Lagrangian Derivation of Brachistocrone?
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Brachistochrone Problem. Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time. The term derives from the Greek (brachistos) "the shortest" and (chronos) "time, delay."
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A little more precisely, suppose the shape of the track is given by the
graph of a function y(x), with y(0) = 0, y(L) = L (we orient the y axis down-
wards). For which function y(x) is the time T of descent minimized? Galileo
experimented with objects rolling down tracks of different shapes, and com-
ments made in a book published in 1638 suggest he may have believed the
optimal track would have the shape of a quarter-circle from (0,0) to (L,L),
which is not the right answer. The problem of the determining the brachis-
tochrone (shortest-time curve) was formally posed by Johann Bernouilli in
1696 as a challenge to the mathematicians of his day. Bernouilli himself
found the solution (using a physical argument partly suggested by Fermat’s
‘least-time’ derivation of Snell’s law of refraction in geometrical optics), and
Newton had a different proof that it was the correct one. This problem
is widely regarded as the founding problem of the ‘calculus of variations’
(finding the curve, or surface, minimizing a given integral), and the solution
described below is in the spirit of the approach developed by L.Euler (in
1736) and J-L. Lagrange (in 1755) to deal with general problems of this
kind.
The first thing to do is to express the time of descent as an integral
involving y(x). Let v =
ds
dt be the speed of the bead along the track, where
ds is arc length along the graph of y(x). From conservation of energy, at
height y below y = 0, we have:
1
2
v
2 = gy, or v =
p
2gy.
The time of descent T is therefore given by:
T =
Z
dt =
Z
ds
v
=
1
√
2g
Z p
dx2 + dy2
√y
HöPe ïT hèLps u ❤❤
,
graph of a function y(x), with y(0) = 0, y(L) = L (we orient the y axis down-
wards). For which function y(x) is the time T of descent minimized? Galileo
experimented with objects rolling down tracks of different shapes, and com-
ments made in a book published in 1638 suggest he may have believed the
optimal track would have the shape of a quarter-circle from (0,0) to (L,L),
which is not the right answer. The problem of the determining the brachis-
tochrone (shortest-time curve) was formally posed by Johann Bernouilli in
1696 as a challenge to the mathematicians of his day. Bernouilli himself
found the solution (using a physical argument partly suggested by Fermat’s
‘least-time’ derivation of Snell’s law of refraction in geometrical optics), and
Newton had a different proof that it was the correct one. This problem
is widely regarded as the founding problem of the ‘calculus of variations’
(finding the curve, or surface, minimizing a given integral), and the solution
described below is in the spirit of the approach developed by L.Euler (in
1736) and J-L. Lagrange (in 1755) to deal with general problems of this
kind.
The first thing to do is to express the time of descent as an integral
involving y(x). Let v =
ds
dt be the speed of the bead along the track, where
ds is arc length along the graph of y(x). From conservation of energy, at
height y below y = 0, we have:
1
2
v
2 = gy, or v =
p
2gy.
The time of descent T is therefore given by:
T =
Z
dt =
Z
ds
v
=
1
√
2g
Z p
dx2 + dy2
√y
HöPe ïT hèLps u ❤❤
,
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