3B (Dynamics and Relativity)

In a short time

d t

, the radius vector sweeps an oriented area

d A = \frac{1}{2} r^{2} d θ

. Thus

\frac{d A}{d t} = \frac{h}{2} .

The constant

h

is twice the oriented areal velocity, or the angular momentum per unit mass. Its constancy is Kepler's second law: equal areas are swept in equal times.

a

\overset{r}{¨} = 0

in the radial equation. The balance is

h^{2} = GM a, ∣ \dot{θ} ∣ = \frac{∣ h ∣}{a ^{2}} = \frac{GM}{a ^{3}} .

Hence a circular orbit exists for any

a > 0

with this nonzero

∣ h ∣

. Its period gives Kepler's third law:

T = 2 π \frac{a ^{3}}{GM}, T^{2} = \frac{4 π ^{2} a ^{3}}{GM} .

h

fixed and set

r = a + ρ

. Expanding

\overset{r}{¨} = \frac{h ^{2}}{r ^{3}} - \frac{GM}{r ^{2}}

about the balanced radius gives

\overset{ρ}{¨} = (- \frac{3 h ^{2}}{a ^{4}} + \frac{2 GM}{a ^{3}}) ρ + O (ρ^{2}) = - \frac{GM}{a ^{3}} ρ + O (ρ^{2}) .

The linear radial perturbation therefore obeys

\overset{ρ}{¨} + Ω^{2} ρ = 0, Ω^{2} = GM / a^{3} > 0.

It remains bounded and oscillates, proving stability under the stipulated small perturbations. Equivalently, the specific effective potential

h^{2} / (2 r^{2}) - GM / r

has a strict minimum at

a

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