Question

1. The Brachistochrone problem. This is considered the oldest problem in the Calculus of Variations. It was proposed by Johann Bernoulli in 1696: Given a point A higher than a point B in a vertical plane, determine a (smooth) curve from A to B along which a mass can slide along the curve in minimum time (ignoring friction) with the only external force acting on the particle being gravity. The time to travel from a point A to B is given by the integral t = ?[A to B] 1/v ds, where s is the arc length and v is the speed. The speed at any point is given by a simple application of conservation of energy equating kinetic energy to gravitational potential energy 1/2 mv^2 = mgy, or v = ?(2gy). Since ds = ?(dx^2 + dy^2) = ?(1 + y'^2) dx, where y' = dy/dx. Therefore we have t = ?[A to B] ?(1 + y'^2) / v dx = ?[A to B] ?(1 + y'^2) / ?(2gy) dx. (i) We want to determine the curve leading to the minimal travel time from A to B. Formulate it in terms of a Calculus of Variation problem. (ii) Show that the Euler Equation takes the form y[1 + (dy/dx)^2] = 1/(2gC^2) = k^2. (1) You are not required to show the following, but the solution to (1) is x = 1/2 k^2(? - sin ?), y = 1/2 k^2(1 - cos ?), and this is part of the cycloid curve

          1. The Brachistochrone problem. This is considered
the oldest problem in the Calculus of Variations. It was proposed by Johann Bernoulli
in 1696: Given a point A higher than a point B in a vertical plane, determine a (smooth)
curve from A to B along which a mass can slide along the curve in minimum time (ignoring
friction) with the only external force acting on the particle being gravity.
The time to travel from a point A to B is given by the integral
t = ?[A to B] 1/v ds,
where s is the arc length and v is the speed. The speed at any point is given by a simple
application of conservation of energy equating kinetic energy to gravitational potential
energy
1/2 mv^2 = mgy,
or v = ?(2gy). Since
ds = ?(dx^2 + dy^2) = ?(1 + y'^2) dx,
where y' = dy/dx. Therefore we have
t = ?[A to B] ?(1 + y'^2) / v dx = ?[A to B] ?(1 + y'^2) / ?(2gy) dx.
(i) We want to determine the curve leading to the minimal travel time from A to B.
Formulate it in terms of a Calculus of Variation problem.
(ii) Show that the Euler Equation takes the form
y[1 + (dy/dx)^2] = 1/(2gC^2) = k^2. (1)
You are not required to show the following, but the solution to (1) is
x = 1/2 k^2(? - sin ?),
y = 1/2 k^2(1 - cos ?),
and this is part of the cycloid curve

1. The Brachistochrone problem. This is considered
the oldest problem in the Calculus of Variations. It was proposed by Johann Bernoulli
in 1696: Given a point A higher than a point B in a vertical plane, determine a (smooth)
curve from A to B along which a mass can slide along the curve in minimum time (ignoring
friction) with the only external force acting on the particle being gravity.
The time to travel from a point A to B is given by the integral
t = ?[A to B] 1/v ds,
where s is the arc length and v is the speed. The speed at any point is given by a simple
application of conservation of energy equating kinetic energy to gravitational potential
energy
1/2 mv^2 = mgy,
or v = ?(2gy). Since
ds = ?(dx^2 + dy^2) = ?(1 + y'^2) dx,
where y' = dy/dx. Therefore we have
t = ?[A to B] ?(1 + y'^2) / v dx = ?[A to B] ?(1 + y'^2) / ?(2gy) dx.
(i) We want to determine the curve leading to the minimal travel time from A to B.
Formulate it in terms of a Calculus of Variation problem.
(ii) Show that the Euler Equation takes the form
y[1 + (dy/dx)^2] = 1/(2gC^2) = k^2. (1)
You are not required to show the following, but the solution to (1) is
x = 1/2 k^2(? - sin ?),
y = 1/2 k^2(1 - cos ?),
and this is part of the cycloid curve

Added by Gema M.

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