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Brachistochrone Under Air Resistance

Brachistochrone Under Air Resistance. Christine Lind 2/26/05 SPCVC. Source of Information:. Collaborator:. Brachistochrone Setup:. Initial Point P(x 0 ,y 0 ) Final Point Q(x 1 ,y 1 ) Resistance Force F r Slope Angle . Geometric Constraints. Parametric Approach:

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Brachistochrone Under Air Resistance

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  1. Brachistochrone Under Air Resistance Christine Lind 2/26/05 SPCVC

  2. Source of Information:

  3. Collaborator:

  4. Brachistochrone Setup: • Initial Point P(x0,y0) • Final Point Q(x1,y1) • Resistance Force Fr • Slope Angle 

  5. Geometric Constraints • Parametric Approach: • Start by using arclength (s) as the parameter • Parametrized by arclength: (curves parametrized by arclength have unit speed)

  6. Energy Constraint? • Normally we use conservation of energy to solve for velocity in terms of the other variables • We have a Non-Conservative system, so what do we do?

  7. Energy Constraint? • Energy is lost to work done by the resistance force:

  8. Energy Constraint • Non-conservative system: • Constraint parametrized by time: • Constraint parametrized by arclength:

  9. Problem Formulation: • Boundary & Initial Conditions: • Minimize the time integral: • Other constraints: How do we incorporate them?

  10. Lagrange Multipliers • Introduce multipliers, vector: • Create modified functional: where

  11. Euler-Lagrange Equations • System of E-L equations: • Additional boundary conditions:

  12. Note: Note: Additional Constraints Appear as E-L equations! 7 Euler-Lagrange Equations:

  13. Natural Boundary Conditions: Note: v1 is not necessarily zero, so:

  14. Note: (s),(s) constants (s)=((s)) Lagrange Multipliers - Solved! • Using: • Determine the Lagrange Multipliers:

  15. First Integral • Recall: No explicit s-dependence! • First Integral:

  16. Parametrize by Slope Angle

  17. Still need constraints... Parametrize by Slope Angle • Define f() to be the inverse function of (s): • f() continuously differentiable, monotonic • Now we minimize:

  18. Modified Functional • Transform modified problem in terms of :

  19. First Integral! (Old Equations) 7 Euler-Lagrange Equations

  20. Same Natural B.C.’s & Lagrange Multipliers

  21. Solve for v() • Using Lagrange Multipliers and First Integral: • Obtain:

  22. Solve for Initial Angle 0 • Evaluate at 0: • Obtain Implicit Equation for initial slope angle:

  23. Solving for f() • Rearrange E-L equation: • Obtain ODE: ( Recall that we already have v(), 0, & initial condition f(0) = 0 )

  24. Solving for x() and y() • Integrate the E-L equations • Obtain

  25. Seems like we are done... • What about parameters 1 & v1? Appear everywhere, due to: • How can we solve for them?

  26. Newton’s Method... • Use the equations for x() and y() and the corresponding boundary conditions: • Now we really are done!

  27. Example: Air Resistance

  28.  0 = /2 Example: Air Resistance • Take R(v) = k v (k - coefficient of viscous friction) • Newtonian fluid • first order approx. for air resistance • Let x0 = 0, y0 = 0, v0 = 0,

  29. Solve for v() Quadratic Formula: ( take the negative root to satisfy v(0) = 0 )

  30. Many Calculations...

  31. (Straight Line) (Cycloid) Results:

  32. Conclusions • Different approach to the Brachistochrone • parametrization by the slope angle  • use of Lagrange Multipliers • Gained: • analytical solution for non-conservative velocity-dependent frictional force • Lost ( due to definition s = f() ): • ability to descibe free-fall and cyclic motion

  33. Questions ?

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