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: d d t K E = d d t ( 1 2 m v 2 ) = m v d v d t This looks really close to Newton's second law F = m a but there is an extra "v" in there. Am I missing something here?

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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:      d          d      t            K    E    =      d          d      t            (          1      2        m          v      2        )    =  m  v            d      v              d      t      This looks really close to Newton&#039;s second law   F  =  m  a but there is an extra &quot;v&quot; in there. Am I missing something here?

tilsjaskak6 · Accepted Answer

For conservative systems, it is true that the force can be expressed as minus the gradient of the potential energy:                    (1)                              F                (                  x                )        =        −        ∇        V        (                  x                )        ,            which can be though of as the defining property of a conservative system.The gradient   ∇ reduces for one-dimensional systems to the derivative with respect to the space coordinate, i.e. you have in this simple case                    (2)                    F        =        −                              d            V                                d            x                          .            Taking as an example the case of a mass m in the gravitational field of the earth, you have the potential energy                    (3)                    V        (        z        )        =        m        g        z        ,            where z is the distance from the ground. The force in the z direction is then given by      F    g    =  −            d      V      (      z      )              d      z        =  −  m  g  ,which is what you would expect.

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