logiski9s

2022-07-16

In my lecture today my professor briefly mentioned that force is the derivative of energy but I did not really get what he meant by that. I tried to express it mathematically:
$\frac{d}{dt}{K}_{E}=\frac{d}{dt}\left(\frac{1}{2}m{v}^{2}\right)=mv\frac{dv}{dt}$
This looks really close to Newton's second law $F=ma$ but there is an extra "v" in there. Am I missing something here?

Expert

For conservative systems, it is true that the force can be expressed as minus the gradient of the potential energy:
$\begin{array}{}\text{(1)}& \mathbf{\text{F}}\left(\mathbf{\text{x}}\right)=-\mathrm{\nabla }V\left(\mathbf{\text{x}}\right),\end{array}$
which can be though of as the defining property of a conservative system.
The gradient $\mathrm{\nabla }$ reduces for one-dimensional systems to the derivative with respect to the space coordinate, i.e. you have in this simple case
$\begin{array}{}\text{(2)}& F=-\frac{dV}{dx}.\end{array}$
Taking as an example the case of a mass m in the gravitational field of the earth, you have the potential energy
$\begin{array}{}\text{(3)}& V\left(z\right)=mgz,\end{array}$
where z is the distance from the ground. The force in the z direction is then given by
${F}_{g}=-\frac{dV\left(z\right)}{dz}=-mg,$
which is what you would expect.

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