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: F_{s} = m
a_{s} = m dv_{s}/dt.

Chain rule of calculus: m a_{s} = m dv_{s}/dt = m
(dv_{s}/ds) (ds/dt) = m v_{s} (dv_{s}/ds).

The force is: F_{s} = - mg sin θ =
- mg dy/ds.

Newton's 2nd law becomes: m v_{s} dv_{s} = - mg dy.

Integrating: ½ m
v_{s}^{2} + mgh = constant. **Mechanical energy
is conserved!**

Identify mgh as the
gravtational potential energy.