Gen. Math. Notes, Vol. 3, No. 2, April 2011, pp.50-58
c
ISSN 2219-7184; Copyright °ICSRS
Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
Analytical Solution of Differential Equation
Associated with Simple Pendulum
M.I. Qureshi1 and Kaleem A. Quraishi2
1
Department of Applied Sciences and Humanities,
Faculty of Engineering and Technology,
Jamia Millia Islamia (A Central University), New Delhi-110025(India)
E-mail: miqureshi delhi@yahoo.co.in
2
Mathematics Section, Mewat Engineering College(Wakf),
Palla, Nuh, Mewat-122107, Haryana(India)
E-mail: kaleemspn@yahoo.co.in
(Received: 29-1-11/ Accepted: 23-3-11)
Abstract
In the present work, we provide the exact equation of motion of a simple
pendulum of arbitrary amplitude. For first time, a new and exact expression
is obtained for the time “t” of swinging of a simple pendulum from the vertical
position to an arbitrary angular position “θ”. The time period “T ” of such a
pendulum is also exactly expressible in terms of hypergeometric functions.
Keywords: Pochhammer’s symbol; Gauss ordinary hypergeometric function; Kampé de Fériet’s double hypergeometric function.
1
Introduction and Preliminaries
In our investigations we shall apply the following results.
The Pochhammer’s Symbol
The Pochhammer’s symbol or shifted factorial or generalized factorial function is defined by
½
Γ(h + r)
1
, if r = 0
=
(h)r =
h(h
+
1)(h
+
2)
·
·
·
(h
+
r
−
1),
if r = 1, 2, 3, · · ·
Γ(h)
51
Analytical Solution of Differential Equation...
where h 6= 0, −1, −2, −3, · · · and the notation Γ denotes the gamma function.
Reduction Formula
We know that
Z
− sin(n−1) (x) cos(x) n − 1
In = sinn (x) dx =
+
In−2 , n ≥ 2
n
n
(1)
I0 = x, I1 = − cos x
By the successive application of above reduction formula, we can find
Z
m−1
−( 12 )m sin(x) cos(x) X (1)r (sin(x))2r x ( 21 )m
+
+Arbitrary constant
sin (x) dx =
(1)m
(1)m
( 23 )r
r=0
(2)
for m = 0, 1, 2, 3, . . . .
2m
Series Identities
The empty sum
−1
X
F (r) is treated as zero.
(3)
r=0
∞ X
m
X
F (m, r) =
m=0 r=0
∞ m−1
X
X
F (m, r) =
m=0 r=0
=0+
∞ X
m
X
m=0 r=0
∞ X
∞
X
F (m + r, r)
(4)
m=0 r=0
−1
X
F (0, r) +
r=0
F (m + 1, r) =
∞ m−1
X
X
F (m, r) =
m=1 r=0
∞ X
∞
X
F (m + r + 1, r)
(5)
m=0 r=0
provided that involved multiple power series, are absolutely convergent.
Gauss Ordinary Hypergeometric Function
In 1812, C. F. Gauss defined the following function
·
2 F1
¸ X
∞
ab z a(a + 1) b(b + 1) z 2
(a)m (b)m (z)m
a, b ;
=1+
+
+
z =
c;
(c)m m!
c
c(c + 1) 2!
m=0
+
a(a + 1)(a + 2) b(b + 1)(b + 2) z 3
+ · · · ad inf.
c(c + 1)(c + 2) 3!
(6)
52
M.I. Qureshi et al.
It is always convergent for |z| < 1 and the denominator parameter c 6=
0, −1, −2, −3, . . .
Note:
·
¸
·
¸
·
¸
a, b ;
0, b ;
a, 0 ;
0 = 2 F1
z = 2 F1
z =1
(7)
2 F1
c;
c;
c;
Binomial theorem in hypergeometric notation, is given by
(1 − z)
−a
·
¸
∞
X
(a)r z r
a ;
=
= 1 F0
z ; |z| < 1
;
r!
r=0
(8)
Kampé de Fériet’s Double Hypergeometric Function
In 1921, P. Appell’s four double hypergeometric functions F1 , F2 , F3 , F4 and P.
