From: Richard.Mathews@West.Sun.COM (Richard M. Mathews)
Newsgroups: sci.physics
Subject: Re: What is (1/6)M(V^3)?
Date: 23 Jul 1992 01:22:44 GMT
Organization: Sun Microsystems, Inc.
NNTP-Posting-Host: astro
Originator: richard@astro.West.Sun.COM

jcr@milton.u.washington.edu (Mr. EuStress) writes:

These questions are really just a list of non sequiturs, but I will try
to respond to each as best I can.

>  If momentum is MV (Mass times Velocity) and energy is 
>  (1/2)M(V^2) (the integral of MV) what is (1/6)M(V^3)?

It is somewhat coincidence that the Newtonian formula for kinetic energy
is related to momentum by E = integral(P dot dv), so there is no reason
to believe that the integral(E dv) has any significance.  Furthermore,
when you think about it, without the dot product in there, it isn't
really even the same sort of integral anyway.  A better (but still
somewhat obscure) way to express the relationship between E and P is
delta-E = integral(v dot dP).  Even better is to recognize that v is
ds/dt and rearrange things to get delta-E = integral(F dot ds) where F
is dP/dt.  This then shows how force in the direction of motion increases
energy.

>    If time is the forth dimension then is time squared
>  (acceleration) the fifth dimension?
>  Is there any limit to the number of dimensions (degrees 
>  of freedom) that we live in?

No.  First, time squared is not acceleration.  Second, when we say
that time is a fourth dimension we mean that the position of any event
can be uniquely described as a function of four numbers (x, y, z, and
time).  If you like, you can describe position of an event in terms the
numbers x^3, tanh(y), e^z, t^2, but that doesn't make time-squared a
*different* independent dimension from time.  You still need four and
only four numbers to describe when/where an event occurred.  For there
to be a fifth dimension you would have to find that there is somehow a
difference between any two points to which we assign the same four
coordinates (no matter how we set up our coordinate system), so a fifth
number is required to distinguish these points.

Now we do mean a little bit more when we call time a "dimension".  Let's
look at x, y, and z a little more first.  Those are defined in terms of
a particular set of coordinate axes.  If we rotate those axes we will
define position in terms of a different set of 3 numbers, say x', y',
and z'.  A rotation about the z axis, for example, might give
	x' = cos(theta) * x - sin(theta) * y
	y' = cos(theta) * y + sin(theta) * x
	z' = z
Similarly, we find that time can be defined only in terms of a particular
coordinate axis.  We rotate this axis whenever we change the speed or
direction of our motion.  Moving, for example, at a speed v = c * tanh(q)
in the x direction we will find that the world is not most easily described
by the following new coordinates:
	t' = cosh(q) * t - sinh(q) * x
	x' = cosh(q) * x - sinh(q) * t
	y' = y
	z' = z
Relative motion corresponds to a rotation of time into the spatial
directions and spatial directions into the time direction.  In Newtonian
physics, time was a magic coordinate which could be assigned to an event
and everyone would agree on the number.  In the Einsteinian world, the
direction of time is no more special than the direction of "x".

    Richard M. Mathews			 Freedom for Lithuania
Richard.Mathews@West.Sun.COM		        Laisve!
Xref: sparky sci.physics:11902 sci.astro:8430 sci.philosophy.tech:2895 talk.religion.misc:12896