The Double Pendulum
First Year Project
R. Ø. Nielsen (23051992), E. Have (08091993), B. T. Nielsen (15051991)
Supervisors:
Namiko Mitarai, Jörg Helge Müller
Pro ject 2013-14
The Niels Bohr Institute
March 21, 2013
Abstract
The main endeavour of this article is to establish whether the double pendulum exhibits chaotic motion and, if so,
to identify the chaotic regimes corresponding to initial angles θ1 (0) = θ2 (0) ∈ [0, 1.8] and initial angular velocities
θ̇1 (0) = θ̇2 (0) = 0. This is done using experimental as well as theoretical and computer simulation based techniques.
A survey of the needed theoretical results and considerations is included which leads to a derivation of the equations of
motion of the double pendulum using the techniques of analytical mechanics due mainly to Joseph Louis Lagrange(1) .
The numerical solution to this set of dierential equations forms the basis for computer simulations of the evolution of
the pendulum in phase space with the goal of calculating the Lyapunov exponent as a function of initial angle. This
allows us to directly identify the chaotic regimes.
Our ndings indicate that the pendulum behaves non-chaotically for initial angles of θ1 (0) = θ2 (0) ∈ [0, 0.69] while
initial angles of θ1 (0) = θ2 (0) ∈ [0.69, 1.55] lead to chaotic behaviour. There are some indications that a narrow region
of largely non-chaotic behaviour corresponding to θ1 (0) = θ2 (0) ∈ [1.55, 1.60] may exist. This would be an obvious
subject for further investigation in a later article.
Dedicated to Cornelius Lanczos (1893 − 1974)
For his exceptionally clear exposition of the variational principles of mechanics
Number of pages:
Main text: 15 pages
Appendices: 14 pages
(1)
1736-1813
Contents
1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
The Equations of Motion of the Double Pendulum . . . . . . . . . . . . . . . . . . . . .
3
3
The Concept of the Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
4
Chaos in Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
5
The Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
6
The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
6.1
Measuring the Moments of Inertia and the Centre of Gravity of our Double Pendulum
. .
10
6.2
Measuring Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
7.1
Initial conditions and predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
7.2
Comparison of experimentally determined trajectories with simulation
. . . . . . . . . . .
14
7.3
The Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
7
8
A The Lagrangian Formalism
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
B The Fundamental Lemma of the Calculus of Variations . . . . . . . . . . . . . . . . . .
19
C Equations of Motion for the Simple Double Pendulum: Where It All Started . . . .
20
D Derivation of the Equations of Motion for the Simple Double Pendulum from Newtonian Mechanics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
E On the Constancy of the Mechanical Energy . . . . . . . . . . . . . . . . . . . . . . . . .
25
F
26
MatLab Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
R. Ø. Nielsen, E. Have, B. T. Nielsen
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The Double Pendulum
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Introduction
The double pendulum is of considerable interest as a model system exhibiting deterministic chaotic behaviour.
The motion of the pendulum is governed by a set of coupled dierential equations and is
determined entirely by these. Nonetheless our ability to predict the behaviour of the double pendulum
is very limited in certain
regimes,
i.e.
for certain initial conditions.
This is due to the extreme sensi-
tivity towards even small perturbations. To qualitatively investigate this phenomenon we shall compare
experimental results obtained by video analysis to theoretically based computer simulations. To assess
the degree of chaoticity we shall determine the principal Lyapunov exponent of the system a quantity
which measures the rate of separation of nearby trajectories in phase space and thus serves as a useful
measure of chaotic behaviour.
Before we delve into the experimental results and the outcome of the simulations an exposition of the
necessary theoretical results is in order. Much of our work, including the following derivation, is based
on the principles of Lagrangian Mechanics and the Calculus of Variations. For an introduction to the key
concepts and mathematical results of Lagrangian Mechanics, please refer to Appendices A and B.
It is worth noting that much of the theoretical material which is collected in the appendices is essential
for the full appreciation of this article.
Figure 1: The double pendulum used in this project.
R. Ø. Nielsen, E. Have, B. T. Nielsen
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The Equations of Motion of the Double Pendulum
We wish to derive the equations of motion for a double pendulum which is composed of two physical
pendula of arbitrary shape and mass distribution.
We will do this using the methods of Lagrangian
(2) .
mechanics
Figure 2: Schematic representation of the compound double pendulum.
The positions of the centres of mass of the two pendula in Cartesian coordinates are given by:
where
d1
and
of suspension.
d2
x1 = d1 sin(θ1 ),
(2.1)
y1 = d1 cos(θ1 ),
(2.2)
x2 = h1 sin(θ1 ) + d2 sin(θ2 ),
(2.3)
y2 = h1 cos(θ1 ) + d2 cos(θ2 ),
(2.4)
are the distances to the centres of mass of the two pendula from their respective points
h1
is the distance between the two points of suspension.
We notice that the four Cartesian coordinates can be expressed as functions of
θ1
and
θ2
(as seen on
Figure 2) leading us to choose these two variables as the generalized coordinates of the system.
The
number of generalized coordinates is, in our case, equal to the number of degrees of freedom of our
(3) [1]
pendulum, namely 2
(2)
[2].
A more concise Lagrangian derivation of the equations of motion for the simpler, idealized double pendulum consisting
of point masses connected by massless rods can be found in Appendix C. The same equations can be derived from Newtonian
mechanics, albeit not as easily. This is discussed in Appendix D
(3)
Actually denitions dier; some authors dene the number of degrees of freedom as minimum number of generalized
coordinates required to describe the evolution of the system, while others dene it as the number of parameters needed to
describe the initial conguration of the system, which would be four in our case.[4]. The rst of these two denitions is
mainly used in mechanics while the second denition has found widespread use in system dynamics. We opted for the rst
one.
R. Ø. Nielsen, E. Have, B. T. Nielsen
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We wish to derive an expression for the total kinetic energy of the double pendulum. The upper pendulum
always rotates about a stationary axis and as a consequence it has only rotational kinetic energy:
1
2
T1 = I1 θ˙1 ,
2
where
I1
(2.5)
is the moment of inertia of the upper pendulum about an axis through the point of suspension
and perpendicular to the pendulum rod. The lower pendulum, on the other hand, rotates about an axis
through its point of attachment on the upper pendulum.
The calculation of the kinetic energy
T2
of
the lower pendulum is simplest if the motion is separated into a translation of the center of mass and a
rotation
around
the center of mass.
1
1
2
T2 = I2,cm θ˙2 + m2 (ẋ22 + ẏ22 ),
2
2
Where
(2.6)
I2,cm is the moment of inertia of the second pendulum about an axis through its center of mass, per-
pendicular to the pendulum rod. By dierentiation of the Cartesian coordinates the following expressions
for the velocity of the center of mass are obtained:
ẋ2 = h1 θ̇1 cos(θ1 ) + d2 θ̇2 cos(θ2 ),
(2.7)
ẏ2 = −h1 θ̇1 sin(θ1 ) − d2 θ̇2 sin(θ2 ).
(2.8)
We are in fact mainly interested in the squares of the velocities:
2 2
ẋ22 = h1 θ̇1 cos(θ1 ) + d2 θ̇2 cos(θ2 ) + 2h1 θ̇1 cos(θ1 )d2 θ̇2 cos(θ2 )
= h21 θ̇12 cos2 (θ1 ) + d22 θ̇22 cos2 (θ2 ) + 2h1 d2 θ̇1 θ̇2 cos(θ1 ) cos(θ2 ),
ẏ22 = h21 θ̇12 sin2 (θ1 ) + d22 θ̇22 sin2 (θ2 ) + 2h1 d2 θ̇1 θ̇2 sin(θ1 ) sin(θ2 ).
