another way of using bernoulli`s method

529
Revista Mexicana de Física 32 No. 3 (19&6) 529.536
ANOTHER WAY OF USING
BERNOULLI'S METHOD
Carlos
Universidade
Farina
Federal do Río de Janeiro
Instituto de Ftsica
Cidade Un¡versjt~ria
- Ilha do Fundao
Rio de Janeiro - CEP.: 21.944 - BRASil
(recibido enero 7, 1986; aceptado mayo 20, 1986)
ABSTRACl'
The Bernoulli's method, used in brachistochrone problems, is
presented in a different way. Instead of the Snell's law we use the ray
equation, whose integration aeems easier when sorne kind of symmetry i5
present.
We apply it in two examples: the classical brachistochrone
problem and in the cases with gravitational potential energies of the forro
V(r) = arn.
RESlJt.1EN
Se presenta de una manera diferente el método de Bernovilli, que
es usado para el problema de la braquistócrona.
En lugar de la ley de
Snell, usamos la ecuación de los rayos, cuya integración parece ser más
fácil cuando existe algún tipo de simetría.
Se aplica a dos ejemplos: el
problema clásico de la braquistócrona y para potenciales gravitacionales
del tipo v(r) = arn,
530
Reeently, P.K. Aravind(l) used Bernoulli's
brachistochrone
of a particle
moving inside a homogeneous
of matter of mass M and radius R.
mechanics
analogy.
the trajectory
PrincipIe).
Besides
= ~ ' with
v calculated
it a11ow5 a new way of attacking
from the energy conservatían
the problem,
theorem with the
the
since Snell t s law can be obtained from Fennat I s
equations.
In this note we apply this method in a difíerent
with the eikonal equation(3):
=
(Fennat1s
approach of solving the Euler-Lagrange
vía the Euler-Lagrange
('15)'
ball
aptíes light chooses
time of travel
In reality this is only a way of shortening
mathematical steps,
start
spherical
i5 based on an optics-
Instead, a11 ane has to use is the Snell's law and that
adequate potential.
princiñle
the minimal
it5 beauty
which avoid the conventional
n
This method
The Teason is that in geometrical
which requires
equations(2).
method to find the
way.
We will
n'
(1)
As is well known, even without the knowledge of the eikonal S it is possible
from Eq. (1) to obtain the ray equation(4):
(2)
where
r
is the position
oí an arbitrary point in the trajectory
and ds is the differential
is very difficult
Even though the ray equation
to solve in the general case, it permits
way of using the method.
direeUy
of the are length.
and integration
The symetries
oí light
a more systematic
of eaeh problem ean be implemented
of Eq. (2) often beeOl!l<"inmediate.
Let us see
in two examples.
We íirst solve the cla5sic brachistochrone
moving
in a constant
symnetry.
vertical
Thus, n
distance
the initial
gravitational
= ~
field.
is a ftmction of one variable
Erom a given horizontal
point of mtion.
problem oí a particle
111i5 case has of course aplane
only, say y, the
plane to the origin O, taken at
Wethen write Eq. (2) in the fonn
531
d
'"
as (n (Yh)
where
=
dn ':'
J
ay
(3)
... as
dr and J':'are nnit
t =
directíon,
respectively.
vectors along the ray directionand t1E vertical
MJltiplying both sides of (3) vertically
by
j
we
obtain
d
as
"
i!
(nj "T)
( 4)
which implies
n sine
where
e
=
(S)
ete
is the angle between
c
sine =
v
=
[~]'
and
t.
Substituting
n
=
£
we have
v
(6)
(C = cte)
From the conservation
v'
j
+
theorem for the energy oí the particle
[~]'
where x is the horizontal
=
we have that
(7)
2gy
coordinate of the particle.
From (6) and
(7)
we can write
sin'e
=
(8)
C' 2gy
Using the trigonometric identity
y(e)
R(l- cos2e).
with
2sen2a
=
1- cos2e
1
R =-4gC'
The express ion for x(e) can be obtained from
we get
(9)
53Z
dx
<lt
Zg Cy
=
(10)
which derives directly from (6) and (7) if we rnultiply both numerator and
denominator of the L.H.S. of (6) by v. Inserting (9) in (10) we easily see
that
dxdO
de <lt
=
[1]
\IT
(1 - cosZO)
(11)
where we assumed a time pararnetrization
conveniently
this parametrization.
far
e.
