Question

Lagrangian mechanics: 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)?

Answer 1

That is, if you assume
\[T=0.5m\dot{x}^2\]
and
\[U=U\]
, and plug this into the Euler-Lagrange equation (and Newton's 2nd law), you get
\[m \ddot{x}=F=-\frac{\partial \mathbf{U}}{\partial \mathbf{x}}\]
. Is this the only fundamental way to get to this equation, or is it only one example of many equally fundamental ones?

Answer 2

Also- is the relation derivable *not* assuming \[T=0.5m\dot{x}^2\]?