(2.9)
(2.10)
ẋ22 + ẏ22 = h21 θ̇12 + d22 θ̇22 + 2h1 d2 θ̇1 θ̇2 cos(θ1 ) cos(θ2 ) + 2h1 d2 θ̇1 θ̇2 sin(θ1 ) sin(θ2 )
= h21 θ̇12 + d22 θ̇22 + 2h1 d2 θ̇1 θ̇2 (cos(θ1 ) cos(θ2 ) + sin(θ1 ) sin(θ2 ))
= h21 θ̇12 + d22 θ̇22 + 2h1 d2 θ̇1 θ̇2 cos(θ1 − θ2 ),
where we have used the trigonometric relation
cos(x) cos(y) + sin(x) sin(y) = cos(x − y).
(2.11)
R. Ø. Nielsen, E. Have, B. T. Nielsen
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This leads to the following expression for the total kinetic energy:
1
2
T = I1 θ˙1 +
2
1 ˙2
= I 1 θ1 +
2
where
m2
1
2
I2,cm θ˙2 +
2
1
2
I2,cm θ˙2 +
2
1
m2 (ẋ22 + ẏ22 )
2
1
1
m2 h21 θ̇12 + m2 d22 θ̇22 + m2 h1 d2 θ̇1 θ̇2 cos(θ1 − θ2 ),
2
2
is the mass of the lower pendulum.
potential energy
V
(2.12)
We wish to obtain an expression for the gravitational
of the system as well. Trigonometric considerations lead to:
V = −m1 gy1,cm − m2 gy2,cm
= −m1 gd1 cos(θ1 ) − m2 gh1 cos(θ1 ) − m2 gd2 cos(θ2 ),
where
g
is the magnitude of the acceleration due to gravity and
m1
(2.13)
is the mass of the upper pendulum.
θ1 =
π
2 . Other zero points are just as valid since the potential energy is determined only up to an additive
The zero point for the potential energy has been chosen such that the potential energy is zero at
θ2 =
constant[1]. The Lagrangian for the system thus becomes:
L =T −V
1
1
1
1
2
2
= I1 θ˙1 + I2,cm θ˙2 + m2 h21 θ̇12 + m2 d22 θ̇22 + m2 h1 d2 θ̇1 θ̇2 cos(θ1 − θ2 )
2
2
2
2
+ m1 gd1 cos(θ1 ) + m2 gh1 cos(θ1 ) + m2 gd2 cos(θ2 ).
(2.14)
The Euler-Lagrange equations for the system then take the form
d ∂L
∂L
=
,
∂θ1
dt ∂ θ̇1
∂L
d ∂L
.
=
∂θ2
dt ∂ θ̇2
(2.15)
(2.16)
The desired derivatives are calculated below.
∂L
= −m2 h1 d2 θ̇1 θ̇2 sin(θ1 − θ2 ) − m1 gd1 sin(θ1 ) − m2 gh1 sin(θ1 ),
∂θ1
∂L
= I1 θ˙1 + m2 h21 θ̇1 + m2 h1 d2 θ̇2 cos(θ1 − θ2 ),
∂ θ̇1
d ∂L
= I1 θ¨1 + m2 h21 θ̈1 + m2 h1 d2 θ̈2 cos(θ1 − θ2 ) − m2 h1 d2 θ̇2 (θ̇1 − θ̇2 ) sin(θ1 − θ2 ).
dt ∂ θ̇1
(2.17)
(2.18)
(2.19)
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
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∂L
= m2 h1 d2 θ̇1 θ̇2 sin(θ1 − θ2 ) − m2 gd2 sin(θ2 ),
∂θ2
∂L
= I2,cm θ˙2 + m2 d22 θ̇2 + m2 h1 d2 θ̇1 cos(θ1 − θ2 ),
∂ θ̇2
d ∂L
= I2,cm θ¨2 + m2 d22 θ̈2 + m2 h1 d2 θ̈1 cos(θ1 − θ2 ) − m2 h1 d2 θ̇1 (θ̇1 − θ̇2 ) sin(θ1 − θ2 ).
dt ∂ θ̇2
(2.20)
(2.21)
(2.22)
This nally allows us to write the equations of motion for the double pendulum:
−m2 h1 d2 θ̇1 θ̇2 sin(θ1 − θ2 ) − m1 gd1 sin(θ1 ) − m2 gh1 sin(θ1 )
= I1 θ¨1 + m2 h21 θ̈1 + m2 h1 d2 θ̈2 cos(θ1 − θ2 ) − m2 h1 d2 θ̇2 (θ̇1 − θ̇2 ) sin(θ1 − θ2 ).
(2.23)
and
m2 h1 d2 θ̇1 θ̇2 sin(θ1 − θ2 ) − m2 gd2 sin(θ2 )
= I2,cm θ¨2 + m2 d22 θ̈2 + m2 h1 d2 θ̈1 cos(θ1 − θ2 ) − m2 h1 d2 θ̇1 (θ̇1 − θ̇2 ) sin(θ1 − θ2 ).
(2.24)
For a rst substantiation of these equations, please refer to Appendix E where the constancy of the energy
is veried.
3
The Concept of the Phase Space
It is often fruitful to study the evolution of a mechanical system in what is known as
phase space.
space is an abstract space characterized by having on the axes the generalized momenta
to the generalized coordinates
qi
pi
Phase
corresponding
as well as these coordinates themselves. The generalized momenta are
dened somewhat dierently from the momenta known from Newtonian mechanics:
pi =
∂L
.
∂ q̇i
(3.1)
For the double pendulum these take the form:
∂L
,
∂ θ̇1
∂L
.
p2 =
∂ θ̇2
p1 =
(3.2)
(3.3)
It is worth noting that the generalized momenta do not necessarily have units of mass times velocity but
(4) .
can take other units for instance units corresponding to angular momenta
(4)
In fact there are no general limitations on the dimensions of the generalized momenta[7].
Expressions for
p1
and
p2
R. Ø. Nielsen, E. Have, B. T. Nielsen
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have already been derived in equations (2.18) and (2.21).
One system which behaves particularly simply in phase space is the ordinary physical pendulum (in the
small oscillation limit
sin(φ) ≈ φ,
ellipses in phase space.
where
φ
is the angle to the vertical). This system is known to describe
That this is the case can be seen by considering the Lagrangian for such a
(5) :
system
1
L = I φ̇2 + mgd cos(φ),
2
∂L
= −mgd sin(φ) ≈ −mgdφ,
∂φ
∂L
= I φ̇.
∂ φ̇
Since it is known that
We see that the
φ(t) = A cos(ωt)
phase space trajectory
(3.4)
(3.5)
(3.6)
is a solution the the resulting E-L equation, we get:
φ(t) = A cos(ωt),
(3.7)
p(t) = −IAω sin(ωt).
(3.8)
of the system is an ellipse parametrised by the time
t
(one direct
way of recognizing this is that the equations above represent a circle which has been scaled by dierent
scalars along the two axes, so to speak). It is clear that the harmonic oscillator must evolve similarly in
phase space since the pendulum in the small angle limit is nothing but an angular oscillator.
4
Chaos in Mechanical Systems
The double pendulum is what is called a Hamiltonian system, which essentially means that the total
mechanical energy of the system is constant in time, or
conserved.
This is of course only approximately
true over short periods of time in the real world as there is some disspation due to air resistance and internal
friction in the bearings. Our system is also in principle deterministic, given exact initial conditions. Given
these facts one might ask how it can undergo chaotic motion, and in what sense it may be considered
chaotic.
These questions nessecitate a brief discussion of what is meant by
chaos.
One denition is that chaotic
systems are highly sensitive to initial conditions. In practice this means that if the initial conditions of the
double pendulum (initial angles and angular velocities) are altered only slightly, the consequent evolution
of the double pendulum will dier drastically.
