Here's the question you clicked on:
henpen
Is the definition \-[\frac{\partial \mathbf{U}}{\partial \mathbf{x}}=\mathbf{F}\] arrived at only by minimising action (that is, using the Euler-Lagrange equation) and F=ma (that is, is there any other equally thorough way of doing it)?
That is, if you assume \[T=0.5mv^2\] and \[U=U\], and plug this into the Euler-Lagrange equation, you get \[m\ddot{x}=\frac{dU}{dx}\]. Is this the only fundamental way to get to this equation, or is it only one example of many equally fundamental ones?
*with a minus sign, of course