motion in central-force field

Let us consider a body with mass 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 =

(rr⁰) =

r⁰ + r

s⁰,

(1)

where r⁰ and s⁰ are the unit vectors in the direction of r and of r rotated 90 degrees anticlockwise (r⁰ = i cos φ + j sin φ, whence ⃗r0
dt = (−i sin φ + j cos φ) dφ-
dt = dφ
dt s⁰). Thus the kinetic energy of the body is

| | (( ) ( ) ) 1- ||d⃗r||2 1- dr- 2 d-φ 2 Ek = 2m |dt| = 2 m dt + r dt .

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

⃗ d⃗r- 2dφ- 0 0 L = ⃗r×m dt = mr dt ⃗r × ⃗s

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

mr2 dφ- = G, dt

whence

(2)

The central force F := − -k
r2 r⁰ (where k is a constant) has the scalar potential U(r) = − k
r . Thus the total energy E = E_k+U(r) of the body, which is constant, may be written

( ) ( ) ( ) 1- dr- 2 1- 2 -G-- 2 k- m- dr- 2 --G2-- k- E = 2m dt + 2 mr mr2 − r = 2 dt + 2mr2 − r.

This equation may be revised to

( dr )2 G2 2k k2 2E k2 --- + --2-2 − ----+ --2 = --- + -2, dt m r mr G m G

i.e.

( ) ( ) dr- 2 k- G--- 2 2 dt + G − mr = q

where

∘ --------------- ( 2) q := 2- E + mk--- m 2G2

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

−

= q cos ψ, dr-
dt

= q sin ψ.

(3)

Differentiation of the first of these equations implies

-G-- dr- dψ- dr- dψ- mr2 ⋅ dt = − q sin ψ dt = − dt ⋅ dt ,

whence, by (2),

dψ- -G-- dφ- dt = − mr2 = − dt .

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 =

2
----(------G----------)-
km 1 − Gqk-cos(C − φ )

(4)

where

2 p := -G--, 𝜀 := Gq-. km k

The result (4) shows that the trajectory of the body in the gravitational field of one point-like sink is always a conic section whose focus contains the sink causing the field.

As for the type of the conic, the most interesting one is an ellipse. It occurs when 𝜀 < 1. This condition is easily seen to be equivalent 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.

"motion in central-force field" is owned by pahio.
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