Brachistochrone Under Air Resistance

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Presentation transcript:

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

Source of Information:

Collaborator:

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

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

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?

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

Energy Constraint Non-conservative system: Constraint parametrized by time: Constraint parametrized by arclength:

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

Lagrange Multipliers Introduce multipliers, vector: Create modified functional: where

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

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

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

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

First Integral Recall: No explicit s-dependence! First Integral:

Roadmap to Solving Problem: Given the first integral, can solve for v(): Then use E-L equation to solve for (s): Then integrate E-L equations for x(s), y(s): Done? - still need l, v1, 1... - maybe we could do better a different way...

Parametrize by Slope Angle

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

Modified Functional Transform modified problem in terms of :

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

Same Natural B.C.’s & Lagrange Multipliers

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

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

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

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

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

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

Example: Air Resistance

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,  0 = /2

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

Many Calculations...

Results: (Straight Line) (Cycloid)

Conclusions Different approach to the Brachistochrone Gained: 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

Questions ?