1
Newton’s Second Law
The form of Newton’s second law for three separate cases will be generated using quaternion operators acting on
position quaternions. In classical mechanics, time and space are decoupled. One way that can be achieved algebraically
is by having a time operator cat only on space, or by space operator only act on a scalar function. I call this the ”2
zero” rule: if there are two zeros in the generator of a law in physics, the law is classical.
Newton’s 2nd Law for an Inertial Reference Frame in Cartesian Coordinates
Define a position quaternion as a function of time.
R
t
,
R
Operate on this once with the differential operator to get the velocity quaternion.
V
d
dt
,
0
t
,
R
1
,
.
R
Operate on the velocity to get the classical inertial acceleration quaternion.
A
d
dt
,
0
1
,
.
R
0
,
R
This is the standard form for acceleration in Newton’s second law in an inertial reference frame. Because the reference
frame is inertial, the first term is zero.
Newton’s 2nd Law in Polar Coordinates for a Central Force in a Plane
Repeat this process, but this time start with polar coordinates.
R
t
,
r Cos
,
r Sin
,
0
The velocity in a plane.
V
d
dt
,
0
t
,
r Cos
,
r Sin
,
0
1
,
.
r Cos
r Sin
.
,
.
r Sin
r Cos
.
,
0
Acceleration in a plane.
A
d
dt
,
0
1
,
.
r Cos
r Sin
.
,
.
r Sin
r Cos
.
,
0
0
,
2
.
r Sin
.
r Cos
.
2
r
Cos
r Sin
,
2
.
r Cos
.
r Sin
.
2
r
Sin
r Cos
,
0
Not a pretty sight. For a central force,
.
L/
2
, and
0. Make these substitution and rotate the quaternion to
get rid of the theta dependence.
A
Cos
,
0
,
0
,
Sin
d
dt
,
0
2
t
,
r Cos
,
r Sin
,
0
0
,
L
2
m
2
r
3
r
,
2 L
.
r
m r
2
,
0
The second term is the acceleration in the radial direction, the third is acceleration in the theta direction for a central
force in polar coordinates.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 123
 Force, Frame of reference, Simple Harmonic Oscillator, Central Force

Click to edit the document details