wendy wrote:An object at constant acceleration requires a constant force to it.
d d d
F = --- (m_2 --- (m_1 --- (m_0 * x) ) ) )
dt dt dt
Obviously though, if acceleration were constant when the net force is zero, objects could easily reach arbitrarily high speeds.
PWrong wrote:If we really wanted to restrict arbitrarily high speeds, we could possibly "wrap" velocity the way we wrap a space around a torus.
If you go past the edge of a torus-shaped universe, you come back on the other side at the same speed. This doesn't break any laws, because that's just the way space is wrapped.
Similarly, if you go above a certain speed, you simply stop instantly, but keep the same acceleration. Hopefully this wouldn't cause any contradiction. You could think of it as a bit like an ordinary circular speedometer. If you keep accelerating, the needle will go all the way round, come back to 0 and keep rising. Everything in the car would experience the same thing, because no "force" is acting, so you wouldn't hit the windscreen.
It looks just like a Taylor expansion. This should be easy to generalize to a universe with n time derivatives.
Speaking of which, in a "F = mj" universe, the Principle of Relativity may have to be modified. If acceleration is constant, then jerk is zero, and hence net force would be zero.
I think I'm also going to try looking at what happens to work-energy in a F = mj universe. I suspect that will be more challenging.
PWrong wrote:I'm sure that relativity would have to be modified along with everything else. I won't formally learn relativity for a few more weeks, although I've had a look in my textbook. It seems a lot easier mathematically than conceptually, since all you do is multiply everything by
gamma = 1/sqrt(1- v^2/c^2).
This apparently works for time, displacement, momentum, force and total energy.
It should be possible to derive the formulas relativity in an 3T universe using just Einsteins first postulate, and the 3T version of the second postulate:
"The acceleration of light is the same in all inertial frames of reference,
and this is independent of the motion of the source."
PWrong wrote:A likely definition for Work would be the integral of force with respect to distance. But that has units of [W3] = kg m^2 s^-3. I can't see any nice formula for energy using that.
Energy conservation comes from url=http://en.wikipedia.org/wiki/Noether%27s_theorem]Noether's Theorem[/url], which I don't think I'll understand for at least a couple of years.[/url]
Sorry, maybe I wasn't clear. I was talking about the "Principle of Relativity" (something that was known by Galileo in the 1600s), not Einstein's "Theory of Relativity".
Edit: Come to think of it though, I don't recall hearing a version of integration by parts, with the dot product replacing ordinary multiplication, for vector-valued functions.
That's why I prefer sticking with "Integral of (F dot dr)". I'm pretty sure that conservative vector fields should still have a property called potential that depends on position. In another five months, I will have finished a course on vector calculus, and I should have the skills to prove it.
PWrong wrote:W = m * Integral (r'''(t) dot r'(t) dt)
Hmm, maybe integration by parts will do the trick!
Presumably, we would want v = 0 and a = 0 when the object comes to rest (the assumption here is that in order for an object to have no kinetic energy, both it's velocity and it's acceleration must be zero).
PWrong wrote:Your assumption sounds valid, but it's unlikely for any object to have zero velocity and zero acceleration at the same time. a = dv/dt, so when a=0, v is at a maximum or minimum.
More importantly, the assumption doesn't agree with K = mva.
If K = 1/3 mva = 0,
then v=0 OR a=0,
but not neccessarily both.
PWrong, you mentioned earlier in this thread the possibility of a universe whose evolution is governed by an infinite order differential equation in time. Recently though, I have to come to see why this probably not possible. The argument is surprisingly simple.
PWrong wrote:That's a shame. I think I understand your argument. For the taylor series to converge, the force would have to vanish.
Thus, dynamics would be obselete.
This would appear to contradict the principle that nature should make no distinction between the past and the future.
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