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Over-determined systems, too many degrees of freedom. What's moving?

  1. Sep 4, 2009 #1

    pL1

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    Hello,

    consider this:

    http://www.alice-dsl.net/l.hansen/overdetermination.png [Broken]

    See this image? Every point in space can be reached for the mass by the springs motion plus the springs rotation and then there's the pendulums rotation additionally. Not solvable analytically but numerically with initial values?

    Then what's with the gravitational force on the mass?. It could simply compress the spring in vertical initial position. Or in the current position: what does the tangential force to the mass do? Does it initiate a pendulum rotation or does it split into x and y force initiating a spring compression and a spring rotation? Then the force is used up and there would be no rotation. Consider that rotation by tangential force is only an assumption. However infintely small you go, the rotation always deviates from a straight line immediately thus no straight force could initiate a rotation. Only as a last resort. But now we have an overdetermined system?

    Or does the force distribute equally onto every reactant?

    Thoughts appreciated!

    Thanks pL1
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Sep 5, 2009 #2

    pL1

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    We know the pendulum in gravitational field. Take a pendulum in the gravitational field with a rigid yet massless connection between mass and rotation axis with the rotation axis being the top of a vertical spring. The spring is massless and confined to vertical movement. Let the connection be horizontal initially. How's the mass moving? In particular, does the gravitation initialize rotation or spring movement or both ? Now the answer is after the resulting force on the mass and thereby on any rigidly connected part has been calculated the movement which starts is the movement where energy consumption is minimal. That is where potential energy consumption is minimal, so where work done is minimal. At the beginning the resulting force due to the gravitational force and the spring force is alongside the spring movement as well as having a tangential component. Shortly the equilibrium location of the gravitational spring (that is a mass on a vertical spring in the gravitational field) will be reached after which the spring force is greater than the gravitational force. New motion (deviation from a straight line or deviation from constant speed) occurs according to acceleration which a force provides. The direction of resulting force is given thus the movement should be given. However the movement is confined allowing only either of the above. Now the resulting acceleration is the component of the resulting force alongside the non-restricted motion possibility. And here are two non-restricted motion possibilities. The resulting force and its components act on both mass and spring (due to the rigid connection) completely at the same time. However from the possible new motions (the rotation could be going up initially for instance) only that new motion of every possible new motion with the minimal energy consumption of the energy consumptions of these every possible new motions will occur. This has the consequence that while both downward rotation and spring movement will initiate, downward spring movement will come to a halt shortly and the rotation will be poised to stop because the rotational movement isn't any longer parallel the tangential part of the resulting force which means there is no longer nothing to work against (work was saved until now) rather the resulting force is antiparallel the current rotation thus work would have to be done. Having a new motion in the opposite direction for both spring a pendulum will therefore occur. New motion. The old motion has a velocity though! Thus the new velocity is the old velocity plus the new acceleration and this may well result in a new location for the mass due to spring and pendulum motion to which work has been done! This happens to be sensible for certain forces: We leave the system on its own without connecting it to any outside: Now the consumed energy that was saved also gets used up again hopefully resulting in net consumed energy difference of 0. Otherwise the system would loose or gain the values of the energy integral which you know is responsible for force (negative gradient ..) and thus acceleration and thus movement. If nothing would move anymore nothing would be and never again. On the other hand if everything moves infinitely fast processing resources would get into trouble.
    The theory here is the one Lagrange used when was saying: ∫ F(r) dr (r is the location of the mass, dr are the infinitesimal segements of the mass motion curve and F is the force at r) is the consumed energy and the motion curve (aka the movement) will be thus that s ∫ F(r) dr = 0 that means the consumed energy of one motion across every possible motion should be minimal (extremal) from all the consumed energy of the possible motions and this is the motion the mass follows.
    Giving a reason for this behaviour I can say "nature is lazy".
    Thanks to nature we don't have to care for the second derivative: the (lowest) extremum found with first derivative = 0 is a minimum.
    However there is at least one system where energy comsumptions extremum is a maximum. Whether the systems motion was actually according to the motion of the maximum energy consumption I didn't find out. However there potentially is one system in nature where motion follows according to spending as much energy as possible. I. ?
    Solving systems numerically we now know how to do. Finding the next acceleration via the minimum energy consumtion theorem may be easier than numerically solving the differential equation system which we can write down.
     
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