decomposable into independent orthogonal components, so you end up with an energy term that is shaped according to:
And you end up with f being proportional to v squared.Note: the components do not need to be independent and orthogonal for this to hold.