A
conservative force is a
forceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) by a conservative force is zero.
It is possible to define a numerical value of
potentialA scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...
at every point in space for a conservative force. When an object moves from one location to another, the force changes the
potential energyIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
of the object by an amount that does not depend on the path taken.
GravityGravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
is an example of a conservative force, while
frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of 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 energyIn physics, mechanical energy is the sum of potential energy and kinetic energy present in the components of a mechanical system. It is the energy associated with the motion and position of an object. The law of conservation of energy states that in an isolated system that is only subject to...
. Suppose a particle starts at point A, and there is a constant 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 forceIn mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law...
, magnetic force (according to some definitions, see below) and electric force (at least in a time-independent magnetic field, see
Faraday's law of inductionFaraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...
for details) are examples of conservative forces, while
frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
and air drag are classical examples of non-conservative forces.
For non-conservative forces, the mechanical energy that is lost (not conserved) has to go somewhere else, by
conservation of energyThe nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
. Usually the energy is turned into
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
, for example the heat generated by friction. In addition to heat, friction also often produces some
soundSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
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
wakeA wake is the region of recirculating flow immediately behind a moving or stationary solid body, caused by the flow of surrounding fluid around the body.-Fluid dynamics:...
. These and other energy losses are irreversible because of the
second law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
.
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. Also the work done by a conservative force is equal to the negative of change in potential energy during that process. For a proof of that, let's 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 (or within a simply-connected volume of 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 zero:
- 2. There is zero net work (W) 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 gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of a potentialIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
, 
:
| Proof that these three conditions are equivalent when F is a force field |
1 implies 2: Let C be any simple closed path (i.e., a path that starts and ends at the same point and has no self-intersections), and consider a surface S of which C is the boundary. Then Stokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
says that
If the curl of F is zero the left hand side is zero - therefore statement 2 is true.
2 implies 3: Assume that statement 2 holds. Let c be a simple curve from the origin to a point  and define a function
The fact that this function is well-defined (independent of the choice of c) follows from statement 2. Anyway, from the fundamental theorem of calculusThe first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
, it follows that
So statement 2 implies statement 3.
3 implies 1: Finally, assume that the third statement is true. A well-known vector calculus identity states that the curl of the gradient of any function is 0. (See proof.) Therefore, if the third statement is true, then the first statement must be true as well.
This shows that statement 1 implies 2, 2 implies 3, and 3 implies 1. Therefore all three are equivalent, Q.E.D.Q.E.D. is an initialism of the Latin phrase , which translates as "which was to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation — and in the setting-out —...
.
(The equivalence of 1 and 3 is also known as (one aspect of) Helmholtz's theoremIn physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational vector field and a...
.) |
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 (in a time-independent magnetic field, see
Faraday's lawFaraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...
), and
spring forceIn mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law...
.
Many forces (particularly those that depend on velocity) are not
force fields. In these cases, the above three conditions are not mathematically equivalent. For example, the magnetic force satisfies condition 2 (since the work done by a magnetic field on a charged particle is always zero), but does not satisfy condition 3, and condition 1 is not even defined (the force is not a vector field, so one cannot evaluate its curl). 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
frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
, do not satisfy any of the three conditions, and therefore are unambiguously nonconservative.
Nonconservative forces
Nonconservative forces can only arise in classical physics due to neglected
degrees of freedomA degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
. 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
frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
and non-elastic material stress.
However,
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
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 pseudotensorIn the theory of general relativity, a stress–energy–momentum pseudotensor, such as the Landau–Lifshitz pseudotensor, is an extension of the non-gravitational stress–energy tensor which incorporates the energy-momentum of gravity. It allows the energy-momentum of a system of gravitating matter to...
.