classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

, a central force on an object is a

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

that is directed towards or away from a point called center of force.

\mathbf(\mathbf) = F( \mathbf )

where F is a force vector, ''F'' is a scalar valued force function (whose absolute value gives the magnitude of the force and is positive if the force is outward and negative if the force is inward), r is the

position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...

, , , r, , is its length, and

\hat = \mathbf r / \, \mathbf r\,

is the corresponding

unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...

. Not all central force fields are

conservative Conservatism is a cultural, social, and political philosophy and ideology that seeks to promote and preserve traditional institutions, customs, and values. The central tenets of conservatism may vary in relation to the culture and civiliza ...

or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant. Examples of spherically symmetric central forces include the

Coulomb force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the ''electrostatic ...

and the force of gravity.

Properties

Central forces that are conservative can always be expressed as the negative

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

of a

potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...

\mathbf(\mathbf) = - \mathbf V(\mathbf) \; \text V(\mathbf) = \int_^ F(r)\,\mathrmr

(the upper bound of integration is arbitrary, as the potential is defined up to an additive constant). In a conservative field, the total

mechanical energy In physical sciences, mechanical energy is the sum of macroscopic potential and kinetic energies. The principle of conservation of mechanical energy states that if an isolated system is subject only to conservative forces, then the mechanical ...

( kinetic and potential) is conserved:

E = \tfrac m , \mathbf, ^2 + \tfrac I , \boldsymbol, ^2 + V(\mathbf) = \text

(where 'ṙ' denotes the

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

of 'r' with respect to time, that is the

velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

,'I' denotes

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...

of that body and 'ω' denotes

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

), and in a central force field, so is the

angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

\mathbf = \mathbf \times m\mathbf = \text

because the

torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...

exerted by the force is zero. As a consequence, the body moves on the plane perpendicular to the angular momentum vector and containing the origin, and obeys Kepler's second law. (If the angular momentum is zero, the body moves along the line joining it with the origin.) It can also be shown that an object that moves under the influence of ''any'' central force obeys Kepler's second law. However, the first and third laws depend on the inverse-square nature of

Newton's law of universal gravitation Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product ...

and do not hold in general for other central forces. As a consequence of being conservative, these specific central force fields are irrotational, that is, its

curl cURL (pronounced like "curl", ) is a free and open source computer program for transferring data to and from Internet servers. It can download a URL from a web server over HTTP, and supports a variety of other network protocols, URI scheme ...

is zero, ''except at the origin'':

\nabla\times\mathbf (\mathbf) = \mathbf .

Examples

Gravitational force and

are two familiar examples with

F( \mathbf )

being proportional to 1/''r''² only. An object in such a force field with negative

F( \mathbf )

(corresponding to an attractive force) obeys

Kepler's laws of planetary motion In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler in 1609 (except the third law, which was fully published in 1619), describe the orbits of planets around the Sun. These laws replaced circular orbits and epicycles in ...

. The force field of a spatial

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...

is central with

F( \mathbf )

proportional to ''r'' only and negative. By

Bertrand's theorem In classical mechanics, Bertrand's theorem states that among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits. The ...

, these two,

F( \mathbf ) = -k/r^2

and

F( \mathbf{r} ) = -kr

, are the only possible central force fields where all bounded orbits are stable closed orbits. However, there exist other force fields, which have some closed orbits.

References

Force Classical mechanics

Properties

Examples

See also

References