Computer Algebra Recipes

196 CHAPTER 4. NONLINEAR ODE MODELS
museum surface should it be connected to, in order to minimize the time of
descent? What is the time of descent and what speed would I have acquired
on reaching the slanted museum roof? Fortunately, the museum roof has many
windows to let in light and I could arrange for the appropriate window to be
left open. A few relevant facts are as follows. The museum roof slants at 45 ±
to the horizontal. The street is 50 meters wide, and the vertical section of the
museum wall adjoining the slanted roof is 50 meters tall."
\You're crazy," Mike replies, \but you do pose an interesting mathematical
problem. I can tackle the solution using the Euler{Lagrange equation. Referring
to Figure 4.16, let's choose to measure x to the right and y downward from the
point at which the wire is attached to the stock exchange building. We need
to ¯nd a general expression for the time of descent along the wire. Let the
equation of the wire be y(x). Neglecting friction and equating the increase
in kinetic energy of a falling mass to its decrease in potential energy yields a
speed v = p 2 gy,wheregistheacceleration due to gravity. But v = ds=dt,
where ds = p 1+(dy=dx) 2 dx is an element of arc length along the wire. If the
(unknown) coordinates of the contact point on the museum roof are (x1 , y1 ),
the time of descent will be given by
Z p
x1
1+(dy=dx) 2
T = p dx:
0 2 gy
Since the factor p 2 g will cancel out, the integrand to use in the Euler{Lagrange
equation can be taken to be
F =( p 1+(dy=dx) 2 )= p y:
I will now use Maple to solve the problem. Loading the necessary library
packages, I enter the integrand F .
> restart: with(VariationalCalculus): with(plots):
> F:=sqrt((1+diff(y(x),x)^2)/y(x));
v
u
t
F :=
1+
μ
d
dx y(x)
2
y(x)
The EulerLagrange command is applied to F , and the ¯rst integral result, involving
the constant K1, is selected from the output (not shown) and simpli¯ed
in ode.
> EL:=EulerLagrange(F,x,y(x));
> ode:=simplify(select(has,EL,K[1])[]);
1
ode := v
u
t 1+
μ
d
dx y(x)
= K1
2
y(x)
y(x)
A general solution to the ¯rst-order nonlinear ode is sought.