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Posers and Puzzles

Posers and Puzzles

  1. Standard member royalchicken
    CHAOS GHOST!!!
    03 Feb '06 00:11 / 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?
  2. Subscriber sonhouse
    Fast and Curious
    03 Feb '06 00:30
    Originally posted by royalchicken
    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?
    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 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.
  3. Standard member DoctorScribbles
    BWA Soldier
    03 Feb '06 01:38
    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?
  4. Subscriber sonhouse
    Fast and Curious
    03 Feb '06 02:32
    Originally posted by DoctorScribbles
    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?
    lets see, you substituted Doodoo for poopoo, which of course would
    instantly get modded.
  5. Donation richjohnson
    TANSTAAFL
    03 Feb '06 03:11
    Originally posted by DoctorScribbles
    Can you give an example of a simpler oscillation equation and a description of its corresponding mechanical realization?
    The motion of simple oscillator (i.e., a mass m hanging from an ideal spring having a spring consant k) can be described as:
    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.
  6. Standard member royalchicken
    CHAOS GHOST!!!
    03 Feb '06 09:01
    Originally posted by sonhouse
    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.
    I 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.
  7. Standard member royalchicken
    CHAOS GHOST!!!
    03 Feb '06 09:02
    Originally posted by richjohnson


    The equation in RC's original post seems to describe an oscillator which is governed by a restoring force which decreases over time.
    Right! Can you think of what such a thing might be?
  8. Subscriber sonhouse
    Fast and Curious
    03 Feb '06 14:28
    Originally posted by royalchicken
    Right! Can you think of what such a thing might be?
    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.
  9. Standard member PBE6
    Bananarama
    03 Feb '06 16:15
    Originally posted by sonhouse
    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.
    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.
  10. Standard member royalchicken
    CHAOS GHOST!!!
    03 Feb '06 17:17 / 1 edit
    Originally posted by sonhouse
    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.
    A 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 ).
  11. Donation richjohnson
    TANSTAAFL
    03 Feb '06 19:49
    Originally posted by PBE6
    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.
    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.
  12. Standard member PBE6
    Bananarama
    03 Feb '06 21:48 / 1 edit
    Originally posted by richjohnson
    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.
    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?).
  13. Standard member royalchicken
    CHAOS GHOST!!!
    03 Feb '06 22:29 / 1 edit
    Originally posted by PBE6
    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 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).

    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.
  14. Subscriber sonhouse
    Fast and Curious
    04 Feb '06 00:02
    Originally posted by richjohnson
    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.
    Dang, that sounds like the force that is causing the accelerated
    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.
  15. Standard member royalchicken
    CHAOS GHOST!!!
    04 Feb '06 17:56
    I found a more complicated version of this in a book today. This equation basically represents a 'quantum harmonic oscillator', and E is a unitless version of the energy (probably why the lecturer who set it as a problem used the letter E). The allowable energy levels are of the form k+1/2 where k is any integer, which explains why E must be odd. I doubt that equations requiring E to be of other forms have physical interpretations.