鹿児島大学リポジトリ

A Generalization of the Classical

Brachistochrone Curve A Preliminary Study

著者

Isokawa Yukinao

journal or

publication title

Bulletin of the Faculty of Education,

Kagoshima University. Natural science

volume

61

page range

9-15

Professor of Kagoshima University, Faculty of Education

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Yukinao ISOKAWA October 27

，

2009 Abstract The classical Brachistochrone curve is one of fastest descent on a vertical plane. In this paper curves on fastest descent on surfaces that can be represenもedby elliptic integrals are studied. In the first part it is shown that such curves have to Iie on spheres or cones. In the second p訂tcurves on cones are studied in detai!and it is found that th田ecurves approach to cycloids酪 conesbecome cylinders gradually

1 Introduction

A Brachistochrone curve

，

or curve of fastest descent

，

is the curve between two points that is cover巴din the least time by a body that starts at the first point with zero speed and is constrained to move乱longthe curve to the second point

，

under the action of constant gravity and assuming no friction.

The problem of Brachistochrone curve is first posed by Johann Bernoulli

，

and is solved simultaneously by himself and several his contemporaries ([1)).Itis very old

，

but has conti -nously atrracted many people until now ([2]

，

[3]). In this paper we study brachistochrone curves on surfaces of rotation. Consider a surface of rotation defined by _{、 、 ‘ ‘} . -- -- E E ' ' ' ' ' m γ m y 活 n E ・ l pugu タ 命 ) ) zz ( ( T T J i t -t ¥

一 一

r Then we have

、

₁ I t

，

s -/ ψ'V 四 岬 n E ・ l ρ し g u ) ) : んグム却 ( ( 1 1 伊 ' 伊 ' J ' t S E E - -

_{、 、}

一 一

r 一 Z 向 。 ス ぴ μ

I

-r(z)sin<p¥

ニ =

I

r(z)cosψl δψ

¥

"

'

o

.

.

r

J

wh悦 rl

=

竿

Ac叫 ngly az

E

=

(

去

y

=

尺

F

=

説

=

o

，

G

=

(

2

)

2

=

r

f

We

∞

nsider a cur町 thatlies on the abo刊 surface

，

(ψ(u)

，

z(u))

，

where u is a paramβter. Then an infinitesimal length of the curve is given by

ds=V

E

_グ

2+

町 ど

+Gz

悩 =Jr2<pf2

+

(ri+ 似 伽 Z 一 u ， Gτd

一 一

z d n a ψ

一

u J u -d u

一 一

ψ ρ U 予 よ e h w

鹿児島大学教育学部研究紀要　自然科学編　第 61 巻　（2010）

10

Suppose that the desired curve starts at a五xedpoint ψ = 0

，

z = Zoand reaches a fixed point ψ=ψ1，Z Zl・Whena body passes through a plane of heightz

，

then its speed becomes

j

2

京

花

三

J

戸

て

コ

ヲ

苛

)

，

w山he肘I吋egd伽e叩no仰t臨侭t山hegravity a飢蹴氾.cc叫 lera tωo find a curve tぬha抗tmi凶r凶n凶山lI山imi包zeωs

f

恥 川 〆 ) 伽 where

ーゾ

r2cp'2

+

(rf十

-

i

)

:Z1:2 J(cp

，

z

，

cp'

，

z')ー

、

「一一

Zo-z SinceJ does not containψ 巴ヨcplicitly

，

the Euler equation of our variational problem reduces to

三

(

号

)

=

支

=0

Conseque泊tlywe can deduce

θJ 1 r2ψF f 一一一・ γ の = C1

，

θcp' ";io~z where C1 is a constant. Hence the di晶rentialequation follows dcp=

!

!

.

.

.

1

v

Z

o

て

ZV

r1(Z)2

τT

dz r(z)

ゾ

r(z)2-Cf(zo -z) Therefore its general solution can be formally expressed as ド C₁

r

.

:

v

o-z

y

'

1

)

(

Z

芹

T b + C

J r(z)

ゾ

r(z)2-q(zo -z) ) ' ・ 4 • 守_E A (

where C1

，

C2 are constan七swhich can be determined by zo

，

ψ1

，

Zl・

We seek a surface such that the integral that app伺rsin (1.1) can be reduced to an elliptic integral.We write

ベ

z)=

伊

(z). Then we have

f

p

叩

EτT

d

z

=

f

F

.

