motion in central-force field

Let us consider a body with m in a gravitational force field ( exerted by the origin and directed always from the body towards the origin.  Set the plane through the origin and the velocity vector v of the body.  Apparently, the body is forced to move constantly in this plane, i.e. there is a question of a planar motion.  We want to derive the trajectory of the body.

Equip the plane of the motion with a polar coordinate system r,φ and denote the position vector of the body by r.  Then the velocity vector is

v=drdt=ddt(rr 0)=drdtr 0+rdφdts 0, (1)

where r 0 and s 0 are the unit vectorsMathworldPlanetmath in the direction of r and of r rotated 90 degrees anticlockwise (r 0=icosφ+jsinφ,  whence  r 0dt=(-isinφ+jcosφ)dφdt=dφdts 0).  Thus the kinetic energy of the body is


Because the gravitational force on the body is exerted along the position vector, its moment is 0 and therefore the angular momentum

L=r×mdrdt=mr2dφdtr 0×s 0

of the body is constant; thus its magnitude is a constant,



dφdt=Gmr2. (2)

The central force  F:=-kr2r 0  (where k is a constant) has the scalar potentialMathworldPlanetmathU(r)=-kr.  Thus the total energy  E=Ek+U(r) of the body, which is constant, may be written


This equation may be revised to






is a constant.  We introduce still an auxiliary angle ψ such that

kG-Gmr=qcosψ,drdt=qsinψ. (3)

DifferentiationMathworldPlanetmath of the first of these equations implies


whence, by (2),


This means that  ψ=C-φ, where the constant C is determined by the initial conditions.  We can then solve r from the first of the equations (3), obtaining

r=G2km(1-Gqkcos(C-φ))=p1-εcos(φ-C), (4)



By the entry, the result (4) shows that the trajectory of the body in the gravitational field ( of one point-like sink is always a conic sectionMathworldPlanetmath whose focus the sink causing the field.

As for the of the conic, the most interesting one is an ellipseMathworldPlanetmath.  It occurs, by the entry, when  ε<1.  This condition is easily seen to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath with a negative total energy E of the body.

One can say that any planet revolves around the Sun along an ellipse having the Sun in one of its foci — this is Kepler’s first law.


  • 1 Я. Б. Зельдович &  А. Д. Мышкис: Элементы  прикладной  математики.  Издательство  ‘‘Наука’’.  Москва (1976).
Title motion in central-force field
Canonical name MotionInCentralforceField
Date of creation 2013-03-22 18:52:41
Last modified on 2013-03-22 18:52:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Derivation
Classification msc 15A72
Classification msc 51N20
Synonym Kepler’s first law
Related topic CommonEquationOfConics
Related topic PropertiesOfEllipse