THEORETICAL DYNAMICS — PHY–5246
HOMEWORK 9
(October 20, 2002)
Due on Monday, October 29, 2002
PROBLEM 25
Imagine yourself being Kepler around the year 1610 and having compiled information
on the motion of all the planets you decide to make a table that includes: a) the name
of the planet, b) the period of revolution of the planet around the Sun (in years), c)
the average distance (or semi-major axis) from the planet to the Sun (in A.U.), and
d) the ratio of the period square to the average distance cubed. After having found
this amazing correlation (Kepler’s third law) compute the mass of the Sun. Although
not yet discover at the time, you may use Newton’s Law of Universal gravitation with
G=6.673 ×10−11 m3kg−1s−2.
PROBLEM 26
a) Starting from Euler-Lagrange’s equation of motion for the radial motion of a
particle of mass mmoving in a central potential, i.e., m¨r=−V0
eff (r), obtain
the differential equation for the orbit:
d2u
dθ2+u=−m
l2
f(1/u)
u2,
where u≡1/r and f(r) = −V0(r). The above equation suggests how to obtain
the orbit of the particle r=r(θ) if the force law f=f(r) is known. Conversely,
if the orbit is known, one can deduce the force law.
b) Consider the equation of the orbit in Cartesian coordinates:
x2
a2+y2
b2= 1 .
Obtain the force law f=f(r).
c) Consider the equation of the orbit in Cartesian coordinates:
(x−c)2
a2+y2
b2= 1 ,
where c=√a2−b2. Obtain the force law f=f(r).
