Conservative force

# Conservative force

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A conservative force is a force
Force
In 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 potential
Scalar potential
A 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 energy
Potential energy
In 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.

Gravity
Gravitation
Gravitation, 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 friction
Friction
Friction 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 energy
Mechanical energy
In 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 force
Hooke's law
In 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 induction
Faraday's law of induction
Faraday'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 friction
Friction
Friction 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 energy
Conservation of energy
The 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 heat
Heat
In 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 sound
Sound
Sound 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 wake
Wake
A 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 thermodynamics
Second law of thermodynamics
The 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 potential
Potential energy
In 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...

, :

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 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 force
Hooke's law
In 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 friction
Friction
Friction 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 freedom
Degrees of freedom (physics and chemistry)
A 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 friction
Friction
Friction 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 relativity
General relativity
General 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 pseudotensor
Stress-energy-momentum pseudotensor
In 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...

.