My title - Departamento de Matemática da Universidade do Minho

7 OSCILLATIONS AND CYCLES 54
Observe that if one of the bodies is much bigger than the other (like the Sun and the Earth), say
m 1 ≫ m 2 , then the center of mass nearly coincides with the position q 1 of the bigger body, while
the reduced mass m is essentially the mass m 2 of the smaller one (hence it looks like the Earth
moving around the Sun, as Galileo had suggested).
Central forces.
Consider the Newton equation
m¨r = F (|r|) ˆr
describing the motion of a particle (planet) of mass m in a central force field F . Conservation
of angular momentum implies that the motion is planar, hence we may take r ∈ R 2 . In polar
coordinates r = ρe iθ , the equations reed
¨ρ − ρ ˙θ 2 = F (ρ)/m
ρ¨θ + 2 ˙ρ ˙θ = 0 .
The second equation says that the “areal velocity” (“velocidade areal”) l = ρ 2 ˙θ is a constant of
the motion (Kepler’s second law).
Taking Newton’s gravitational force F (ρ) = − GmM
ρ
(where M is the mass of the Sun and G is
2
the gravitational constant), the first equation may be written as
m¨ρ = − ∂ ∂ρ V l (ρ) ,
where we defined the ”effective potential energy” as V l (ρ) = 1 2 m l2
ρ
− G mM 2 ρ
is
E = 1 2 m ˙ρ2 + 1 2 m l2
ρ 2 − GmM ρ .
. The conserved energy
Kepler’s effective potential and some energy level sets.
Now we set ρ = 1/x and look for a differential equation for x as a function of θ. Computation
shows that dx/dθ = − ˙ρ/l, and, using conservation of l, that d 2 x/dθ 2 = −ρ 2 ¨ρ/l 2 . There follows
that the first Newton equation reads
we get
d 2 x
dθ 2 + x = − 1
l 2 x 2 m F (1/x) .
d 2 x
dθ 2 + x = −GM l 2 .