The potential energy in relation to conservative force.

When the conservative force F\left(x\right) (for simplicity, in one dimension) does work W on a particle within the system, the change in potential energy \triangle U of the system is equal to the negative of the work done by the conservative force, i.e.,

\;\;\;\;\;\;\;\;\;\;\;\triangle U=-W

But, \;\;\;\;\;\;W=\int_{x_i}^{x_f}F\left(x\right)\;dx

\therefore\;\;\;\;\;\;\;\;\triangle U=-\int_{x_i}^{x_f}F\left(x\right)\;dx

Differentiating the above equation, we get

\frac{\operatorname dU\;\left(x\right)}{\operatorname dx}=-F\left(x\right)

or, F\left(x\right)=-\frac{\operatorname dU\;\left(x\right)}{\operatorname dx}

Hence potential energy U may be defined as a function whose negative gradient gives the force. Conversely, we may define conservative force as a force which is equal to the negative gradient of the potential energy U.

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