I was looking at the usual derivation of conservation of energy for a ball thrown in the air. One starts with F=ma, rewrite as F=m dv/dt = m dv/dy dy/dt = m v dv/dy, and then rearrange to Fdy = mv dv. Let the force F be gravity (-mg) and integrate the rearranged version of Newton's Law from y1 to y2 and v1 to v2 to get
mgy1 + KE1 = mgy2 + KE2
One can do a similar derivation using t. Start with F = m dv/dt, rearrange to Fdt=mdv (a vector equation), let the force be -mgj (j is unit vector along vertical) and integrate from t1 to t2 and v1 to v2 (vectors). One obtains
mgt1 j + mv1 = mgt2 j +mv2
or
mgt1 j + p1 = mgt2 j + p2 (vector p's)
This is formally similar to the conservation of energy equation, the difference being that it is a vector equation and that it involves momentum and a term mgt analogous to mgy. The energy equation conserves the sum of KE and a positional energy. The other equation conserves the sum of momentum and a temporal term.
Is anyone aware of an attempt to develop a momentum-time conservation approach to physics similar to the familiar KE-position conservation approach - i.e., an attempt to avoid an impulse-momentum approach by using a conservation law involving momentum and a temporal potential of sorts similar to an avoidance of a work-KE approach by using conservation of KE and positional energies? I would assume that the biggest impediment would be finding suitable forces that are functions of time instead of position.