## henpen Group Title 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)? one year ago one year ago

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