We are free to choose
Choosing ane so that
(IZ)
we see that
x(O)
=
[4~] (ZO "senZo)
is a solution oí (11).
(13)
All we have to do now is to calculate
A. This can be done if we force Eq. (7) to be valid.
(IZ) and (13) in (7) we obtain
_I_{A'sen'(ZO)
4C'g'
+
C'g'cos'(ZO)\
= _1_ (1 -
+
the constant
Inserting Eqs. (9),
_1_{1_ Zcos(ZO)}
4C'
(14)
cosZO)
ZC'
Frem (14) we see that A must be given by
A = gC
(15)
Substituting (15) in (13) we obtain
x(O) = R(ZO - senZO)
(16)
533
=
where we used that R
Equations
1 __
4gC'
(9) and (16) are the parametric
confinning a well known result
Now, we are going to treat more general
whose potentials
for a cicloid,
gravitational
fields,
are given by
(17)
= al"'
Ver)
where o is a constant,
zera intcger
number.
case fer (17), with a
determine
r the distance
P.K. Aravind(l)
and n
= ~
the brachistocfi~nes
=
from the centcr
oí force and n a non
in
solved a particular
Z.
his article
The differential
for these potentials
RameshÜlander(5) via the Eu1er-Lagrangc equations.
reobtain
equations
for this prob1em(3).
these differential
cquations
equations
were obtained
which
by
In this artic1e
with the Bernoulli's
method
we
as exposed
befare.
Since central potentials have spherical synanetry the "index oí
refraction"
in this case is a funetian
oí
T
alone and we can write Eq. (2)
in the fonn
d
A)
Os (n(r)T
where
=
dn A
Or r
(18)
? is the unit vector
in thc
r
direction.
Mu1tip1ying both sides of
(18) vectoria11y by ; wc obtain
(19)
From the last equation
we havc
(20)
n r sinB :: ctc
where
theorem
B is the angle betwcen
gives the relation
1- and
T.
In this case the energy
conservation
534
n(r) =
c
=
v~r
J
{2a
= c-
(21)
m
Substituting (21) in (20) we obtain
r sinB
(k
k
where q is the distance
=
cte)
(22)
from the point ~here the particle was initially at
rest and the center of force (see Fig. 1). To obtain the differentia1
equations
for the brachistochrones
we easily see that
rd~
sina "-
(23)
ds
and
[~]'
+
(24)
r' [~]'= 1
From (23) and (24) we find
(25)
lnserting (25) into (22) we finally have
,
~ = } {r' . k'(qn . rn)}'
(26)
which coincides with Eq. A.5 of R. Chander's(5) paper if we identify our k
with his constant of integratían
o.
In this article we used Bernoulli's
manner.
method
in a littIe different
Instead oí the Snellts law, we started with the eikonal equation
and ray equation.
It seems easier to use the sYJIIIletries of each problem
535
Q
I
I
\
I
\
I
I
\
p
Ir
\
dr /,?'~
\
\
I
\
ds
\
\
q\
7
\
\
\
\
\
\
\
\
Fig.
1.
Brachistochrone
between peints Q and P in a radial
field with canter oí force at o.
gravitational
536
in the integration of this last equation.
Thus, in applying the method to
other cases (wi th axial synrnetry for instance)
or in more complicated
brachistochrone problems this new point oí view may a150 be used.
Besides
its charm, the simplicity of the Bernoulli' s method makes it understandable
by any undergraduated student.
students used it more often.
Therefore, it would be interesting
if those
REFERENCIAS
1.
2.
3.
P.K. Aravind, Am. J. Phy~., 49(9) (1981).
K.R. Symon, Mee~,
Readin9, Mass, Adisson-Wesley (1972).
H. Goldstein, Cla4~~al
Meehani~.Readin9, Mas., Addison-Wesley (1950).
4. S. Borowitz, Fundamento~ de Mee~~a C~ea.
Editorial Reverté, S.A.
(1973).
5. Ramesh Chander, A.J .•Phy~•• 49(9) (1977).
/
I