Humbert’s seven confluent double hypergeometric functions Φ1 , Φ2 , Φ3 , Ψ1 , Ψ2 , Ξ1 , Ξ2
were unified and generalized by J. Kampé de Fériet.
We recall the definition of general double hypergeometric functions of Kampé
de Fériet in the slightly modified notation of Srivastava and Panda [8,pp.423424 (26,27); see also 9,p.23(1.2,1.3)]


(aA ) : (bB ); (dD ) ;
A:B;D 
FE:G;H
x , y
(eE ) : (gG ); (hH ) ;
∞ X
∞
X
[(aA )]m+n [(bB )]m [(dD )]n xm y n
=
[(eE )]m+n [(gG )]m [(hH )]n m! n!
m=0 n=0
(9)
∞
∞
X
[(aA )]m [(bB )]m xm X [(aA )]n [(dD )]n y n
=1+
+
+
[(eE )]m [(gG )]m m! n=1 [(eE )]n [(hH )]n n!
m=1
∞ X
∞
X
[(aA )]m+n [(bB )]m [(dD )]n xm y n
+
[(eE )]m+n [(gG )]m [(hH )]n m! n!
m=1 n=1
where (aA ) denotes the array of A parameters a1 , a2 , . . . , aA , [(aA )]m =
(10)
A
Q
(aj )m
j=1
with similar interpretation for others and denominator parameters are neither
zero nor negative integers. For convergence conditions of double series (9), we
have
(i) A + B < E + G + 1, A + D < E + H + 1, for |x| < ∞, |y| < ∞
or, (ii) A + B = E + G + 1, A + D = E + H + 1 and
½
1
1
|x| A−E + |y| A−E < 1 if A > E
max {|x|, |y|} < 1
if A ≤ E
Analytical Solution of Differential Equation...
2
53
Incomplete Elliptic Integral Related with Simple Pendulum
Figure 1 shows a simple pendulum, where its motion along an arc is considered.
The motion may be described in terms of the angle θ. We intend to find an
exact expression for θ as a function of time t; subject to a given set of initial
conditions.
Suppose θ0 is the maximum angular displacement of pendulum bob (towards
right or left) from the vertical position(PA).
Figure 1
Let t = 0 be identified with the vertical position(i.e., with θ = 0)
mL2 θ̇2
At any arbitrary time t, the energy consists of Kinetic Energy =
2
(11)
and Potential Energy = mg(L − Lcosθ)
(12)
By the law of conservation of energy, the total of the two terms is constant.
Total Energy = Kinetic Energy + Potential Energy = a constant
54
M.I. Qureshi et al.
Since the kinetic energy is zero, when the angular displacement is maximum
(i.e., θ0 ), we may write,
or
mL2 θ̇2
+ mgL(1 − cos θ) = mgL(1 − cos θ0 )
2
(13)
mL2 θ̇2
+ mgL(cos θ0 − cos θ) = 0
2
(14)
therefore,
µ
2
θ̇ =
dθ
dt
¶2
=
2g
(cos θ − cos θ0 )
L
i.e.
r
dθ
2g
=
(cos θ − cos θ0 )
(15)
dt
L
which is the well known differential equation of motion of a simple pendulum.
Let “ t ” be the time of swinging of simple pendulum from vertical position
to an arbitrary angular position θ. It has been assumed that at t = 0, the
angular displacement θ is zero. Then the above differential equation may be
integrated to yield
Ãr !
r Z t
Z y=θ
dy
2g
2g
√
dt =
=
t
(16)
L t=0
L
cos y − cos θ0
y=0
s Z
s Z
L y=θ
dy
L y=θ
dy
√
q
Hence t =
=
2g y=0
2g y=0
cos y − cos θ0
2 sin2 ( θ0 ) − 2 sin2 ( y )
s Z
1 L y=θ
dy
q
t=
2 g y=0
sin2 ( θ20 ) − sin2 ( y2 )
2
2
(17)
The integral involved in equation (17) is the incomplete elliptic integral of the
first kind.
