Any discussion of energy must be prefaced with one of the fundamental statements
of physics: energy is always conserved. This guiding principle forms the basis
for many branches of physics. That said, though total energy in a system cannot
change in total amount, energy can change forms. Electrical energy can
turn into mechanical energy; mechanical energy can turn into heat. However,
since at this point we are only familiar with mechanical energy, for now we can
only use the principle of the conservation of energy if no energy is converted
to other forms. That is, for our purposes, all mechanical energy must remain
mechanical energy. In order to know when mechanical energy is conserved, we
must define those forces that do conserve mechanical energy.

Definition of a Conservative Force

So exactly what kinds of forces conserve mechanical energy? To answer this we
consider particles traveling in closed loops under the influence of the forces
in question. In other words, a closed loop describes a "round trip", during
which the particle is under influence of the force. Many systems produce closed
loops, such as a ball bouncing up and down, or a mass on a spring. If a
conservative force acts upon the particle during this closed loop, the
velocity of the particle at the beginning and the end of the loop must be the
same. Why? Because if the velocity is any different, the kinetic energy of
the particle will be different, meaning that mechanical energy must not have
been conserved. Thus we come to our first statement about conservative forces:

If a body is under the action of a force that does no net
work during any closed loop, then the
force is conservative. If work is done, the force is
nonconservative.

In other words, a particle located at the same physical location in a closed
loop must have the same kinetic energy at all times if it is within a
conservative system. This fact is the fundamental definition of a conservative
force. Though we will derive other properties of conservative forces from this
statement, it remains the most important one to keep in mind.

Since the work over a closed loop must be zero for conservative forces, what
other properties can we state? Let's break the path of a closed loop into two
separate paths:

Since the path in part a) is a closed loop, we know that the total work around
the loop must be zero if the force in question is conservative: W_{1} + W_{2} = 0.
Now compare the two different paths taken from A to B in part b). The work on
the first path is the same as part a), simply W_{1}. The direction of travel on
the second path is reversed in b), implying that the work done over the path is
negated, or equal to - W_{2}. But we know from a) that - W_{2} = W_{1}. Thus the
work done over path 1 and path 2 in part b) is the same! This concept, called
path independence, is incredibly useful, as we will soon see. Stated
verbally:

The work done by a conservative force in moving a body from an initial
location to a final location is independent of the path taken between the two
points

Let's examine the implications of this statement. Consider a particle moving
between two points in an odd shaped path. Our old definition of work demands
that we evaluate the work done at each part of the odd path in order to evaluate
the total work done over the journey, and thus the change in kinetic energy and
velocity. With this just-stated principle of conservative forces, however, we
can use any path we like: a straight line, a circular arc, or a path in
which the work done on the particle is constant. Though our first statement
about conservative forces is powerful, this second statement proves to be the
most applicable: we will use this concept to solve numerous problems in the
sections to come.

Examples of Conservative and Nonconservative Forces

Such abstract principles might be confusing. In order to clarify these two very
important concepts, we will examine two forces: gravity, a conservative force,
and friction, a nonconservative one.