These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics
by Prof. Chuck Keeton, and modified by Profs. Saurabh Jha and Eric Gawiser. All rights
reserved.
c
2011
Lecture 4: Deriving Kepler’s Laws
This discussion is drawn from Chapters 1–2 of Carroll & Ostlie.
I.
Mathematical Prelude: Cartesian and Polar Coordinates
In Cartesian coordinates the position vector (measured from the origin of the coordinate
system) of an object in two dimensions is:
~
r
=
x
ˆ
x
+
y
ˆ
y
This is sometimes written as
~
r
=
r
x
ˆ
x
+
r
y
ˆ
y
or
~
r
=
x
ˆ
i
+
y
ˆ
j
or
~
r
=
x
ˆ
e
x
+
y
ˆ
e
y
, but we will try
to stick with the notation above.
ˆ
x
and ˆ
y
are
unit vectors
, i.e.

ˆ
x

=

ˆ
y

= 1, and they always point in the direction of the x
and yaxis, respectively, as shown in Figure
??
.
r
!
y
x
!
r
^
!
^
!
x
^
y
^
!
!
2
r
!
2
x
^
y
^
r
^
!
^
Figure 1: A moving object with positions
~
r
and
~
r
2
can be described by projection onto unit
vectors in Cartesian (ˆ
x
and ˆ
y
) and polar (ˆ
r
and
ˆ
θ
) coordinates.
If our object moves in two dimensions as a function of time, then we can describe its motion
~
r
(
t
) =
x
(
t
)ˆ
x
+
y
(
t
)ˆ
y
. In general I won’t explicitly write out the fact that
~
r
,
x
, and
y
are
functions of time.
1
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We can derive the velocity of the object:
~v
≡
d~
r
dt
=
dx
dt
ˆ
x
+
dy
dt
ˆ
y
Sometimes this is written
~v
=
˙
~
r
= ˙
x
ˆ
x
+ ˙
y
ˆ
y
, but again, we’ll try to stick with the notation
above.
We can also write the acceleration vector:
~a
≡
d~v
dt
=
d
2
x
dt
2
ˆ
x
+
d
2
y
dt
2
ˆ
y
The forms for the velocity and acceleration vectors in Cartesian coordinates are simple
because the unit vectors ˆ
x
and ˆ
y
are fixed; they don’t change with time, and so (
d
ˆ
x/dt
) =
(
d
ˆ
y/dt
) = 0.
We can also write the position of the object in
polar coordinates
, in which we state the
object’s radius
r
and the angle
θ
, taken by convention to be the angle between the xaxis
and
~
r
. Then we have
x
=
r
cos
θ
y
=
r
sin
θ
r
=
p
x
2
+
y
2
tan
θ
=
y
x
Similarly, we can define
unit vectors in polar coordinates
.
The easier one to picture is ˆ
r
,
which has unit length (

ˆ
r

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 Fall '11
 Gawiser
 Physics, Cartesian Coordinate System, Polar coordinate system, Conic section

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