Here we are interested in the exact solution of equation (17), that can be
obtained by using the hypergeometric approach as follows:
3
Hypergeometric Approach
Using the substitution sin( y2 ) = sin( θ20 ).sinz, we get
¡ ¢
2 sin θ20 cosz dz
dy = q
1 − sin2 ( θ20 )sin2 z
Analytical Solution of Differential Equation...
It may now be noted that: when y = 0 ⇒ z = 0
"
#
θ
sin
2
and when y = θ ⇒ z = sin−1
= z1 (say)
sin θ20
Therefore, (17) reduces to
s Z
¡ ¢
2 sin θ20 cosz dz
1 L z1
q
q
t=
2 g 0
1 − sin2 ( θ20 )sin2 z sin2 ( θ20 ) − sin2 ( θ20 )sin2 z
or
55
(18)
(19)
s Z
L z1
dz
q
t=
g 0
1 − sin2 ( θ20 )sin2 z
(20)
s Z
¸ −1
·
µ ¶
2
L z=z1
θ
0
t=
sin2 z
dz
1 − sin2
g z=0
2
(21)
In hypergeometric notation (8), the integrand of incomplete elliptic integral of
first kind involved in (20) or (21) can be written as
s Z
µ ¶
¸
· 1
θ0
L z=z1
;
2
2
2
sin
sin z dz
t=
1 F0
;
g z=0
2
Using power series form of (8), interchanging the order of summation and
integration, we get
s
£
¡ ¢¤m Z z=z
∞
1
L X ( 12 )m sin2 θ20
t=
sin2m z dz
(22)
g m=0
m!
z=0
Now using the integral (2) in (22), we get
"s ∞
#
L X ( 12 )m (sin θ20 )2m
t=
×
g m=0
m!
"(
×
) (
)#
m−1
−( 12 )m sin(z1 ) cos(z1 ) X (1)r (sin(z1 ))2r
z1 ( 12 )m
+
(1)m
(1)m
( 32 )r
r=0
s
s
∞
L X ( 12 )m ( 21 )m (sin θ20 )2m
L
−
sin(z1 ) cos(z1 )×
= z1
g m=0
m! (1)m
g
∞ m−1
X
X ( 1 )m ( 1 )m (1)r (sin2 ( θ0 ))m (sin(z1 ))2r
2
2
2
×
3
m!
(1)
(
)
m
2 r
m=0 r=0
(23)
(24)
56
M.I. Qureshi et al.
Further using double series identity (5) and hypergeometric notation (6), in
(24) we get
s
s
µ ¶¸
· 1 1
θ
L
L
,
;
0
2 2
t = z1
sin2
−
sin(z1 ) cos(z1 )×
2 F1
1
;
g
2
g
∞ X
∞
X
( 21 )m+r+1 ( 21 )m+r+1 (1)r (sin2 ( θ20 ))m+r+1 (sin(z1 ))2r
(25)
3
(1)
(1)
(
)
m+r+1
m+r+1
r
2
m=0 r=0
s
s
µ ¶¸
· 1 1
µ ¶
L
θ
1
L
θ0
,
;
0
2
2
2 2
= z1
sin
−
sin(z1 ) cos(z1 ) sin
×
2 F1
1
;
g
2
4 g
2
×
∞ X
∞
X
( 32 )m+r ( 32 )m+r (1)r (1)m (1)r (sin2 ( θ20 ))m (sin2 ( θ20 ))r (sin2 (z1 ))r
(2)m+r (2)m+r ( 32 )r m! r!
