A conservative force is a force with the property that the work done in moving a
particle between two points is independent of the taken path.
Equivalently, if a particle travels in a closed loop, the net work done by a
conservative force is zero. A conservative force is dependent only
on the position of the object. If a force is conservative, it is possible to
assign a numerical value for the potential at any point. When an object
moves from one location to another, the force changes the potential energy of
the object by an amount that does not depend on the path taken. If the force
is not conservative, then defining a scalar potential is not possible,
because taking different paths would lead to conflicting potential
differences between the start and end points.
Gravity is an example of a conservative force, while friction is an example of a
non-conservative force. Informal definition
Informally, a conservative force can be thought of as a force that conserves
mechanical energy. Suppose a particle starts at point A, and there is a force
F acting on it. Then the particle is moved around by other forces, and
eventually ends up at A again. Though the particle may still be moving, at
that instant when it passes point A again, it has traveled a closed path. If
the net work done by F at this point is 0, then F passes the closed path test.
Any force that passes the closed path test for all possible closed paths is
classified as a conservative force. The gravitational force, spring force,
magnetic force and electric force are examples of conservative forces, while
friction and air drag are classical examples of non-conservative forces.
For non-conservative forces, the mechanical energy that is lost has to go
somewhere else, by conservation of energy. Usually the energy is turned
into heat, for example the heat generated by friction. In addition to
heat, friction also often produces some sound energy. The water drag on a moving
boat converts the boat's mechanical energy into not only heat and sound
energy, but also wave energy at the edges of its wake. These and other
energy losses are irreversible because of the second law of thermodynamics.
Path independence A direct consequence of the closed path
test is that the work done by a conservative force on a particle moving
between any two points does not depend on the path taken by the particle.
This is illustrated in the figure to the right: The work done by the
gravitational force on an object depends only on its change in height because the
gravitational force is conservative. The work done by a conservative force is
equal to the negative of change in potential energy during that process.
For a proof, imagine two paths 1 and 2, both going from point A to point B. The
variation of energy for the particle, taking path 1 from A to B and then path
2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e.,
the work is independent of the path followed, as long as it goes from A to
B. For example, if a child slides down a
frictionless slide, the work done by the gravitational force on the child from
the top of the slide to the bottom will be the same no matter what the shape of
the slide; it can be straight or it can be a spiral. The amount of work done
only depends on the vertical displacement of the child.
Mathematical description A force field F, defined everywhere in
space, is called a conservative force or conservative vector field if it meets
any of these three equivalent conditions:
1. The curl of F is the zero vector: 2. There is zero net work done by the
force when moving a particle through a trajectory that starts and ends in the
same place: 3. The force can be written as the
negative gradient of a potential, : The term conservative force comes from
the fact that when a conservative force exists, it conserves mechanical energy.
The most familiar conservative forces are gravity, the electric force, and
spring force. Many forces are not force fields. In
these cases, the above three conditions are not mathematically equivalent. For
example, the magnetic force satisfies condition 2, but does not satisfy
condition 3, and condition 1 is not even defined. Accordingly, some authors
classify the magnetic force as conservative, while others do not. The
magnetic force is an unusual case; most velocity-dependent forces, such as
friction, do not satisfy any of the three conditions, and therefore are
unambiguously nonconservative. Nonconservative forces
Nonconservative forces can arise in classical physics due to neglected
degrees of freedom or from time-dependent potentials. For instance,
friction may be treated without resorting to the use of nonconservative
forces by considering the motion of individual molecules; however that means
every molecule's motion must be considered rather than handling it
through statistical methods. For macroscopic systems the nonconservative
approximation is far easier to deal with than millions of degrees of freedom.
Examples of nonconservative forces are friction and non-elastic material
stress. However, general relativity is
non-conservative, as seen in the anomalous precession of Mercury's orbit.
However, general relativity can be shown to conserve a stress-energy-momentum
pseudotensor. References