Force
Conservation

A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space, and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area.

Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector r {\displaystyle \mathbf {r} } emanating from spherically symmetric potentials. Examples of this follow:

For gravity: F g = − G m 1 m 2 r 2 r ^, {\displaystyle \mathbf {F} _{\text{g}}=-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat {\mathbf {r} }},} where G {\displaystyle G} is the gravitational constant, and m n {\displaystyle m_{n}} is the mass of object n.

For electrostatic forces: F e = q 1 q 2 4 π ε 0 r 2 r ^, {\displaystyle \mathbf {F} _{\text{e}}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r^{2}}}{\hat {\mathbf {r} }},} where ε 0 {\displaystyle \varepsilon _{0}} is electric permittivity of free space, and q n {\displaystyle q_{n}} is the electric charge of object n.

For spring forces: F s = − k r r ^, {\displaystyle \mathbf {F} _{\text{s}}=-kr{\hat {\mathbf {r} }},} where k {\displaystyle k} is the spring constant.

For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of microstates. For example, static friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. For any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.

The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. According to the Second law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.
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