Central force
Encyclopedia
In classical mechanics
, a central force is a force whose magnitude only depends on the distance
r of the object from the origin
and is directed along the line joining them:
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
= r/||r|| is the corresponding unit vector.
Equivalently, a force field is central if and only if it is spherically symmetric.
of a potential
:
(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
(kinetic
and potential) is conserved:
(where ṙ denotes the derivative
of r with respect to time, that is the velocity
), and in a central force field, so is the angular momentum
:
because the torque
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.)
As a consequence of being conservative, a central force field is irrotational, that is, its curl is zero, except at the origin:
. An object in such a force field with negative F (corresponding to an attractive force) obeys Kepler's laws of planetary motion
.
The force field of a spatial harmonic oscillator
is central with F(r) proportional to r and negative.
By Bertrand's theorem, these two, F(r) = −k/r2 and F(r) = −kr, are the only possible central force field with stable closed orbits.
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, a central force is a force whose magnitude only depends on the distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
r of the object from the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
and is directed along the line joining them:
where
is the force, F is a vector valued force functionVector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
, F is a scalar valued force function, r is the position vector, ||r|| is its length, and
= r/||r|| is the corresponding unit vector.Equivalently, a force field is central if and only if it is spherically symmetric.
Properties
A central force is a conservative field, that is, it can always be expressed as the negative gradientGradient
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
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
:
(the upper bound of integration is arbitrary, as the potential is defined up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
an additive constant).
In a conservative field, the total 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...
(kinetic
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
and potential) is conserved:
(where ṙ denotes the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of r with respect to time, that is the velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
), and in a central force field, so is the angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
:
because the torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
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
Kepler's laws of planetary motion
In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....
. (If the angular momentum is zero, the body moves along the line joining it with the origin.)
As a consequence of being conservative, a central force field is irrotational, that is, its curl is zero, except at the origin:
Examples
Gravitational force and Coulomb force are two familiar examples with F(r) being proportional to 1/r2Inverse-square law
In physics, an inverse-square law is any physical law stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity....
. An object in such a force field with negative F (corresponding to an attractive force) obeys Kepler's laws of planetary motion
Kepler's laws of planetary motion
In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....
.
The force field of a spatial harmonic oscillator
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 constant....
is central with F(r) proportional to r and negative.
By Bertrand's theorem, these two, F(r) = −k/r2 and F(r) = −kr, are the only possible central force field with stable closed orbits.