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)?

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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)?

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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 \[T=0.5m\dot{x}^2\]?

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