Springy universe

Discussion in 'Physics & Math' started by arfa brane, Apr 29, 2014.

  1. arfa brane call me arf Valued Senior Member

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    If you hang a cable or rope between two points, you get a catenary; if you hang a spring between two points, you get a parabola.
    A parabola is a conic section, a catenary isn't.

    The motion of an ordinary object like a basketball under uniform acceleration is described by \(s = ut + \frac {1} {2}at^2\). Here, t is a parameter that satisfies the equation. If the ball is thrown upwards at an angle, it describes a parabolic path through space.
    Is there a connection between the static hanging spring and the dynamic motion of the basketball? Yes, there is.
     
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  3. arfa brane call me arf Valued Senior Member

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    Let's look at conic sections, but using pairs of planes which are kept parallel.

    There are two basic types of conic section, open and closed curves. The first set contains the circle and ellipse.
    A circle is a section of a right cone with the plane parallel to the base, or generating circle. What if the cone isn't a right cone (where the plane is perpendicular to the symmetry axis), is the section still circular? This is equivalent to asking what happens to a circle drawn on a right circular cone when the cone is made oblique, or what changes to the cone's geometry leave the circle invariant.

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    .

    These changes would necessarily exclude non-circular cones.

    A circle is also considered as a special case of an ellipse whose foci coincide. A plane section gives an ellipse on the cone when the plane is not parallel to the base.

    The two open curves, the parabola and hyperbola, are distinguished by having different 'eccentricities'.

    In this diagram

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    . . . the parabola lies on a chord of the generating circle such that its highest point lies on the other side of the cone, or the cone's axis passes through the slicing plane. The hyperbola lies on a chord such that its highest point lies on the same side of the cone, or symmetry axis.

    So . . . what happens when we construct slicing planes parallel to those for the parabola and hyperbola on the cone? What if the plane lies on a chord which is a diameter of the generating circle?
     
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  5. Randwolf Ignorance killed the cat Valued Senior Member

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    Way out of my field, but what does this mean?

    "If you roll a parabola along a straight line, its focus traces out a catenary."
     
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  7. 1100f Banned Registered Senior Member

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    807
    You don't get a parabola.
    Look here http://en.wikipedia.org/wiki/Catenary#Suspension_bridge_curve (at nearly the end of the page you have a chapter: "Elastic catenary" where the chain of the catenary is replaced by an elastic and the shape is realy not a parabola.
     
  8. exchemist Valued Senior Member

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    11,201
    Snap! I was going to say just the same.
     
  9. arfa brane call me arf Valued Senior Member

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    http://en.wikipedia.org/wiki/Parabola
    Hmmm . . .
     
  10. arfa brane call me arf Valued Senior Member

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    Back to the topic:

    First, a note about catenaries and parabolas; these are clearly related and both can be used to construct "theories" of least energy or least action. But we can consider mathematical "ideal" catenaries and parabolas, whereas in the physical world, these are within some function that transforms one into the other. A catenary can be formed from an ideal "cable" by hanging it between two fixed points, if the material is not subject to changes in length under load. A parabola can be formed by hanging an "ideal" spring between two fixed points which does "relax" under load and undergoes changes in length which are not linear (it stretches apart more near the lowest point than at either of the points of attachment).

    So, anyway, since a parabola is a conic section, I was looking at the wiki pages, and a diagram got me thinking about how the plane intersecting the cone forms each type, or class, of curve. For instance, if the elliptic section is made [more] parallel to the parabolic section, it's still an ellipse, but it isn't an ellipse when parallel, but maybe it 'contracts' to a straight line.

    You have to have the slicing plane aligned so it doesn't intersect more than a single point on the generating circle, and doesn't intersect the apex either, then the section is a closed curve on the cone.

    As soon as the plane intersects two distinct points on the boundary, it becomes an open curve, but "when" is it either an hyperbola or parabola?

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    Note in many diagrams, a parabolic section is aligned so the plane is parallel to one side of the cone, or to a second plane which intersects a straight line from the apex to the base. So, in the diagram, if this angle changes, how does the curve change? And so on.
     