This denition of chaos is somewhat qualitative, but it is very useful in practice and it encapsulates what
(5)
We see that the generalized momentum actually represents the angular momentum in this case.
R. Ø. Nielsen, E. Have, B. T. Nielsen
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is important about chaotic systems in physical applications; our limited ability to predict the evolution
of a given system over long periods of time.
5
The Lyapunov Exponent
It is to be expected that the double pendulum behaves chaotically only in certain
regimes,
meaning that
some sets of initial conditions lead to chaotic behaviour, while others do not. It is expected that small
initial angles (meaning that the pendula are displaced only slightly from the vertical) combined with
small initial velocities will lead to non-chaotic behaviour while larger initial angles and velocities lead to
chaotic behaviour. A system is said to evolve chaotically if an initial seperation between two phase space
trajectories grows exponentially with time. If the initial separation vector in phase space is
separation at a time
t
is
δ x(t)
δ x0
and the
we may write
|δ x(t)| = eλt |δ x0 |.
(5.1)
The system is said to behave chaotically if the Lyapunov exponent
λ
exponent less than or equal to zero denotes non-chaotic behaviour.
It is possible to dene separate
is positive, while a Lyapunov
Lyapunov exponents for dierent directions in phase space, and the largest of these exponents is said to
be the
principal
Lyapunov exponent. We calculate the principal Lyapunov in such a way that it does not
correspond to one single direction in phase space more on this later.
It is clear from 5.1 that this denition of the Lyapunov exponent relies on some concept of distance in
phase space.
There are a few seemingly problematic aspects in relation to this.
One is dimensional;
phase space is the space of generalized momenta and coordinates, and the Euclidean norm is bound to
have nonsensical dimensionality in this space, meaning that the dimensions of an expression of the form
p
p2 + q 2
are undened. The second issue is that we have no a priori reason to assume that phase space
is Euclidean.
These issues do not concern us, however, since we are only interested in the sign of the
Lyapunov exponent, which is not aected by the imposed metric[3].
We have determined the principal Lyapunov exponent for the system as a function of initial conditions
by means of a computer simulation. The relevant script,
lyap.m, is included in Appendix F.
The method we have employed (which is loosely based on the methods described in [3] and [8]) involves
simulating two trajectories in phase space separated by a very small initial separation
letting the two trajectories evolve for a short, xed, amount of time
trajectories is denoted
δX
i
∆t.
δ Xi0
of length
and
The resulting separation of the
. The procedure is illustrated in Figure 3. This provides us with an average
R. Ø. Nielsen, E. Have, B. T. Nielsen
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Figure 3: Schematic presentation of the method for determining the principal Lyapunov exponent.
value
λi
for the Lyapunov exponent in this short interval
λi = ln
|δ Xi |
|δ Xi0 |
.
(5.2)
The next Lyapunov exponent is calculated by the same method. We let the rst trajectory start where it
ended during the previous short simulation, but we let the new initial separation be given by:
=
δ Xi+1
0
δ Xi
.
|δ Xi |
(5.3)
The reason we choose to separate the two trajectories along this particular direction is that we wish to
measure the
principal
Lyapunov exponent, meaning the exponent associated with the direction of the
highest rate of separation. Once again we let
λi+1 = ln
|δ Xi+1 |
|δ Xi+1
0 |
.
(5.4)
This process is repeated many times until the total simulation time reaches
the i'th Lyapunov exponents
λi
T
(we chose 80s) after which
are averaged so as to give the principal Lyapunov exponent for the entire
trajectory:
λ=
N
1X
λi .
T
(5.5)
i=1
We see that this denition of the principal Lyapunov exponent leads to an exponent which does not
correspond to one single direction in phase space, since the separation vectors
δ Xi0
are not, in general,
parallel. They are chosen such that the initial separation is along the (approximate) direction in which
the separation between the two trajectories locally grows the fastest.
discussed in Section 7.3 on page 15.
The actual results obtained are
R. Ø. Nielsen, E. Have, B. T. Nielsen
6
The Double Pendulum
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The Experiment
To ascertain the validity of our theoretical methods there is need for experimental verication.
section details the experiments involving the motion of the double pendulum.
This
The setup to test our
equations of motion consisted of a physical double pendulum with some movable masses (See Figure 1 on
page 2) such that the center of mass can be adjusted. To analyse the behaviour of this physical double
(6) .
pendulum we record its motion with a high speed camera
By video analysis it is possible to track
the trajectory of one or more points on the pendulum, and these data can be analysed using Matlab.
The tracking software used was Tracker
(7) ,
which is capable of auto-tracking.
For this purpose, we
tted the pendula with brightly coloured stickers and illuminated the pendulum which allowed eective
auto-tracking.
6.1
Measuring the Moments of Inertia and the Centre of Gravity of our Double
Pendulum
In order to simulate the motion of our double pendulum, we need to know the moments of inertia and
(8)
the distances from the respective points of attachment to the centres of gravity
for the constituent
pendula. The centres of gravity were found by carefully balancing the pendula until equilibrium occurred.
The moments of inertia were obtained by taking the double pendulum apart and measuring the periods
of small oscillations of the pendula
(9) ,
which, upon insertion into (6.2), yields the sought moments of
inertia. The torque acting on a single pendulum due to gravity is given by
distance from the point of attachment to the centre of gravity,
acceleration due to gravity, and
angles,
ϕ ≈ 0,
ϕ
m
τ = −dmg sin ϕ, where d is the
is the mass of the pendulum,
g
is the
is the angle of the pendulum relative to vertical. However, for small
a series expansion of sine yields the well-known result that sine of small angles may be
approximated as the angle itself, i.e,
sin ϕ ≈ ϕ(10) .
For the torque, we may then write in the small angle
approximation:
τ = I ϕ̈ ≈ −dmgϕ ⇒ ϕ̈ = −
(6)
dmg
ϕ,
I
(6.1)
Using the moderate setting of 300fps
http://www.cabrillo.edu/~dbrown/tracker/
(8)
Which coincide with the centres of mass since the acceleration due to gravity can be treated as constant in this experiment.
(9)
suspended, of course, from the same point as in the double pendulum
(10)
This requires, of course, that the angles are measured in radians. This is assumed throughout the article.
(7)
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
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and thus, by using the denition of torque, and comparing this to the familiar dierential equation
governing periodic phenomena of this kind,
ẍ = −ω 2 x,
mgd
= ω2 =
I
2π
T
we nd:
2
⇒I=
Plugging into this formula, we get the moments of inertia
is included rather than
I2
I1
mgdT 2
.
4π 2
and
(6.2)
I2 . I2
can be found in Table 1.
I2,cm
since what we actually need for the lower pendulum is the moment of inertia
around the center of mass. The reason for this is discussed in Section 2. This necessitates the use of the
Huygens-Steiner Theorem (11) :
I2 = I2,cm + m2 d22 ⇔ I2,cm = I2 − m2 d22 .
The masses of the pendula were measured as well, as were the distances
(6.3)
d1
and
d2
involved in the
equations of motion for the compound double pendulum, (2.23) and (2.24). We may calculate the error
Table 1: Experimental Results
I1 = 0.0862kg · m2 ± 0.0018kg · m2
d1 = 0.148m ± 0.005m
m1 = 0.955kg ± 0.002kg
T1 = 1.316s ± 0.0102s
(12) , σ
I
on the moments of inertia
as follows:
s
σI =
r
=
I2,cm = 0.0177kg · m2 ± 0.0014kg · m2
d2 = 0.0800m ± 0.005m
m2 = 0.630kg ± 0.002kg
T2 = 1.566s ± 0.016s
∂I
∂m
2
2
σm
+
∂I
∂T
2
σT2
+
∂I
∂d
2
σd2
m2 g 2 T 2 2
g 2 d2 T 4 2
m2 g 2 d2 T 2 2
σm +
σT +
σ ,
4
4
16π
4π
16π 4 d
(6.4)
which, upon insertion of the experimentally established values, yields the results shown in Table 1. The
standard deviations of the periods, of which many measurements were made, were obtained using the
familiar expression:
v
u
N
u1 X
σT = t
(Ti − µT )2 ,
N
i=1
where
(11)
(12)
N
is the number of measurements made and
µT
is the average period.
