Take the scalar product of Newton's second law of motion with the particle's velocity where the force is obtained from the potential energy m r ¨ ¨ = F = − − ∇ ∇ U ( r ) / ⋅ ⋅ r ˙ ˙ gives m r ˙ ˙ ⋅ ⋅ r ¨ ¨ = 1 2 m d d t ( r ˙ ˙ ⋅ ⋅ r ˙ ˙ ) = − − r ˙ ˙ ⋅ ⋅ ∇ ∇ U ( r ) = − − ∑ ∑ i = 1 3 d x i d t ∂ ∂ U ∂ ∂ x i = d U d t where summation is assumed over the spatial Cartesian index i = 1 , 2 , 3 and we have used the fact that d d t ( r ˙ ˙ ⋅ ⋅ r ˙ ˙ ) = r ¨ ¨ ⋅ ⋅ r ˙ ˙ + r ˙ ˙ ⋅ ⋅ r ¨ ¨ = 2 r ˙ ˙ ⋅ ⋅ r ¨ ¨ and used the chain rule d d t U ( x , y , z ) = ∑ ∑ i ∂ ∂ U ∂ ∂ x i d x i d t . Rearranging d d t ( 1 2 m r ˙ ˙ 2 + U ( r ) ) = 0 The term in parentheses on the left hand side is a constant, label this with E t o t , the total mechanical energy. Clearly, this is the sum of the kinetic energy and the potential energy.