03 Feb 06
2 edits
It can be shown that the equation x'' + (E - t^2)x = 0 has a solution which converges to 0 for large t if and only if E is an odd integer (I've just had to prove this bit on a coursework -- we can take that as read. In fact, the solution is, for E = 2k+1, P*exp(-t^2/2) where P is a certain kth-degree polynomial which is not that important here).
This suggests a question for the physicists, because that equation seems to describe an oscillator of some kind, although weirder than any I've dealt with. Can someone tell me how an oscillator like this would behave, or what its physical realisation might be?
Also, the bit that intrigued me was the fact that there might be a real, physical situation which can only be realised for certain discrete values of a parameter. The fact that E must be odd for the solution to converge comes from a recurrence relation between the coefficients in a series expansion of the solution, and it should be possible to generate equations like the one above where, rather than being odd, E must be of the form 9k+7 or 4k+2 (or something) in order for such a solution to exist. I wonder if these equations can also be interpreted physically.
Typing the last paragraph gave me a nauseous feeling that this might somehow involve quantum mechanics, but I don't have the physics background to know for sure. Anyone?
03 Feb 06
Originally posted by royalchickenCan you graph it a bit for us? It sounds like it might be a kind
It can be shown that the equation x'' + (E - t^2)x = 0 has a solution which converges to 0 for large t if and only if E is an odd integer (I've just had to prove this bit on a coursework -- we can take that as read. In fact, the solution is, for E = 2k+1, P*exp(-t^2/2) where P is a certain kth-degree polynomial which is not that important here).
Thi ...[text shortened]... volve quantum mechanics, but I don't have the physics background to know for sure. Anyone?
of oscillator called a square wave generator. In electonics, the
square wave is composed of all the odd harmonics, a perfect
square wave is the sum of 1 to infinity but only odd harmonics.
Its a summation of odd harmonics of sin waves.
There is a neat and very cheap math graphics software set called
GPgraph, that might be able to graph out your equation if you
can solve it. If you PM me with your email, I can send it to you,
its pretty neat, I have fun with it all the time, you can get the
wildest graphics with relatively simple equations. Its pretty powerful
for such a cheap program.
03 Feb 06
My last post got moderated. Let me try again, with a cooler tongue.
Beats the doodoo out of me. I doubt it describes something that can be mechanically implemented. However, I'm not sure that I even understand what the equation states.
Can you give an example of a simpler oscillation equation and a description of its corresponding mechanical realization?
03 Feb 06
Originally posted by DoctorScribbleslets see, you substituted Doodoo for poopoo, which of course would
My last post got moderated. Let me try again, with a cooler tongue.
Beats the doodoo out of me. I doubt it describes something that can be mechanically implemented. However, I'm not sure that I even understand what the equation states.
Can you give an example of a simpler oscillation equation and a description of its corresponding mechanical realization?
instantly get modded.🙂
03 Feb 06
Originally posted by DoctorScribblesThe motion of simple oscillator (i.e., a mass m hanging from an ideal spring having a spring consant k) can be described as:
Can you give an example of a simpler oscillation equation and a description of its corresponding mechanical realization?
x" + (k/m)x = 0.
(The mass will oscillate at a frequency equal to (k/m)^1/2.)
The equation in RC's original post seems to describe an oscillator which is governed by a restoring force which decreases over time.
03 Feb 06
Originally posted by sonhouseI did solve it analytically, but a square wave is the sum of odd harmonics in the sense that its Fourier series has only cosine terms. This is a bit different, since the solution is completely aperiodic and converges to 0 for odd E and is divergent for even or non-integer E, when it behaves like x=e^(t^2)/2.
Can you graph it a bit for us? It sounds like it might be a kind
of oscillator called a square wave generator. In electonics, the
square wave is composed of all the odd harmonics, a perfect
square wave is the sum of 1 to infinity but only odd harmonics.
Its a summation of odd harmonics of sin waves.