Also known as the Parallel Axis Theorem, see [6]
T and m are, of course, independent variables, ensuring the validity of the error propagation law.
(6.5)
R. Ø. Nielsen, E. Have, B. T. Nielsen
6.2
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Measuring Chaos
Since chaos is dened as being very sensitive to initial conditions, our method to experimentally determine
the degree of chaoticity of the system relies on measuring the initial conditions and the positions and
velocities after a certain time. In all cases we chose the initial conditions such that
θ̇1 (0) = θ̇2 (0) = 0.
θ1 (0) = θ2 (0)
and
We have, in other words, only studied a limited subset of the set of practically feasible
initial conditions.
We tracked the motion of the pendulum for
10s
for each set of initial conditions. This time was chosen
such that dissipation would be negligible and such that any chaotic tendencies would become apparent.
In total 107 measurements at angles
θ1 (0) = θ2 (0) ∈ [0; 2]
were made. We have plotted the trajectory of
the pendula for one of these measurements (obtained by video analysis) in Figure 4.
Figure 4: Plot of trajectories of pendula in Cartesian coordinates constructed from experimental data.
7
7.1
Results
Initial conditions and predictability
On Figure 5 on the next page we see the experimental plot of initial angle
for the case
θ1 (0) = θ2 (0)
and
θ̇1 (0) = θ̇2 (0) = 0.
We see that for
in initial angle lead to small dierences in nal angle.
θ1 (0)
vs. nal angle
0 ≤ θ1 (0) / 0.7,
θ1 (10s)
small dierences
This indicates a non-chaotic regime, or, in any
case, a regime of such low chaoticity that there is a high degree of predictability for at least the rst
R. Ø. Nielsen, E. Have, B. T. Nielsen
10s.
For
θ1 (0) ∈ [0.7, 1.8]
The Double Pendulum
(13) .
the pendulum exhibits chaotic motion
13/31
If we compare this plot with the
plot (see Figure 6) obtained by a MatLab simulation based on our equations of motion (Appendix F,
initcondsimulation.m)
we see the the plots share the same features, and that the change from the non-
chaotic regime to the chaotic regime occurs where we theoretically predicted it.
Figure 5: A plot of initial θ1 = θ2 vs. nal θ1 . Other initial conditions: θ̇1 = θ̇2 = 0
2
1.5
final value of theta1 [rad]
1
X: 0.6966
Y: 0.3823
0.5
0
−0.5
−1
−1.5
0
0.2
0.4
0.6
0.8
1
Initial value of theta1 [rad]
1.2
1.4
1.6
Figure 6: Simulation of initial θ1 = θ2 vs. nal θ1 . Other initial conditions: θ̇1 = θ̇2 = 0
(13)
Except, possibly, for a narrow interval discussed in section 7.3
1.8
R. Ø. Nielsen, E. Have, B. T. Nielsen
7.2
The Double Pendulum
14/31
Comparison of experimentally determined trajectories with simulation
Our ability to predict the motion of the double pendulum in the non-chaotic regime is most easily demonstrated by a comparative plot of a simulated trajectory and one determined by video analysis. Such a
plot may be found in Figure 7 the initial conditions are
θ1 (0) = θ2 (0) = 0.35
with the pendula starting
from rest.
In the chaotic regime our ability to predict the motion of the pendulum for more than a few seconds
breaks down. This can be clearly seen in Figure 8 on the following page.
Since some dissipation is to be expected, we have limited our analysis to 10 seconds, since we deemed the
eects of dissipation as negligible over such a short amount of time.
X: 0
0.4 Y: 0.3477
Experimental
Simulated
0.3
Angle theta1/rad
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0
2
4
6
Time/s
8
10
12
Figure 7: Comparison of simulated trajectory with trajectory obtained by video analysis. Non-chaotic regime.
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
15/31
X: 0
1.5 Y: 1.351
Experimental
Simulated
Angle theta1/rad
1
0.5
0
−0.5
−1
−1.5
0
2
4
6
Time/s
8
10
12
Figure 8: Comparison of simulated trajectory with trajectory obtained by video analysis. Chaotic regime.
7.3
The Lyapunov Exponent
In Figure 9 on the next page we present a plot of the Lyapunov exponent computed by the method
described in Section 5.
We see that the Lyapunov exponent conrms our earlier nding that chaotic
behaviour becomes apparent at
θ1 ≈ 0.7
(again for the case
note the apparent existence of a narrow interval around
only slightly positive,
θ1 (0) = θ2 (0), θ̇1 (0) = θ̇2 (0) = 0).
θ1 (0) = 1.6,
We also
where the Lyapunov exponent is
(14) .
indicating a region of more predictable behaviour
Unfortunately we do not have
sucient amounts of data close to this initial angle, but it is denitely an interesting observation which
warrants further investigation at some point.
It is, however, interesting to note that this non-chaotic
region shows up on Figure 6 as well and that we see a lump of points in the experimentally based
Figure 5 in the same general area. Lumpiness, of course, indicates predictability as well, since it means
that initial conditions which dier only slightly produce similar nal conditions. However, as noted, more
measurements will be necessary before we can draw any conclusions with condence.
We expect that, given a longer run-time and higher precision in the numerical integration of the dierential
equations, the Lyapunov exponents for small angles would approach zero even more closely.
(14)
Note that this narrow interval of non-chaoticity is also present in Figure 6 on page 13
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
16/31
1.4
1.2
Lyapunov exponent [1/s]
1
0.8
0.6
0.4
X: 0.693
Y: 0.2028
0.2
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Initial angle (theta2=theta1) [rad]
1.4
1.6
1.8
2
Figure 9: The Lyapunov exponent as a function of initial angle (θ1 (0) = θ2 (0), θ̇1 (0) = θ̇2 (0) = 0)
8
Conclusion
We have utilized methods for characterizing chaotic motion, both experimentally and theoretically and
veried the agreement between these.
chaotic motion for
Our ndings suggest that our double pendulum exhibits non-
θ1 (0) = θ2 (0) ∈ [0, 0.69],
while initial angles of
θ1 (0) = θ2 (0) ∈ [0.69, 1.55]
do indeed
lead to chaotic behaviour, although, seemingly, a region of non-chaoticity occurs for initial angles near
1.6,
corroborated by our simulations and by the experiment, to some degree, though our experimental
data in this region is scarce.
We have, to a satisfactory degree of accuracy, been able to reproduce
the results obtained experimentally by simulating the equations of motion derived using the methods
of analytical mechanics, thus arming the validity of our equations. Indeed, the resemblance between
Figure 5 and Figure 6 on page 13 is striking.
R. Ø. Nielsen, E. Have, B. T. Nielsen
A
The Double Pendulum
17/31
The Lagrangian Formalism
The fundamental quantity in the Lagrangian formulation of classical mechanics is the Lagrangian
which may in general depend on the
n generalized coordinates
qi (15) , their derivatives q̇i
L (q̇i , qi , t),
and time t, dened
as the dierence between the kinetic and potential energies of the system:
L (q̇i , qi , t) = T − V.
The governing principle is known as
the
action S
Hamilton's Principle
or
(A.1)
The Principle of Least Action (16) .
