My title - Departamento de Matemática da Universidade do Minho

7 OSCILLATIONS AND CYCLES 53
Phase portrait of a mathematical pendulum (without and with friction)
Jacobi elliptic functions. What does a mathematician/physicist do when he/she face an integral
and doesn’t see how to solve it in terms of known functions He/she gives a name to
it.
√
E+1
Define k =
2
and then x = 1 k
sin(θ/2). The conservation of energy reads
ẋ = √ (1 − x 2 )(1 − k 2 x 2 )
There follows that time is given by the so called Jacobi’s elliptic integral of the first kind
∫
dx
t = √
(1 − x2 )(1 − k 2 x 2 )
The solution, actually the inverse function x = sn(t, k) as a function of t and the parameter k, is
“named” Jacobi elliptic function.
Tis is the beginning of along story. You may want to know that sn, as well its relatives, is a
quotient of products of Jacobi’s theta functions, hence, we are at the intersection between complex
analysis, algebraic geometry, number theory, ...
7.3 Central forces and planetary motions
Bertrand theorem. 17 The unique central forces F (r) such that any bounded orbit is periodic
are the Newtonian force F (r) = k/r 2 and the elastic force F (r) = kr.
Two-body problem. Kepler problem deals with the motion of two point-like bodies (planets
and/or stars) under mutual gravitational interaction. Let m 1 , m 2 > 0 be their masses, and q 1 , q 2 ∈
R 3 their positions, respectively. Gravitational interaction is described by the conservative force
−∇V with potential energy
V (q 1 , q 2 ) = G m 1m 2
|q 1 − q 2 |
where G is the gravitational constant. This force verifies the ”third law of dynamics”, hence the
total linear and angular momentum
and
P = m 1 ˙q 1 + m 2 ˙q 2 M = m 1 q 1 ∧ ˙q 1 + m 2 q 2 ∧ ˙q 2
are conserved. This implies that the center of mass moves at uniform rectilinear speed and that
the motion of the two bodies takes place in a plane orthogonal to the angular momentum M. If
we choose a Galileian reference system where P = 0 and M is parallel to the z-axis (in particular
M is supposed different from the zero vector, a case which leads to a collision ...) , the full system
is described by the single vector q 2 − q 1 in the x-y plane, which we write in polar coordinates
as ρe i2πθ . It turns out that the two-body problem is equivalent to the motion of a single point
mass m = m1m2
m 1+m 2
moving on a plane under the influence of a potential energy V (ρ) = −G m ρ , the
(conserved) energy beeing
E = 1 2 m (
˙ρ 2 + ρ 2 ˙θ2 ) + V (ρ)
17 J. Bertrand, Théorème relatif au mouvement d’un point attiré vers un centre fixe, C. R. Acad. Sci. 77 (1873),
849-853.