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 by dt, 0) acting on R = (1, R dot)

Operate on the velocity to get the classical inertial acceleration quaternion.

A = (d by dt, 0) squared acting on R = (0, R double dot)

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  cosine theta, r  sine theta, 0)

The velocity in a plane.

V = (d by dt, 0) acting on (t, r cosine (theta), r sine (theta, 0)
=

= (1, r dot cosine theta - r sine theta theta dot, r dot sine theta + r
cosine theta theta dot, 0)

Acceleration in a plane.

A = (d by dt, 0) acting on (1, r dot cosine theta - r sine theta theta
dot, r dot sine theta + r cosine theta theta dot, 0)
=

= (0, - 2 sine theta R dot theta dot - cosine theta R (theta dot)squared
+ cosine theta R double dot - R sine theta theta double dot,2 cosine theta R
dot theta dot -R sine theta (theta dot)squared + sine theta R double dot +
cosine theta R theta double dot, 0)

Not a pretty sight. For a central force, = [Graphics:images/s_gr_10.gif], and [Graphics:images/s_gr_11.gif] = 0. Make these substitution and rotate the quaternion to get rid of the theta dependence.

A = (cosine theta, 0, 0, - sine theta) times (d by dt,0) squared acting
on R =

= (0, -L squared over m squared  R cubed + R double dot, 2 L R dot over m R
squared , 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.

Newton's 2nd Law in a Noninertial, Rotating Frame

Consider the "noninertial" case, with the frame rotating at an angular speed omega. The differential time operator is put into the first term of the quaternion, and the three directions for the angular speed are put in the next terms. This quaternion is then multiplied by the position quaternion to get the velocity in a rotating reference frame. Unlike the previous examples where t did not interfere with the calculations, this time it must be set explicitly to zero (I wonder what that means?).

V = (d by dt, Omega) acting on (0,R) = (- omega dot R, R dot + Omega
Cross R)

Operate on the velocity quaternion with the same operator.

A = (d by dt, Omega) acting on (- omega dot R, R dot + Omega Cross
R)

= (- Omega dot dot R, R double dot + 2 Omega Cross R dot + Omega dot  Cross
R - Omega dot R Omega )

The first three terms of the 3-vector are the translational, coriolis, and azimuthal alterations respectively. The last term of the 3-vector may not look like the centrifugal force, but using a vector identity it can be rewritten:

- Omega dot R Omega = - Omega Cross (Omega Cross R) + Omega squared
R

If the angular velocity an the radius are orthogonal, then

Omega Cross (Omega Cross R) = Omega squared R if and only if Omega dot R =
0

The scalar term is not zero. What this implies is not yet clear, but it may be related to the fact that the frame is not inertial.

Implications

Three forms of Newton's second law were generated by choosing appropriate operator quaternions acting on position quaternions. The differential time operator was decoupled from any differential space operators. This may be viewed as an operational definition of "classical" physics.