In
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, a central force on an object is a
force that is directed towards or away from a point called center of force.
:
where
is the force, F is a
vector valued force function, ''F'' is a scalar valued force function, r is the
position vector, , , r, , is its length, and
is the corresponding
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
.
Not all central force fields are
conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
or
spherically symmetric
In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...
. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant.
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 of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a
potential energy:-
:
(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 potential energy and kinetic energy. The principle of conservation of mechanical energy states that if an isolated system is subject only to conservative forces, then the mechanical energy is ...
(
kinetic
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to i ...
and potential) is conserved:
:
(where 'ṙ' denotes the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of 'r' with respect to time, that is the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
,'I' denotes
moment of inertia of that body and 'ω' denotes
angular velocity), and in a central force field, so is the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
:
:
because the
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
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
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits ...
. (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 is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
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 is zero, ''except at the origin'':
:
Examples
Gravitational force and
Coulomb force are two familiar examples with
being
proportional to 1/''r''2 only. An object in such a force field with negative
(corresponding to an attractive force) obeys
Kepler's laws of planetary motion.
The force field of a spatial
harmonic oscillator is central with
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 f ...
, these two,
and ''
'', 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.
Notes
This article uses the definition of central force given in Taylor.
Another common definition (used in ''ScienceWorld''
[
]) adds the constraint that the force be spherically symmetric, i.e.
.
See also
*
Classical central-force problem
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, ...
*
Particle in a spherically symmetric potential
In the quantum mechanics description of a particle in spherical coordinates, a spherically symmetric potential, is a potential that depends only on the distance between the particle and a defined centre point. One example of a spherical potential ...
References
Force
Classical mechanics