There is a neat and very cheap math graphics softwar ...[text shortened]... ildest graphics with relatively simple equations. Its pretty powerful
for such a cheap program.
03 Feb 06
Originally posted by royalchickendampened oscillations in that case, like a spring with foam inside
Right! Can you think of what such a thing might be?
when you dangle a weight on it and let it spring up and down, the
up and down motion is slowly stopped by the foam inside the
spring which absorbs some of the movement each cycle.
03 Feb 06
Originally posted by sonhouseThat's the opposite of an oscillator with a decreasing resistance. Royalchicken is talking about one that "flies off the handle" so to speak for large t, unless E is an odd integer. It would be more akin to a spring made of weak material that loses its ability to pull the mass back into position as it gets used.
dampened oscillations in that case, like a spring with foam inside
when you dangle a weight on it and let it spring up and down, the
up and down motion is slowly stopped by the foam inside the
spring which absorbs some of the movement each cycle.
03 Feb 06
1 edit
Originally posted by sonhouseA dampened oscillator would, in the notation of this thread, have an x' term, which this one does not. See PBE6's comment below (above, even 😛).
dampened oscillations in that case, like a spring with foam inside
when you dangle a weight on it and let it spring up and down, the
up and down motion is slowly stopped by the foam inside the
spring which absorbs some of the movement each cycle.
03 Feb 06
Originally posted by PBE6Exactly, except this 'spring' loses its ability to pull the mass back into its neutral position over time regardless of whether or not it is used, until a time t = E^(1/2), at which point it would begin to push the mass away from its neutral position with increasing force.
That's the opposite of an oscillator with a decreasing resistance. Royalchicken is talking about one that "flies off the handle" so to speak for large t, unless E is an odd integer. It would be more akin to a spring made of weak material that loses its ability to pull the mass back into position as it gets used.
03 Feb 06
1 edit
Originally posted by richjohnsonSounds like a burrito I had once. 😞
Exactly, except this 'spring' loses its ability to pull the mass back into its neutral position over time regardless of whether or not it is used, until a time t = E^(1/2), at which point it would begin to push the mass away from its neutral position with increasing force.
BTW, what's the analytical solution to x" + (E - t^2)x = 0? I can't remember how to do an ordinary differential equation with variable coefficients (that's what this is, right?).
03 Feb 06
1 edit
Originally posted by PBE6For odd E, say E = 2k-1, it's x = exp(-(1/2)t^2)*P_k(x) where P is given by differentiating exp(-t^2) k times and then multiplying by exp(t^2).
Sounds like a burrito I had once. 😞
BTW, what's the analytical solution to x" + (E - t^2)x = 0? I can't remember how to do an ordinary differential equation with variable coefficients (that's what this is, right?).
For nonintegral, or even E, it's the same exponential function times a series whose coefficients a_n obey a silly recurrence relation:
a_n+2 = {[(2n+1)-E]/[(n+1)(n+2)]}*a_n
That explains why E odd gives a polynomial; the series will truncate when 2n+1 = E (that and all subsequent terms will vanish). Proving that that polynomial is the one above, in the form I've given it, is a little tricky. The solution I've given for odd E is subject to the condition that it goes to 0 as t --> infinity.
04 Feb 06
Originally posted by richjohnsonDang, that sounds like the force that is causing the accelerated
Exactly, except this 'spring' loses its ability to pull the mass back into its neutral position over time regardless of whether or not it is used, until a time t = E^(1/2), at which point it would begin to push the mass away from its neutral position with increasing force.
expansion of the universe just found a few years ago and proven
recently. The universe first 'exploded' and blowing itself up like a
balloon and some theories suggested with enough mass in the
universe it would slow down and contract back to a big crunch from the
big bang. But what is happening is there is a negative gravitational
effect, unkonwn as to just what it really is but at a certain point in
the life cycle of the universe, when it was about half its present age,
the expansion sped up and is still speeding up as we speak. I wonder
if that equation could be simulating that sort of thing.