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.
replyAction is linked to spatial symmetry too, and you can find the square there.
replySince 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:
f(√(a^2 + b^2)) = f(a) + f(b)
Note: the components do not need to be independent and orthogonal for this to hold.