Trajectories in 2D

briefly reviews 2 key pieces: “state space” (2D example) and “change vectors”

Summary

  • How do we obey the change vectors?
    • If you are at the point \((X,Y)\)
    • then head in the direction \((X', Y')\)
    • at a speed given by the length of the vector \((X', Y')\), which is \( \left\lvert(X', Y')\right\rvert\).
    • Head in that direction, yes, but for how long?
  • Big problem
    • if you obey it for \(1\) sec you are wrong, but if you obey it for \(.01\) sec you are still wrong: you are ignoring the change vectors along the way. The correct answer to the question “for how long?” is “zero”, which is obviously a problem!
  • This problem was solved by Newton, and was the beginning of calculus.
    • He saw that you could let the “time step” Δt approach 0 and still get a finite value for “the instantaneous change”.
  • Picard-Lindelof theorem
    • There is a theorem (Picard-Lindelof) that says that if we are given the instantaneous changes at every point, then there exists a curve that has exactly those instantaneous changes. (“the one true curve”, “the solution curve”).
  • This is great: the curve exists.
    • But how do we find it? There is the problem. In particular, can we figure out the equation for the curve from the equation for the vector field? The answer is no, 99% of the time. There is no formula for the curve. Then what are we to do? We must find some way of approximating the true curve.