Motion Equation Central Force

The basic equation of motion of a particle moving under the influence of a central force F = F_r e_r is



where, acceleration a = a_r e_r of the particle. We shall now solve the above equation for the spherically symmetric inverse square force:

F_r = – C/r²      (C > 0)

Our task is to find the trajectory of the particle, i.e. a relation between r and θ. (Remember, the motion of the particle is confined in a plane and its position is given by co-ordinates r and θ at any time.) To do that, we use the fact that angular momentum of the particle is a constant of motion:



If we introduce the variable u = 1/r. Hence, we get



Substituting above relation, we find



Taking, Fr = – Cu², we get the simple relation,



The above can be rewritten as



which is the familiar equation of simple harmonic motion; its solution is,

(u – A) = B cos (θ – θ₀)

where, B and θ₀ are constants of integration.

Thus we get the equation of the trajectory as,

U = 1/r = A + B cos (θ – θ₀)

or, p/r = 1 + e cos (θ – θ₀)

where, 1/A, and e = B/A are constants of motion.

It is the standard equation of the conic section, i.e. the equation of the curve formed by the section of a cone by a plane. The constants p is called the orbital parameter and e is called eccentricity. The conic section may be an ellipse (e < 1), a circle (e = 0), a parabola (e = 1), or a hyperbola (e > 1). That is, in general, the path of a particle under an inverse-square force produced by a fixed source (center of force) in space could be bounded (ellipse or circle), or unbounded (parabola or hyperbola) depending upon the value of e.

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