.

j

P'(Z)2十4P(z)む r(z)

、

/r(z)2-Cf(zo -z)

J

2P(z)

ゾ

P(z)-q(zo -z) Accordingly it is問 団ssarythatP(z)is a quardratic polynomial and

ゾ

P'(Z)2

+

4P(z)is a

square of a polynomial.Thus wモassumethatP(z)=αZ2

+

bz

+

c. Then 明 have

p'(Z)2

+

4P(z)= 4α(α+1)z2+4(α+ l)bz

+

(b2

+

4c). Since it is a square of a polynomial of degree one

，

iωdeterminant α(+1)(b2-4ac)mustvanish. Hene it follows thatα = -1 orb2_{-4町 =O. In the former case w} ちhaver(z)=ゾ_Z2十bz+c

，

or ~

I b

¥2

I

b¥

r(z)~

+

I

z -;;

I

=

I

I

;

;

¥ 2/ ¥2/

which is an equation of a sphere. 1吋 lelatter c蹴 wehaver(z)=

.

，

f

a

z

+

合

，

whichis an

equation of a cone. In this preliminary study we treat only a cone. Without loss of generality we may assume that a cone is specified by r(z)=α+ bz.

2 Brachistochrone on a

cone

Suppose that a surface of rotation is a cone r(z)=α+bz

，

where b is supposed to be positive. Then the equation(1.1) be

∞

mes dt.p

C1Nττ

ゾ石士吉

dz α+ bz)

ゾ

α(+bz)2 -Cf(zo -z) Hence

，

changing variable by x =

.

J

石

士

吉

，

weha刊 dt.p Cx2 dx (x2 λ2)

ゾ

(x2一入2)2ー (2μx)2' where we put C =

2

c

1

N

τ

I

、

/

α

+

bzo C1

=一寸

7

一

，

A =

V

-b--'μ = 2b' To solve the differential equation (2.1)

，

we need to evaluate an elliptic integral 1 =

r

_

_

X2dx

-J

(x2 -)..2)J f(x)

，

where f(x)= (x2一入2)2一(2μX)2. In the next section we will evaluate the integral1 and find the following result 1 =

去

[8

+

arc

叫

J

コ

"

2

tan8) -2(1

ー 約 九

n-1(sin8)] where a variable 8 is related to x as 、

J

T

て

k2_sin2 _{8 -11} x=

λ'YI

て古石弓 +

ν

(2.1) (2.2) with parameters11and k defined by (3.1) and (3.2) respectively. ThereforちW巴obtainthe following，th巴orem. Theorem The Brachistochrone on a cone is a curve defined by ψ =

宅+

1 [8

+

arc

叫

。

nd

一一(ト)

•

[匹需;;::2~r

ωhere k and

C

arecりηstantsdete門ninedby zo

，

Z，rψ1・

3 Evaluation of an e

l

l

i

p

t

i

c

i

n

t

e

g

r

a

l

In this section we evaluate an elliptic integral I = f ( Z 2 - Z h (2.4)

鹿児島大学教育学部研究紀要　自然科学編　第 61 巻　（2010） 12 where f(x)= (X2一入2)2一(2μX)2. To evaluate the integral 1， we change variable as x =

入.生色主こと

dn(u，k) +ν' where we define a positive parameter 1/and k by

ν2

y

'

"

;

t

π

天王一入

d

可予+入

and k2

=

_{1 _ 1/4.} Sinc巴(3.1)implies 2入ν μ 1て 1 /2' we have f(x) 入

4

{

(

お

)2

-

l

f

-

{

4

入2

市 能 力

2 Using (3.2)we see (1 -1/2)2dn2 u -(dn2 u _ 1/2)2 (2 -k2)dn2u -dn4u -(1-k2) (1 -dn2 u)(dn2 uー (1-k2)) k2sn2 u . k2cn2 u. Accordingly we g巴t

バ

、

2.. 1.2n~..~~.. y f(x)=二 ー ・ Lニニニニ V

J

¥

-

'

1-1/2 (dn+ν)2・ On the other hand， since

ご~d出n引t戸一k仇2snucn

αu we have

さ

さ

=-2入ν.

竺竺ど巳

du _..- (dnu

+

ν)2

Consequently，仕om(3

め

and(3.5)， it follows that

dx 1ーν2 Jf(x) 2入 仰 That is

，

by (3.3)

，

we obtain dx 1/ .