  11. arfa brane call me arf Valued Senior Member

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    Construct a set of planes parallel to the green area (elliptic section) in the above figure; as these extend down towards the base, eventually the plane intersects points on the circle there. However, the curve is still elliptical in shape, an open ellipse. Equivalently, construct planes parallel to the generating circle, extending upwards; any of these planes that cut through the green area will only divide the ellipse into two parts.

    Intuitively, if you identify diametrically opposite points on the generating circle, then rotate the plane about the inscribed diameter, you get half an ellipse on the cone. Keep rotating the plane towards the vertical, eventually it intersects the apex, where instead of an ellipse you have two straight lines joined at the apex. Exclude this 'discontinuity' and keep rotating the plane. When the plane is parallel to either side of the cone, do we have a parabola? You get elliptic curves (semi-ellipses) everywhere else except when the plane is (again) horizontal.
     
  12. Cheezle Hab SoSlI' Quch! Registered Senior Member

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    745
    Not sure if your question mark is really a question, but yes, only when the plane is parallel to the cone edge is the section a parabola. The parabola is the boundary condition between the hyperbola and the ellipse. The dynamics of gravity illustrate this. If a small body has a relative velocity in the neighborhood of a large body, then the small body's path will be either an ellipse, a parabola, or a hyperbola. The difference is dependent on the velocity of the small body. At exactly the escape velocity the small body follows a parabolic path. Above the escape velocity the small body follows a hyperbolic path. Below the escape velocity the small body follows an elliptic path. So the parabolic path is a very special case where the small body's velocity is exactly the escape velocity. In fact, parabolic paths are only an ideal and don't exist in practical physics, since velocities are never exactly the escape velocity to an infinite precision and large and small are relative rather than ideal conditions.

    The conic sections are something that has been known since the ancient Greeks figured it out. But there are many ways to look at the conic sections. Here is a good video on the subject. You don't need to watch the whole thing. I should be obvious where the conic section info ends.
    However you might enjoy the rest of the video.

    [video=youtube;_mvjOoTieTk]https://www.youtube.com/watch?feature=player_detailpage&v=_mvjOoTieTk#t=285[/video]

    I suspect that wikipedia is talking about hanging springs being approximately a parabola. The same way that high school physics says that falling bodies follow a parabolic path. They don't (unless they are at exactly the escape velocity). An engineer might use the parabolic approximation to arrive at practical solutions.
     
  13. arfa brane call me arf Valued Senior Member

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    Ok, there's an obvious mistake in this: "Intuitively, if you identify diametrically opposite points on the generating circle, then rotate the plane about the inscribed diameter, you get half an ellipse on the cone. Keep rotating the plane towards the vertical, eventually it intersects the apex, where instead of an ellipse you have two straight lines joined at the apex."

    If you rotate the plane towards the apex, the section is a parabola when the plane is parallel to the side of the cone, then between this point and the apex it's an hyperbola (because the plane intersects the other "nappe" or half of the cone). So you start with a circle, then have ellipses until the parabolic section, then hyperbolas. This would also be true if the plane rotated about any chord. It seems that ellipses and hyperbolas are more general.

    Note that circles are 'bounded by' ellipses, whereas parabolas are bounded by ellipses 'from below' and hyperbolas 'from above'.
    There's an interesting homework exercise from a page of course material on Classical Mechanics; the problem involves showing how a static hanging spring is like a particle in motion, by replacing distance with imaginary time.http://math.ucr.edu/home/baez/classical/
    I'm not sure about your high school physics comment. Does a basketball follow a parabolic path when someone throws it, or what?
     
  14. Cheezle Hab SoSlI' Quch! Registered Senior Member

    Messages:
    745
    Trajectories in a uniform gravitational field are parabolic. Uniform means the gravity force vectors are parallel. But we all know that the vectors are not parallel, they all point toward the center of mass of the planet. Newton wrote about this in (I think) the Principia and considered the cases for trajectories in both uniform (Galilean) and his improved theory of gravity.

