Classical Mechanics (Kibble & Berkshire) · Problem 13.15

Hamilton's equations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · calculus-of-variations

Problem

The downhill skier Solve the problem of finding the path of quickest descent for a skier from the point

P(x_0, y_0, z_0)

to the point

Q(x_1, y_1, z_1)

on a snow covered mountain whose profile is given by

z = G(x, y)

, where

G

is a known function. [Assume that the skier starts from rest and that the total energy of the skier is conserved in the descent.] Let

\mathcal{C}

be a path connecting

P

and

Q

. Show that the time taken to descend by this route is given by

T[y] = (2g)^{-1/2}\int_{x_0}^{x_1} \left(\frac{1 + \dot{y}^2 + \left(G_{,x} + G_{,y}\dot{y}\right)^2}{G(x_0, y_0) - G(x, y)}\right)^{1/2} dx

where

y = y(x)

is the projection of

\mathcal{C}

on to the plane

z = 0

\dot{y}

means

dy/dx

, and

G_{,x}

G_{,y}

are the partial derivatives of

G

with respect to

x

y

. Obtain the E-L equation with computer assistance and solve it numerically with the initial conditions

y(x_1) = y_1

y'(x_1) = \lambda

and choose

\lambda

so that the path passes through

P

. [The numerical ODE integrator finds it easier to integrate the equation starting from the bottom. Why?] Plot the quickest route using 3D graphics. The author used the profile of the Cosine Valley resort, for which

G(x,y) = \cos^2(\pi x/2)\cos^2(\pi y/4)

. The skier had to descend from

P(1/3, 0, 3/4)

Q(2, 2, 0)

. You may make up your own mountain profile, but keep it simple.

Related theory

calculus-of-variations

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