We dene
as follows:
Z
tb
S=
L (q̇i , qi , t)dt.
(A.2)
ta
The requirement that the action be stationary is thus:
where
δS
is the
variation
δS = 0,
(A.3)
of the action, to be dened.
The action is a so-called functional, that is, it
depends not only on the value of
on in the range
t ∈ [ta , tb ].
L
at a particular point, but rather on its innitely many values taken
Characteristic of the actual path
q̄i
is then that the action remain unchanged
(to a rst order approximation) with respect to slight variations of the path
xed, that is,
δqi (ta ) = δqi (tb ) = 0.
δqi
with the endpoints held
This is exemplied in Figure 10:
Figure 10: A possible variation of the i'th generalized coordinate.
(15)
These are merely the coordinates with which we describe our system; for example spatial position coordinates such as
(x1 , x2 , . . . , y1 , . . . , z1 , . . . ), or angles, (θ1 , θ2 , . . . ). The coordinates are themselves understood to be functions of time.
(16)
For historical reasons the principle is known as The Principle of Least Action. However there is actually no requirement
that the action must be minimized - all that is required is that the action be stationary.
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
18/31
Analogous to the dierential of a regular function, one may obtain the variation of a functional (for xed
i)
as follows:
δS = S[qi + δqi ] − S[qi ],
(A.4)
substituting for the action the denition (A.2) yields:
Z
tb
L (q̇i + δ q̇i , qi + δqi , t)dt
∂L
∂L
L (q̇i , qi , t) +
=
δ q̇i +
δqi dt
∂ q̇i
∂qi
ta
Z tb ∂L
∂L
= S [qi ] +
δ q̇i +
δqi dt .
∂ q̇i
∂qi
ta
|
{z
}
S[qi + δqi ] =
ta
Z tb
(A.5)
=δS
Now, we need to evaluate the integral. Consider then the rst term, which may readily be integrated by
parts
(17) :
Z
tb
ta
Z tb
Z tb
∂L
d ∂L
d ∂L
∂L tb
δ q̇i dt = δqi
−
δqi dt = −
δqi dt,
∂ q̇i
∂ q̇i ta
ta dt ∂ q̇i
ta dt ∂ q̇i
due to the restrictions imposed upon the endpoints (δqi (ta )
expression for
δS
= δqi (tb ) = 0).
(A.6)
Plugging this result into our
found in (A.5), we nd:
Z
tb
δS =
δqi
ta
∂L
d ∂L
−
∂qi
dt ∂ q̇i
dt.
(A.7)
Equating this to zero, we see that this implies the following equality (in fact due to the fundamental
lemma of the calculus of variations see Appendix B):
∂L
d ∂L
−
= 0,
∂qi
dt ∂ q̇i
which is known as the
(A.8)
Euler-Lagrange equation.
We note that for the action to be truly stationary, it must be stationary with respect to each of the
variables on which it depends.
coordinates
qi
It follows that the Euler-Lagrange equation must hold for any of the
one might choose to consider.
One may also derive the Euler-Lagrange equation from what is known as
(17)
D'Alembert's Principle [5], which
This is actually hinged on the fact that variation and dierentiation are permutable processes, that is that the derivative
of the variation is equal to the variation of the derivative as is proved in [5]
R. Ø. Nielsen, E. Have, B. T. Nielsen
states that
P
F
− ma =
0.
−ma
The Double Pendulum
19/31
is called the inertial force, and is included such that the sum of the
impressed forces is zero. This makes it possible to use the well-known methods of statics (particularly
dV = 0).
We see that this indicates that the E-L equation follows from Newton's laws in inertial systems.
It is very interesting to note that our derivation of the Euler-Lagrange equation is an example of a
way of looking at mechanical problems which is quite dierent from what one usually sees in courses
on Newtonian mechanics.
Usually, in vectorial mechanics, one is given a set of initial conditions in
the form of the coordinates and velocities at one specic time,
t0 ,
from which one can extrapolate the
subsequent motion given knowledge of the forces involved, by the use of Newton's laws. It is, in other
words, understood that the motion must obey Newton's second law at every point along the trajectory.
In this sense, Newton's laws can be said to constitute a local principle.
Our derivation takes a dierent approach in that we are given two sets of initial conditions; a set of
position coordinates
qi (ta )
at the time
ta
and another set of position coordinates at time
tb .
We then
formulate a principle the principle of least action which takes the form of a requirement on an integral
which is dependent on the shape of the entire trajectory. In other words, we know only where we are at
two distinct times, and how long it took us to move from one point to the other. It is reassuring to see
that this somewhat holistic viewpoint gives rise to a local principle after all, namely the Euler-Lagrange
equation
B
(18)
which must hold for every point along the trajectory.
The Fundamental Lemma of the Calculus of Variations
Also known as Lagrange's Lemma, this theorem states that:
(The Fundamental Lemma of the Calculus of Variations): Let f : [a, b] → R be continuous
R
on the interval [a, b] ⊂ R and assume that ∀η(x) ∈ C 0 [a, b] | η(a) = η(b) = 0 : ab f (x)η(x)dx = 0. Then
f (x) = 0 for all x ∈ [a, b].
Lemma B.1
Assume that the lemma does not hold, i.e., for some x0 ∈ [a, b] : f (x0 ) 6= 0. Due to the continuity
of η(x) on the interval [a, b], there must be some interval I ⊂ [a, b] for which f (x) is either positive or
negative, depending on its original value in x0 , for x ∈ I . Consider then the specic function η(x) =
f (x)(x − a)(b − x), which is positive for x ∈ (a, b) and zero for x = a ∧ x = b, hence yielding:
Proof
Zb
f (x)2 (x − a)(b − x)dx > 0,
(B.1)
a
thus contradicting the assumption that the integral were to vanish for all functions η . We have strictly
speaking only established that ∀x ∈ (a, b) : f (x) = 0, but as f ∈ C 0 [a, b], it must also vanish at the
(18)
Or, rather, one E-L equation for each generalized coordinate qi .
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
20/31
endpoints, and thence we arrive at the desired result: f (x) = 0, ∀x ∈ [a, b]
An alternative, more informal proof which utilizes the Dirac
δ
function may be given as well:
Assume that the lemma does not hold, i.e., for some ξ ∈ [a, b] : f (ξ) 6= 0. Choosing η(x) = δ(x − ξ)
where δ is the Dirac delta function, we thus have
Proof
Zb
Zb
f (x)δ(x − ξ)dx = f (ξ) 6= 0
f (x)η(x)dx =
a
(B.2)
a
contradicting the assumption that the integral were to vanish for all functions η . We may thus conclude
that the desired result holds, namely that ∀x ∈ [a, b] : f (x) = 0.
C
Equations of Motion for the Simple Double Pendulum: Where It All
Started
Consider the simple double pendulum depicted in the gure below:
Figure 11: Schematics of the Simple Double Pendulum
Taking the ordinate axis to be positive when pointing upwards, we may immediately write the following
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
relations between the Cartesian coordinates
(x, y)
and our generalised coordinates,
21/31
θ1
and
θ2 :
x1 = l1 sin θ1 ,
(C.1)
y1 = l2 + l1 (1 − cos θ1 ),
x2 = l1 sin θ1 + l2 sin θ2 ,
y2 = l2 − l2 cos θ2 + l2 − l1 cos θ1
The total kinetic energy,
T , is, of course, merely the sum of the respective kinetic energies of the constituent
pendula:
1
1
1
T = m1 ẋ21 + ẏ12 + m2 ẋ22 + ẏ22 = m1 l12 cos2 θ1 θ̇12 + l12 sin2 θ1 θ̇12
2 2
2
2 2 1
+ m2
l1 cos θ1 θ̇1 + l2 cos θ2 θ̇2 + l1 sin θ1 θ̇1 + l2 sin θ2 θ̇2
2
1
1
= m1 l12 θ̇12 + m2 (l12 cos2 θ1 θ̇12 + l22 cos2 θ2 θ̇22 + 2l1 l2 θ̇1 θ̇2 cos θ1 cos θ2
2
2
+ l22 sin2 θ2 θ̇22 + l12 sin2 θ1 θ̇22 + 2l1 l2 θ̇1 θ̇2 sin θ1 sin θ2 )
1
1
= m1 l12 θ̇12 + m2 l12 θ̇12 + l22 θ̇22 + 2l1 l2 θ̇1 θ̇2 (cos θ1 cos θ2 + sin θ1 sin θ2 ) .