    For all practical purposes, an area the size of a basketball court's gravity can be considered very close to uniform. So using parabolic calculations is going to be very accurate for basketball sized projectiles traveling at basketball speeds. Which is a good thing because parabolas are much easy to calculate than ellipses. But it is an approximation and ignores the the reality that a basketball will follow an elliptical path, esp if the trajectory is extended through the earth rather than just hitting the floor. In the case of modern military artillery, trajectories can travel many miles and the calculations include elliptical trajectories.

    From http://plato.stanford.edu/entries/newton-principia/
    The retrospective view of the Principia has been different in the aftermath of Einstein's special and general theories of relativity from what it was throughout the nineteenth century. Newtonian theory is now seen to hold only to high approximation in limited circumstances in much the way that Galileo's and Huygens's results for motion under uniform gravity came to be seen as holding only to high approximation in the aftermath of Newtonian inverse-square gravity. In the middle of the nineteenth century, however, when there was no reason to think that any confuting discrepancy between Newtonian theory and observation was ever going to emerge, the Principia was viewed as the exemplar of perfection in empirical science in much the way that Euclid's Elements had been viewed as the exemplar of perfection in mathematics at the beginning of the seventeenth century. Because of the extent to which Einsteinian theory was grounded historically on Newtonian science, the Principia has retained its unique seminal position in the history of physics in our post-Newtonian era. Perhaps more strikingly, because of the logical relationship between Newtonian and Einsteinian theory — Einstein showed that Newtonian gravity holds as a limit-case of general relativity in just the way Newton showed (in Book 1, Section 10) that Galilean uniform gravity holds as a limit-case of inverse-square gravity — even though the Principia can no longer be regarded as an exemplar of perfection, it is still widely regarded by physicists as an exemplar of empirical science at its best.​
     
  15. LaurieAG Registered Senior Member

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    543
    The cycloid arc is an interesting curve based on a circle that has a more specific function with respect to gravity, as in the Tautochrone Curve, although the cycloid curve is traced onto a cylindrical surface not a conical surface. Galileo thought the cycloid arc would be adaptable for bridge arches in 1640.

    http://en.wikipedia.org/wiki/Cycloid

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    http://en.wikipedia.org/wiki/Tautochrone_curve

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    http://www.parabola.unsw.edu.au/vol29_no3/vol29_no3_4.pdf

    The cycloid arc also has some interesting mathamatical properties over one complete rotation:-

    (1) the area above the cycloid curve equals the area of the circle that drew the curve
    (2) the area below the cycloid curve equals 3 times the area of the circle that drew the curve
    (3) The length of the cycloid curve equals 8 times the radius of the circle that drew the curve
    (4) The total area of the cycloid plot (cylinder excluding ends) equals the surface area of a sphere of the same radius of the circle that drew the curve.
     
  16. arfa brane call me arf Valued Senior Member

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    If you aren't following the line of thought here, it's this: hanging a spring between two spatially separated points gives you a shape to work with; this shape (a parabola or a 'good' approximation) means you have a way to integrate the potential energy at each point. Then, according to Baez et al. you can transform this integral into a dynamic form which describes a particle with kinetic energy but preserves the shape of the curve.

    So I'm thinking, what if the particle is in free fall, and the curve is a straight line, or what if you throw a basketball straight up? What's the static spring equivalent, and how do we relate the shape of the path/spring to a conic section? Recall that any object falling straight towards the earth from relatively low height, has an equation of motion that could be considered a degree 2 polynomial in t, namely \(s\,=\, ut\,+\, \frac {1} {2}at^2 \), for some constant s.
     
  17. arfa brane call me arf Valued Senior Member

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    7,656
    Well, of course when you throw an object like a basketball straight up, or drop it from a height a la Galileo, it describes a more or less straight line through space, but it has a polynomial in time, a curve. Moreover (don't you just love that word?) as throwing an object straight up, or not, demonstrates, the motion is continuous; a basketball doesn't stop moving at its greatest height, nor does a pendulum.

    So how can we relate this continuous curve of "motion" to the observed fact that an object projected at a dead vertical angle to a relatively small height, returns along the same path?
    I suppose we can consider a cone with a small generating circle to height ratio, a "skinny" cone so a parabola lying on it will look like a straight line, or we imagine the object "drawing" a curve on a graph of space vs time (a spacetime diagram!), where the time axis "moves" at a constant rate, and this "time motion" makes the skinny cone look "fatter". Something like that.

