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Question about vector fields

  1. Jun 22, 2012 #1
    Let's say we have the acceleration vector, A, which gives us the acceleration of some particle as a function of its position in a three-dimensional space.

    Let's say that we also know the starting point of the function, say P.
    Let's say we also know the starting velocity, V.

    Can we determine the path of the object? What about its velocity, speed, acceleration all as a function of time? How much information can we get knowing only the field of acceleration and the starting point?

    I would think that knowing the starting point and knowing the acceleration allows you to immediately set up a differential equation involving all these vectors which we can solve to get more info, but I can't imagine how this would be done. It's only a guess, so I am probably wrong.

    Ideas? Please discuss. Also, if this is in the wrong forum, please move it to the right forum so the right people can answer it.

    BiP
     
    Last edited: Jun 22, 2012
  2. jcsd
  3. Jun 22, 2012 #2
    You might need for instance a starting velocity. In one dimensional motion, or differential equations, the acceleration and an initial position is not enough, one typically is also given the initial velocity, which is sufficient. (I'm ignoring any possible subtleties like is the acceleration nice, which would only make things worse.)

    If you want a counterexample to your claim, take acceleration=6, initial position is 7. Then x(t)=7+vt+3t2, for v any number. This motion satisfies acc=6 and initial position=7, but there are infinitely many trajectories, one for each value of v.
     
  4. Jun 22, 2012 #3
    I see. Let's suppose now that we do indeed know the initial velocity. I will edit my original post to add this new fact.

    BiP
     
  5. Jun 22, 2012 #4

    chiro

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    For this, you have three different attributes: acceleration, velocity, and displacement which are in vector form. If you have these you should be able to determine the full path, but you will need the initial conditions of starting point and starting velocity which I think you said you have. Without these initial conditions you will either misestimate the velocity and thus screw up the displacement or misestimate the starting position and misestimate the final position at some time (or both).

    For example if you under-estimate your starting velocity, then the real particle will move faster than your calculated trajectory. If your over-estimate it, it will move slower.

    One thing to take into account though is the nature of the system. For example you might have a system which converges to a particular value for the three attributes irrespective of your initial condition. If this is the case then it means that the long-term behaviour of the particle will always be a good approximation irrespective of the initial conditions.

    An example of this for a simple function is say a y = f(x) function which approaches 0 in an inverse exponential manner like say a Newtonian cooling problem, irrespective of the initial condition.

    But mathematically if you want to justify this, simply use the mathematical representation of getting displacment from acceleration (at least explicitly) and then see that there are two unknowns with respect to the initial value: this is the mathematical explanation while some physical ones are mentioned above.
     
  6. Jun 22, 2012 #5
    Let's not assume convergence, and let's say we know all variables in the initial time state. What do we do from there?

    BiP
     
  7. Jun 22, 2012 #6

    chiro

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    Well for an exact model we need the exact initial conditions.

    For your own interest, you might want to study the issues that are faced in introducing errors which are a topic of numerical analysis. The propagation of error and its own analysis is used when attempting to model systems with erroneous or approximated measurements, and this is relevant to your situation.

    In the end though you will have to supply information that corresponds to initial velocities in some form or another whether its approximated, a good guess, or exact. To understand how this will propagate in terms of the model, you can refer to the numerical analysis techniques specifically in relation to error analysis and propagation. If your model is very well defined, then these error properties become highly specific.

    Typically this kind of thing is introduced in a differential equations class or any numeric analysis classes, as well as in some elements of a linear algebra class.
     
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