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Conservative force
 
 
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A conservative force is defined as a force
Force

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
 with the following property: when an object moves from one location to another, the force changes the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
 of the object by an amount that does not depend on the path taken.

rmally, 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 constant force F acting on it.






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A conservative force is defined as a force
Force

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
 with the following property: when an object moves from one location to another, the force changes the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
 of the object by an amount that does not depend on the path taken.

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 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 is classified as a conservative force.

The gravitational force, spring force
Hooke's law

In mechanics, and physics, Hooke's law of theory of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit....
, magnetic force (according to some definitions, see below) and electric force (at least in a time-independent magnetic field, see Faraday's law of induction
Faraday's law of induction

Faraday's law of induction describes a basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators....
 for details) are examples of conservative forces, while friction and air drag are classical examples of non-conservative forces (the energy is transferred to the air as heat and cannot be retrieved)

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
Force field (physics)

Originally a term coined by Michael Faraday to provide an intuitive paradigm, but theoretical construct , for the behavior of electromagnetic fields, the term force field refers to the Line of force one object exerts on another object or a collection of other objects....
 F is called conservative if it meets any of these equivalent (proof
Conservative force/Proofs

Properties of conservative forcesThe main page states that the following three properties are equivalent for conservative forces:#The Curl of F is zero:...
) conditions:

1. The curl of F is zero:


2. The work, W, is zero for any simple closed path:


3. The force can be written as the gradient
Gradient

In vector calculus, the gradient of a scalar field is a vector field which 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 potential, :


Conservative force fields are curl-less as a direct consequence of Helmholtz decomposition
Helmholtz decomposition

In 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 function, Schwartz space vector field can be resolved into irrotational vector field and solenoidal component vector fields....
. 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 law
Faraday's law of induction

Faraday's law of induction describes a basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators....
), and spring force
Hooke's law

In mechanics, and physics, Hooke's law of theory of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit....
.

Many forces (particularly those that depend on velocity) are not force fields
Force field (physics)

Originally a term coined by Michael Faraday to provide an intuitive paradigm, but theoretical construct , for the behavior of electromagnetic fields, the term force field refers to the Line of force one object exerts on another object or a collection of other objects....
. 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. It should be emphasized that 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 arise due to neglected degrees of freedom
Degrees of freedom (physics and chemistry)

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters....
. 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
Friction

File:Friction alt.svgFriction is the force resisting the relative lateral motion of solid surfaces, fluid layers, or material elements in contact....
 and non-elastic material stress
Stress (physics)

In continuum mechanics, stress is a measure of the average amount of force exerted per unit area. It is a measure of the intensity of the total internal forces acting within a body across imaginary internal surfaces, as a reaction to external applied forces and body forces....
.

See also

  • Conservative vector field


Footnotes and references