m=0 r=0
(26)
·
¸
θ
sin
2
Using z1 = sin−1
from (18) and writing double power series of (26) in
sin θ20
hypergeometric notation (9), after simplification we get
√
¸
· 3 3
¸
· 1 1
2 − α2
α
T (0)z1
T
(0)α
,
;
,
:
1
;
1,
1
;
0
2:1;2
2
2
2
2 2
t=
α −
F2:0;1 2 2
α , α
2 F1
1 ; 0
2, 2 :
; 32 ; 0
2π
8π
(27)
where

α = sin( 2θ )




α0 = sin( θ20q
)
(28)
T (0) = 2π Lg




and z1 = sin−1 ( αα0 )
×
where 0 ≤ θ ≤ θ0 ≤
π
2
and θ = 0 at t = 0.
The equation (27) is the exact solution of equation of motion (17)
of a simple pendulum yielding an implicit function θ(t). The above
result for arbitrary time t is always convergent [see convergence conditions of
Kampé de Fériet’s double hypergeometric function (9) and Gauss ordinary
hypergeometric function 2 F1 (6)].
The result (27) is not available in literature, though many approximate expressions for time period “T ” are available[1-7].
4
Some Deductions
(a) Exact Time Period for Arbitrary Amplitude
57
Analytical Solution of Differential Equation...
To find the time period, we use equation (27). In (27), when θ = θ0 then
z1 = sin−1 (1) = π2 and t = T4 (The bob has completed one fourth of an oscillation).
The exact time period T of simple pendulum of given length L, is therefore
given by
às !
· 1 1
· 1 1
µ ¶¸
µ ¶¸
L
θ
θ0
,
;
,
;
0
2
2
2 2
T (θ0 ) = 2π
sin
= T (0)2 F1 2 2
sin
2 F1
1
;
1
;
g
2
2
(29)
where “g” is the acceleration due to gravity and 2 F1 is the Gauss ordinary
hypergeometric function given by (6). Here θ0 is the maximum angular displacement of the pendulum from vertical position (and corresponds to t = T4 ).
Obviously 0 ≤ θ0 ≤ π2 .
(b) Simple Harmonic Approximation
When θ0 ≈ 0 then (29) reduces to
s
lim T (θ0 ) ≈ 2π
θ0 →0
L
≈ T (0)
g
(30)
which is the well known approximate formula for the time period of a simple
pendulum.
References
[1] B. Augusto, F. Jorge, O. Manuel, G. Sergi and B.J. Guillermo, Higher
accurate approximate solutions for the simple pendulum in terms of elementary functions, Eur. J. Phys., 31 (2010), L65-L70.
[2] T.H. Fay, The pendulum equation, Int. J. Math. Educ. Sci. Technol., 33
(2002), 505-519.
[3] W.P. Ganley, Simple pendulum approximation, Am. J. Phys., 53 (1985),
73-76.
[4] R.B. Kidd and S.I. Fogg, A simple formula for the large-angle pendulum
period, Phys. Teach., 40 (2002), 81-83.
[5] F.M.S. Lima, Simple ‘Log Formulae’ for pendulum motion valid for any
amplitude, Eur. J. Phys., 29 (2008), 1091-1098.
[6] M.I. Molina, Simple linearization of the simple pendulum for any amplitude, Phys. Teach., 35 (1997), 489-490.
58
M.I. Qureshi et al.
[7] R.R. Parwani, An approximate expression for the large-angle period of a
simple pendulum, Eur. J. Phys., 25 (2004), 37-39.
[8] H.M. Srivastava and R. Panda, An integral representation for the product
of two Jacobi polynomials, J. London Math. Soc., 12(2) (1976), 419-425.
[9] H.M. Srivastava and M.A. Pathan, Some bilateral generating functions
for the extended Jacobi polynomial-I., Comment. Math. Univ. St. Paul.,
28(1) (1979), 23-30.