chapter  1
20 Pages

From Kepler problem to skyrmions

The classical Kepler motion can be described with the Newtonian equations of motion

r¨ + K

r2 r

r = 0. (1.1)

Here, K is a constant, r denotes the position vector of the material point whose motion is described, with respect to the center of force, and a dot over a symbol means derivative with respect to time. The constant K does not depend on quantities related to the point in motion, but only in cases when electric forces are involved. We can simplify the algebra by confining the geometry to the plane of motion, where the coordinates of the point in motion are ξ and η (see [1]). Eq. (1.1) is then equivalent to the system

·· ξ +K


r2 = 0,

·· η +K


r2 = 0 (1.2)

with r and ϕ the polar coordinates of the plane with respect to the attraction center. The magnitude of the rate of area swept by the position vector of the particle is then given by

a˙ ≡ ξη˙ − ηξ˙ = r2ϕ˙. (1.3)

This constant of motion allows us an elegant integration of the system (1.2) with the analytical form of the trajectory as a direct

outcome. First we define the complex variable

z ≡ ξ + iη = reiϕ. (1.4)

so that (1.2) can be written in the form

·· z +K

r2 = 0 (1.5)

Now, use (1.3) to eliminate, such that

z¨ +K eiϕ

a˙ ϕ˙ = 0, z˙ = i

( K eiϕ

a˙ + w

) , (1.6)

where w ≡ w1 + iw2 is a complex constant of integration to be determined by the initial conditions of the problem. The analytical equation of motion can be then extracted directly from (1.3) by using (1.6). In polar coordinates of the plane of motion the result is

r = K

a˙ + w1 cosϕ+ w2 sinϕ. (1.7)

The shape of this trajectory is best pictured by going back to Cartesian coordinates, where we have, instead of (1.7) the seconddegree curve a conic:( K2

) ξ2 − 2w1w2ξη +

( K2

) η2 + 2

· a(w1ξ + w2η) = a˙

(1.8) The center of this conic is not the center of the force, but has the


ξ0 = − a˙w1 ∆

, η0 = − · aw2 ∆

,∆ ≡ ( K

)2 − w21 − w22. (1.9)

In cases where ∆ = 0, the center of this trajectory is at infinity: the trajectory is a parabola. We have here the ballistic cases, where the basic motion is parabolic.