2
2
Now, the following relation proves useful,
∀θ1 , θ2 ∈ R : cos θ1 cos θ2 + sin θ1 sin θ2 = cos(θ1 − θ2 ),
(C.2)
in that
it allows us to rearrange (C.2) as follows:
1
1
T = m1 l12 θ̇12 + m2 l12 θ̇12 + l22 θ̇22 + 2l1 l2 θ̇1 θ̇2 cos(θ1 − θ2 ) .
2
2
The potential energy
V
of the double pendulum is given by:
V = m1 gy1 + m2 gy2 = m1 g (l2 + l1 − l1 cos θ1 ) + m2 g (l2 − l2 cos θ2 + l1 − l1 cos θ1 ) ,
thus yielding, for the Lagrangian
L,
the following expression:
L =T −V
1
1
= l12 θ̇12 (m1 + m2 ) + m2 l22 θ̇22 + m2 l1 l2 θ̇1 θ̇2 cos(θ1 − θ2 )
2
2
− m1 g (l2 + l1 + l1 cos θ1 ) − m2 g (l2 − l2 cos θ2 + l1 − l1 cos θ1 ) .
(C.3)
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
22/31
The rst equation of motion is obtained by simply plugging into the Euler-Lagrange equation:
∂L
d ∂L
= 0,
−
dt ∂ θ̇1
∂θ1
(C.4)
and so we need only compute the derivatives:
i
d ∂L
d h2 2
=
l1 θ̇1 (m1 + m2 ) + m1 l1 l2 θ̇2 cos(θ1 − θ2 )
dt ∂ θ̇1
dt
= l12 θ̈1 (m1 + m2 ) + m2 l1 l2 θ̈2 cos (θ1 − θ2 ) − m2 l1 l2 θ̇2 sin (θ1 − θ2 ) θ̇1 − θ̇2 ,
∂L
= −m2 l1 l2 θ˙1 θ˙2 sin (θ1 − θ2 ) − m1 gl1 sin θ1 − m2 gl1 sin θ1
∂θ1
= gl1 sin θ1 (m1 + m2 ) − m2 l1 l2 θ˙1 θ˙2 sin (θ1 − θ2 ) .
(C.5)
(C.6)
Upon combining (C.5) and (C.6) as dictated by (C.4), we obtain the rst equation of motion:
l12 θ̈1 + gl1 sin θ1 (m1 + m2 ) + m2 l1 l2 θ̈2 cos (θ1 − θ2 ) + θ̇22 sin (θ1 − θ2 ) = 0.
(C.7)
Similarly, the second equation may be found as follows:
i
d ∂L
d h
=
m2 l22 θ̇2 + m2 l1 l2 θ̇1 cos (θ1 − θ2 )
dt ∂ θ̇2
dt
h
i
= m2 l22 θ̈2 + m2 l1 l2 θ̈1 cos (θ1 − θ2 ) − θ̇1 sin (θ1 − θ2 ) θ̇1 − θ̇2 ,
∂L
= m2 l1 l2 θ̇1 θ̇2 sin (θ1 − θ2 ) − m2 gl2 sin θ2 ,
∂θ2
∴
h
i
m2 l22 θ̈2 + gl2 sin θ2 + m2 l1 l2 θ̈1 cos (θ1 − θ2 ) − θ̇12 sin (θ1 − θ2 ) = 0.
And so we have derived the equations of motion for the simple double pendulum.
(C.8)
R. Ø. Nielsen, E. Have, B. T. Nielsen
D
The Double Pendulum
23/31
Derivation of the Equations of Motion for the Simple Double Pendulum from Newtonian Mechanics
For the simple double pendulum, we have previously obtained the following expressions:
x1 = l1 sin θ1 ,
(D.1)
y1 = l2 + l1 (1 − cos θ1 ),
x2 = l1 sin θ1 + l2 sin θ2 ,
y2 = l2 (1 − cos θ2 ) + l1 (1 − cos θ1 )
To derive the equations of motion using the Newtonian framework, we need to know the forces acting on
both pendula. Figure 12 shows these forces.
Figure 12: Forces on the Simple Double Pendulum
From the diagram one immediately obtains the equations
(19)
using Newton's Second Law,
P
F
= ma:
X
F1,x = m1 ẍ1 = −T1 sin θ1 + T2 sin θ2 ,
(D.2)
X
F1,y = m1 ÿ1 = T1 cos θ1 − T2 cos θ2 − m1 g,
(D.3)
X
F2,x = m2 ẍ2 = −T2 sin θ2 ,
(D.4)
X
F2,y = m2 ÿ2 = T2 cos θ2 − m2 g.
(D.5)
We then make the observation that forces acting on the bob pertaining to the second pendulum, i.e.,
m2 ,
T2
does not depend on the tension
multiplied by cosine and sine of
(19)
T1 ,
θ2 ,
while the forces on the rst bob
does
depend on the tension
respectively. From (D.4) and (D.5) we may acquire exactly such
It is rather unfortunate that the letter T is used to denote both kinetic energy and string tension (or rather rod tension
we are, of course, assuming that the "pendulum strings" are perfectly rigid).
R. Ø. Nielsen, E. Have, B. T. Nielsen
The Double Pendulum
24/31
expressions:
T2 sin θ2 = −m2 ẍ2
T2 cos θ2 = m2 (ÿ2 + g) ,
which, upon substitution into (D.2) and (D.3), yields:
We may eliminate
(D.7) by
sin θ1 ,
T1
m1 ẍ1 = −T1 sin θ1 − m2 ẍ2 ,
(D.6)
m1 ÿ1 = T1 cos θ1 − m2 (ÿ2 + g) − m1 g.
(D.7)
entirely from these equations by observing that if we multiply (D.6) by
the factors containing
T1
− cos θ1
and
will be equal, thus, upon rearrangement, yielding:
− cos θ1 (m1 ẍ1 + m2 ẍ2 ) = sin θ1 (m1 ÿ1 + m2 ÿ2 + (m1 + m2 ) g) .
(D.8)
T2
become equal if we
We can do almost exactly the same for (D.4) and (D.5): the factors containing
multiply the former by
− cos θ2
and the latter by
sin θ2 :
)
− cos θ2 m2 ẍ2 = T2 sin θ2 cos θ2 ,
sin θ2 m2 ÿ2 = T2 sin θ2 cos θ2 − sin θ2 m2 g
Now, all that remains is to nd expressions for
⇒ − cos θ2 m2 ẍ2 = sin θ2 (m2 ÿ2 + m2 g) .
ẍ, ẍ2 , ÿ1 , ÿ2 .
(D.9)
Dierentiating (D.1) w.r.t time twice yields:
ẍ1 = θ̈1 l1 cos θ1 − l1 sin θ1 θ̇12 ,
ẍ2 = θ̈1 l1 sin θ1 + l1 cos θ1 θ̇12 ,
ÿ1 = θ̈1 l1 cos θ1 − l1 sin θ1 θ̇12 + θ̈2 l2 cos θ2 − l2 sin θ2 θ̇22 ,
ÿ2 = θ̈1 l1 sin θ1 +
l1 cos θ1 θ̈12
+ θ̈2 l2 sin θ2 +
(D.10)
l2 cos θ2 θ̇22 .