J

f(x)=

-

p

.

au Now

，

since 2 ν

一ω

+

ν q L u n 守 d l 一 釦 一 一 qa-U

一

E 一 一 日 n

一

_d 守 d 一 L -U 44-d 法 一 一 -内 4

一 、

A

d

一

一

-m 7 e

- z

(3.1) (3.2) (3.3) (3.4) (3.5) (3.6)

we have 1=

土

I

I

dnudu

+

ν2

I

ご

と -

2vul 4μ LJ J anu Then， by change of variable as snu = sinB， we have cnu dn u du = cosB. Hence du = cosBdB = cosBdB cosBd刀 u=

一一一一=一一一一一一一=一一一一一一

_{cnu v'lて石} =dB. 2 U

v1て五百

Accordingly we see Similarly we can see that

f

du

f

M

t

L

f

d O l dnu =./ dn

τ

2

= ) 1

ー仇れ=万ヲ

arctan(

v

1

コ

2

'

tanB) Therefore we obtain (2.2).

4 Asymptotic curve a

s

k

tends to zero

As b tends to zero with both αand C1 remaining unchag巴d

，

we have

Then and 入~ 2y'a‘/i.. μ~;;百、 U

2F(~f

-

~

H

ア

+

t

入 4

、

/ σ r k~

=

1-v告 白2一向一品=三、 b μ

、

ιI Hence we reasonably conjecture that the asymptotic Brachistochrone curve ask vanishes is the same as that on a cylinder. In this section we will con命m this conjecture. Let us start with the Taylor expansion of arctan function: (z -ZO)2 -2zo ぽctanz~ ar伽 nzo+(z-zo)

・一一+一一一・/，

， 寸 ぢ i十z5' 2 (1

+

Z5)2 asz -Zovanishes. Since we set 1庁てk2白 l-Ef-K4. 2 8' !k2 _k4¥ 0= tanB

，

z -Zo =-1一 +:';;-1 ¥ 2 8 J in the above Taylor expansion. Then we get arctan(

v

1

て

PtanB)

。_

向

k2

~山

OC056-K41momO(1+2dO)

・

(4.1)

2 ---- 8

鹿児島大学教育学部研究紀要　自然科学編　第 61 巻　（2010） 14 Next we approximate the first kind of elliptic integral as follows f ι

急

百

=f[1+jtsinhjfs

山

+

.

.

.

]

d8 ゆ+ Z f山 d8+

ぞ

J

ゆ 山 謝 + 9k4 ( . 2 ¥ 田 ゆ +τ(ゆ-sin仰 吋 )+ -~~い一山ゆ m ゆ -55in匂 c叫) Hence we can derive an asymptotic formula for the inverse of sn snー1Z =

んいぐ

l μ

州包

ω

仇d的2

勺

)

=

l

ar目 悶 …C由s

問

…

z+

子

(

紅

csinz-z

口 )

+

誓

(

紅

白

inz-z

口 一

;

Z

3

日 )

In this asymptotic formula， settingz = sin，()w巴have sn-1 _{(sin())田。+ぺ(()-sin()cos())} +k4

占

(

8

-

8

山 s()

ー

ト

山

問

。

)

Accordingly we get (1 -k2₎

_{む 内}

_in8)

~

(

)

-

吋

(

)-

_sin()c州 一寸k4

土

_{伊()+5( 山 8恥c}

_∞

_{Oωs8侃6s山in句(}3_)c

_∞

_O田s8

_め

_μ4.2 64 By (4.1) and (4.2) we can deduce ( ) + arctan(

ji

て

k2tan())ー2(1-k2)~ sn-1 _(sin()) 1.4 見工(()十sin8cω()(1-2 sin2 ())) 32 1.4 ι(48 + sin4()) 128 Therefore we obtain の ぴ 4 n g u

+

η " v A せ α

一 昨

ψ _(4.3) On the other hand

，

since

F

丙 万 一 同

5(1-2sin20)=

∞

ミ

s2() and

J

て玉石弓

+

ν

田 2

，

we have 入k2 Z同 一 一cos28同 一 一cos2(). 8 --- 2Cl

Therefore we obtain _{_2} -z田 ニ 寸(1

+

∞

s4B) 百七

T

By (4.3)and (4.4)we obtain the following result. (4.4) Corollary The Brachistochrone curve on a cone approaches to a cycloid on a cylinder.

References

[1JStruHは D. (1969)A Source Book in Mathematics

，

1200-1800 Harvard University Press.

[2JErlichson

，

H (1988)Galileo's work on swiftest descent企oma circle and how he almost proved the circle itself was the minimum time path it Amer.Math.Monthly 105 338・347.

[3]Erlichson

，

H (1999)Johann Bernoulli's brachistochrone solution using Fermat's prin -ciple of least timeEur.J.Phys.20 299-304.

鹿児島大学教育学部研究紀要　自然科学編　第 61 巻　（2010）