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.