Plugging these values into (D.8) and (D.9) results in the full equations of motion for the simple double
pendulum. Feeding these equations into MatLab or another similar program would be perfectly simple,
and so we have derived the equations of motion using only the Newtonian framework. The computational
superiority of the Lagrangian method is now obvious, and it is often pointed out that deriving these very
equations using only Newton's equation, albeit possible, proves an arduous task. Nevertheless, it can be
done without too much trouble, as we have shown. Useful though the Lagrangian and, by extension,
the Hamiltonian - formalisms may be, they will, of course, do nothing (classically) that the Newtonian
formalism won't! [7]
R. Ø. Nielsen, E. Have, B. T. Nielsen
E
The Double Pendulum
25/31
On the Constancy of the Mechanical Energy
To verify the equations of motion, (2.23) and (2.24), we may determine whether they give rise to a constant
energy function, as they should in a Hamiltonian system such as ours.
The energy is given by the sum of the potential and kinetic energies:
E =T +V
1
1
1
1
2
2
= I1 θ˙1 + I2,cm θ˙2 + m2 h21 θ̇12 + m2 d22 θ̇22 + m2 h1 d2 θ̇1 θ̇2 cos(θ1 − θ2 )
2
2
2
2
− m1 gd1 cos(θ1 ) − m2 gh1 cos(θ1 ) − m2 gd2 cos(θ2 )
(E.1)
(E.2)
To test whether our equations of motion guarantee the constancy of this function with respect to time, we
entered the equations into MatLab and computed the energy as a function of time. Using this method,
some numerical noise is to be expected, however we found that the energy is indeed conserved, as can
be seen in Figure 13. The energy is constant to at least ve signicant gures over a period of
witnessed by MatLab. The MatLab script which produced this plot,
energy.m,
30s
as
is included in Appendix
F.
The initial conditions were chosen entirely at random. However, to calculate the energy we needed some
physical constants pertaining to the double pendulum (moments of inertia, locations of centers of gravity,
masses and so on). These were determined by the methods described in section 6.1. The actual initial
conditions chosen were
θ1 (0) = 1.3, θ2 (0) = 1.2, θ̇1 (0) = 0.3s−1 , θ̇2 (0) = −0.1s−1 .
please bear in mind that the zero point of the potential energy is chosen at
When viewing the plot,
θ1 = θ2 = π/2.
Energy/time. Initial energy:−1.2104 Final energy:−1.2104
−1.2104
−1.2104
−1.2104
Energy [J]
−1.2104
−1.2104
−1.2104
−1.2104
−1.2104
−1.2104
0
5
10
15
Time [s]
20
25
30
Figure 13: Mechanical energy as a function of time. Notice the scale on the axis of ordinates.
R. Ø. Nielsen, E. Have, B. T. Nielsen
F
1
3
5
7
9
The Double Pendulum
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MatLab Scripts
f u n c t i o n xdot = odesample ( t , x )
% P h y s i c a l c o n s t a n t s o f the double pendulum i n SI u n i t s
I1 =0.08623;
I2 =0.017681;
m1=0.955;
m2=0.63;
g =9.82;
h1 =0.4035;
d1 =14.8/100;
d2 =8/100;
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%%Here ar e the d i f f e r e n t i a l e q u a t i o n s ( transformed i n t o a system o f f o u r
%%f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s ) .
xdot=z e r o s ( 4 , 1 ) ;
xdot ( 1 )=x ( 3 ) ; % Theta1−dot
xdot ( 2 )=x ( 4 ) ; % Theta2−dot
xdot ( 3 )=−(m2∗ h1 ∗ d2 ∗ x ( 4 ) ^2 ∗ s i n ( x ( 1 )−x ( 2 ) ) ∗ I2+m2^2 ∗ h1 ∗ d2^3 ∗ x ( 4 ) ^2 ∗ s i n ( x ( 1 )−x ( 2 ) )+m1∗ g ∗ d1 ∗ s i n ( x ( 1 ) ) ∗
I2+m1∗ g ∗ d1 ∗ s i n ( x ( 1 ) ) ∗m2∗ d2^2+m2∗ g ∗ h1 ∗ s i n ( x ( 1 ) ) ∗ I2+m2^2 ∗ g ∗ h1 ∗ s i n ( x ( 1 ) ) ∗ d2^2+m2^2 ∗ h1^2 ∗ d2^2 ∗ cos
( x ( 1 )−x ( 2 ) ) ∗ x ( 3 ) ^2 ∗ s i n ( x ( 1 )−x ( 2 ) )−m2^2 ∗ h1 ∗ d2^2 ∗ cos ( x ( 1 )−x ( 2 ) ) ∗ g ∗ s i n ( x ( 2 ) ) ) /( −m2^2 ∗ h1^2 ∗ d2^2 ∗
cos ( x ( 1 )−x ( 2 ) )^2+I1 ∗ I2+I1 ∗m2∗ d2^2+m2∗ h1^2 ∗ I2+m2^2 ∗ h1^2 ∗ d2 ^2) ; %theta1 −double −dot
xdot ( 4 ) =(h1^2 ∗ cos ( x ( 1 )−x ( 2 ) ) ∗m2∗ d2 ∗ x ( 4 ) ^2 ∗ s i n ( x ( 1 )−x ( 2 ) )+h1 ∗ cos ( x ( 1 )−x ( 2 ) ) ∗m1∗ g ∗ d1 ∗ s i n ( x ( 1 ) )+h1
^2 ∗ cos ( x ( 1 )−x ( 2 ) ) ∗m2∗ g ∗ s i n ( x ( 1 ) )+h1 ∗ x ( 3 ) ^2 ∗ s i n ( x ( 1 )−x ( 2 ) ) ∗ I1+h1^3 ∗ x ( 3 ) ^2 ∗ s i n ( x ( 1 )−x ( 2 ) ) ∗m2−g ∗
s i n ( x ( 2 ) ) ∗ I1 −g ∗ s i n ( x ( 2 ) ) ∗m2∗ h1 ^2) ∗m2∗ d2/( −m2^2 ∗ h1^2 ∗ d2^2 ∗ cos ( x ( 1 )−x ( 2 ) )^2+I1 ∗ I2+I1 ∗m2∗ d2^2+m2
∗ h1^2 ∗ I2+m2^2 ∗ h1^2 ∗ d2 ^2) ; %theta2 −double −dot
end
../Matlab/odesample.m
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clear all ;
% The purpose o f t h i s s c r i p t i s to i l l u s t r a t e the s e n s i t i v i t y to i n i t i a l
% c o n d i t i o n s by r e c o r d i n g f i n a l a n g l e as f u n c t i o n o f i n i t i a l a n g l e
t f i n a l =10.01; % S imu la tio n time
n p o i n t s =10000; % S im ula ti on data p o i n t s
t i n t e r v a l=t f i n a l / n p o i n t s ; % Time i n t e r v a l between p o i n t s .
