Conservation of Mechanical Energy
We have just established that ΔU =  W, and we know from the Work
Energy Theorem that ΔK = W. Relating the two equations, we see that
ΔU =  ΔK and thus ΔU + ΔK = 0. Stated verbally,
the sum of the change in kinetic and potential energy must always equal
zero. By the associative property, we can also write that:
Thus the sum of U and K must be a constant. This constant, denoted by E, is
defined as the total mechanical energy of a conservative system. We can
now generate a mathematical expression for the conservation of mechanical
energy:
This statement is true for all conservative systems, and thus for all systems in
which U is defined.
With this equation we have completed our proof of the
conservation of mechanical energy within conservative systems. The relation
between U, K and E is elegantly simple, and is derived from our concepts of
work, kinetic energy, and conservative forces. Such a relation is also a
valuable tool in solving physical problems. Given an initial state in which we
know both K and U, and asked to calculate one of these quantities in some final
state, we simply equate the sums at each state: U_{o} + K_{o} = U_{f} + K_{f}. Such a
relation further bypasses our kinematics laws, and makes calculations in
conservative systems quite simple.
Using Calculus to find Potential Energy
Our calculation of the gravitational potential energy was quite easy. Such an
easy calculation will not always be the case, and calculus can be a great help
in generating an expression for the potential energy of a conservative system.
Recall that work is defined in calculus as W = F(x)dx. Thus
the change in potential is simply the negative of this integral.
To demonstrate how to calculate potential energy using vector calculus we shall
do so for a massspring system. Consider a mass on a spring, at equilibrium at
x = 0. Recall that the force exerted by the spring, which is a conservative
force, is: F_{s} =  kx, where k is the spring constant. Let us also assign an
arbitrary value to the potential at the equilibrium point: U(0) = 0. We can now
use our relation between potential and work to find the potential of the system
a distance x from the origin:
U(
x)  0 = 
(
kx)
dx
Implying that
U(x) = kx^{2} 

This equation is true for all x. A calculation of the same form can be
completed for any conservative system, and we thus have a universal method for
calculating potential energy.
Though Newtonian mechanics provide an axiomatic basis for the study of
mechanics, our concept of energy is more universal: energy applies not only to
mechanics, but to electricity, waves, astrophysics, and even quantum
mechanics. Energy pops up again and again in physics, and
the conservation of energy remains one of the fundamental ideas of physics.