Newton's Law and Energy Conservation

Energy conservation follows from Newtons' second law. As an example, consider a particle moving in a gravitational field along the s-axis that is inclined at an angle θ with respect to the horizontal:

Newtons' 2nd law:   Fs = m as = m dvs/dt.

Chain rule of calculus:  m as = m dvs/dt = m (dvs/ds) (ds/dt) = m vs (dvs/ds).

The force is:   Fs = - mg sin θ = - mg dy/ds.

Newton's 2nd law becomes:   m vs dvs = - mg dy.

Integrating:   ½ m vs2 + mgh = constant.    Mechanical energy is conserved!

Identify mgh as the gravtational potential energy.