    Which would mean the static spring is also on a skinny cone, so that imples it's a really loose kind of spring, with a very low k. So most of the potential is going to be in mgx where x is some point along the spring. Likewise, dropping an object from a height is analogous to hanging a loose spring from one end.
     
  18. arfa brane call me arf Valued Senior Member

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    M'kay, Baez stuff is about a lot more than using a Wick rotation. I think the wiki on this is fairly succinct:


    What they're saying here is that the result has, or "is in" one less dimension, the y dimension. In the homework example I referred to, Baez says "One of the stranger aspects of Lagrangian dynamics is how it turns into statics when we replace the time coordinate t by it . . .".
    But the wiki page says you replace distance, x, with it, and the posted solutions to the homework all do this too.​
     
  19. eram Sciengineer Valued Senior Member

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    1,877
    arf, what is your question exactly?
     
  20. arfa brane call me arf Valued Senior Member

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    What I was hoping was that someone would ask a question about this correspondence between a static 'extended' spring and a dynamic 'non-extended' particle.

    There has been a bit of discussion, or what qualifies as discussion at this forum, about the nature of time and space.
    So, if there are 'laws' of physics that describe reality, the equations--integrals over the length of a static spring and over the length of a path through space--must be telling us something about just that. But what? The wiki article says we can "trade" one dimension of space with time, but I'm not that happy with that idea, because the trade reduces the number of spatial dimensions by one, replacing it with imaginary time, or rather, mathematically the product of a real number and i.

    Note that the square root of -1, when it first appeared, was considered a mathematical trick that led to solutions. What does this trick with time say about what time might actually be?
    The operation that replaces dx with dit leaves something invariant, so it's an identity, but acting on what exactly?
     
  21. arfa brane call me arf Valued Senior Member

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    If you look further into the Lagrangian approach, you should find out it's based on the calculus of variations, and something called the variational principle.
    The wiki page on that subject says it's a "scientific principle", and that, "For example, to answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy".

    Maybe this principle is something you could explain to your grandmother, maybe not. The idea, though, is something like nature "prefers" to maximise potential rather than kinetic energy, over time. Roughly speaking, objects with a lot of kinetic energy have a high velocity because that minimises the time they spend in that state, if they accelerate this is "done quickly"; objects like a static spring hanging between two points (or from just one) have maximum potential hence spend more time in the "same" state.

    This "same" state for a static spring is a curve which is time-independent, or motion-independent. For a particle it's the critical path, which path again does not depend on the time taken along it, the shape of the path does not depend on the actual motion--it's "equation-dependent".

    The "stuff" in the shape of the path is energy, and this is what happens to it under a Wick rotation.
     
  22. arfa brane call me arf Valued Senior Member

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    7,656
    So you have to ask, do questions about the existence of time and "before" the big bang event, in the context of imaginary time make the questions about when did the square root of -1 exist? Does that even make sense, since it's a mathematical thing, a "true or false" statement about numbers.

    Does physics need to explain what time and space are, or does it do ok just assuming they "exist" and have certain properties we can measure, it really just shifts the context to one where we assume we can make "reasonable" measurements, or build machines that do this. However, it seems that any kind of machine we can reasonably build has to have certain dimensions, it must be able to collect, store and transform "information" and hence, energy. Can energy exist in less than three dimensions of space?

    Note also how we have, in the case of a discrete particle following a one-dimensional path through space, the notion of locality everywhere along this path. In the case of a spring suspended from both ends, local energy or tension is the same everywhere along the length of the spring, otherwise it wouldn't be motionless.

    Where I was going with that is in what sense can we say either a discrete particle or a spring can "collect, store, and transform" information? And in how many dimensions?
     
  23. arfa brane call me arf Valued Senior Member

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    7,656
    About locality: the action is an integral over the path length, so the locality of a test particle is necessarily defined by differential equations of motion. The correspondence between the static spring and a dynamic particle is the way energy is 'spread' over space and time according to some principle of least action or shortest path.
     

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