M=1000; % Number o f s i m u l a t i o n s to run
max_angle =1.8
f o r i =1:M;
% Inital conditions
i n i t i a l ( 1 )=i ∗ ( max_angle/M) ;
i n i t i a l ( 2 )=i ∗ ( max_angle/M) ; % t h e t a 1 ( 0 )=t h e t a 2 ( 0 )
i n i t i a l ( 3 ) =0; % I n i t i a l v e l o c i t i e s =0
i n i t i a l ( 4 ) =0;
% Numerical ODE s o l u t i o n :
[ t , y]= ode45 ( @odesample , 0 : t i n t e r v a l : t f i n a l , i n i t i a l ) ;
% Record i n i t i a l c o n d i t i o n f o r p l o t t i n g
r e c o r d i n i t ( i )= i n i t i a l ( 1 ) ;
% Record f i n a l a n g l e t h e t a 1
r e c o r d f i n a l ( i )=y ( n p o i n t s +1 ,1) ;
R. Ø. Nielsen, E. Have, B. T. Nielsen
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i % P r i n t i t e r a t i o n number to CLI .
end
plot ( recordinit , recordfinal , ' . ' )
x l a b e l ( ' I n i t i a l value o f t h e t a 1 [ rad ] ' )
y l a b e l ( ' f i n a l value o f t h e t a 1 [ rad ] ' )
../Matlab/initcondsimulation.m
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clear all ;
I1 =0.08623;
I2 =0.017681;
m1=0.955;
m2=0.63;
g =9.82;
h1 =0.4035;
d1 =14.8/100;
d2 =8/100;
% Total time o f s i m u l a t i o n s
T = 80;
% Length o f i n i t i a l phase space s e p a r a t i o n
D = 1E− 4;
% Degree o f " f i n e n e s s " o f each p a r t i a l s i m u l a t i o n ( that o f l e n g t h dt )
Npart = 1 0 0 0 ;
% Length o f each p a r t i a l s i m u l a t i o n
dt =0.1;
% M i s the v a r i a b l e which i s incremented c a u s i n g the i n i t i a l a n g l e
% to change .
M=200;
% Vector which w i l l c o n t a i n t h e t a 0 and Lyapunov exponent
LYAPS=z e r o s (M+1 ,2) ;
max_angle =1.8;
f o r i =0:M
t h e t a 0=i ∗ ( max_angle/M) ;
% I n i t i a l i n i t i a l c o n d i t i o n s ( ahem ) :
i n i t i a l 1 (1) = theta0 ;
i n i t i a l 1 (2) = theta0 ;
i n i t i a l 1 (3) = 0;
i n i t i a l 1 (4) = 0;
initial2 = initial1 ;
f i n a l 1=i n i t i a l 1 ;
f i n a l 2=i n i t i a l 2 ;
t =0;
L=0;
while t < T
i n i t i a l 1=f i n a l 1 ;
i f norm ( f i n a l 2 − f i n a l 1 ) > 0
i n i t i a l 2=f i n a l 1 +(( f i n a l 2 − f i n a l 1 ) . / norm ( f i n a l 2 − f i n a l 1 ) ) . ∗ D;
end
i f norm ( f i n a l 2 − f i n a l 1 ) == 0
% I f the two phase space v e c t o r s ( f i n a l 1 , f i n a l 2 ) c o i n c i d e , we
% cannot d e f i n e any r e a s o n a b l e d i r e c t i o n i n which to s e p a r a t e
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% the two t r a j e c t o r i e s , so we j u s t choose the d i r e c t i o n
% given by the u n i t v e c t o r (1/ s q r t ( 2 ) ) ∗ [ 1 , 1 , 0 , 0 ]
i n i t i a l 2=f i n a l 1+D∗ (1/ s q r t ( 2 ) ) ∗ [ 1 , 1 , 0 , 0 ] ;
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end
[ t1 , y1]= ode45 ( @odesample , 0 : dt / Npart : dt , i n i t i a l 1 ) ;
[ t2 , y2]= ode45 ( @odesample , 0 : dt / Npart : dt , i n i t i a l 2 ) ;
l a s t e l e m=l e n g t h ( t1 ) ;
f i n a l 1=y1 ( l a s t e l e m , : ) ;
f i n a l 2=y2 ( l a s t e l e m , : ) ;
% C o n f i g u r a t i o n space −> Phase Space t r a n s f o r m a t i o n o f c o o r d i n a t e s
i n i t i a l P 1 ( 1 )=i n i t i a l 1 ( 1 ) ;
i n i t i a l P 1 ( 2 )=i n i t i a l 1 ( 2 ) ;
i n i t i a l P 1 ( 3 )=I1 ∗ i n i t i a l 1 ( 3 )+m2∗ h1^2 ∗ i n i t i a l 1 ( 3 )+m2∗ h1 ∗ d2 ∗ i n i t i a l 1 ( 4 ) ∗ cos ( i n i t i a l 1 ( 1 )−
i n i t i a l 1 (2) ) ;
i n i t i a l P 1 ( 4 )=I2 ∗ i n i t i a l 1 ( 4 )+m2∗ d2^2 ∗ i n i t i a l 1 ( 4 )+m2∗ h1 ∗ d2 ∗ i n i t i a l 1 ( 3 ) ∗ cos ( i n i t i a l 1 ( 1 )−
i n i t i a l 1 (2) ) ;
% C o n f i g u r a t i o n space −> Phase Space t r a n s f o r m a t i o n o f c o o r d i n a t e s
i n i t i a l P 2 ( 1 )=i n i t i a l 2 ( 1 ) ;
i n i t i a l P 2 ( 2 )=i n i t i a l 2 ( 2 ) ;
i n i t i a l P 2 ( 3 )=I1 ∗ i n i t i a l 2 ( 3 )+m2∗ h1^2 ∗ i n i t i a l 2 ( 3 )+m2∗ h1 ∗ d2 ∗ i n i t i a l 2 ( 4 ) ∗ cos ( i n i t i a l 2 ( 1 )−
i n i t i a l 2 (2) ) ;
i n i t i a l P 2 ( 4 )=I2 ∗ i n i t i a l 2 ( 4 )+m2∗ d2^2 ∗ i n i t i a l 2 ( 4 )+m2∗ h1 ∗ d2 ∗ i n i t i a l 2 ( 3 ) ∗ cos ( i n i t i a l 2 ( 1 )−
i n i t i a l 2 (2) ) ;
% C o n f i g u r a t i o n space −> Phase Space t r a n s f o r m a t i o n o f c o o r d i n a t e s
f i n a l P 1 ( 1 )=f i n a l 1 ( 1 ) ;
f i n a l P 1 ( 2 )=f i n a l 1 ( 2 ) ;
f i n a l P 1 ( 3 )=I1 ∗ f i n a l 1 ( 3 )+m2∗ h1^2 ∗ f i n a l 1 ( 3 )+m2∗ h1 ∗ d2 ∗ f i n a l 1 ( 4 ) ∗ cos ( f i n a l 1 ( 1 )− f i n a l 1 ( 2 ) ) ;
f i n a l P 1 ( 4 )=I2 ∗ f i n a l 1 ( 4 )+m2∗ d2^2 ∗ f i n a l 1 ( 4 )+m2∗ h1 ∗ d2 ∗ f i n a l 1 ( 3 ) ∗ cos ( f i n a l 1 ( 1 )− f i n a l 1 ( 2 ) ) ;
% C o n f i g u r a t i o n space −> Phase Space t r a n s f o r m a t i o n o f c o o r d i n a t e s
f i n a l P 2 ( 1 )=f i n a l 2 ( 1 ) ;
f i n a l P 2 ( 2 )=f i n a l 2 ( 2 ) ;
f i n a l P 2 ( 3 )=I1 ∗ f i n a l 2 ( 3 )+m2∗ h1^2 ∗ f i n a l 2 ( 3 )+m2∗ h1 ∗ d2 ∗ f i n a l 2 ( 4 ) ∗ cos ( f i n a l 2 ( 1 )− f i n a l 2 ( 2 ) ) ;
f i n a l P 2 ( 4 )=I2 ∗ f i n a l 2 ( 4 )+m2∗ d2^2 ∗ f i n a l 2 ( 4 )+m2∗ h1 ∗ d2 ∗ f i n a l 2 ( 3 ) ∗ cos ( f i n a l 2 ( 1 )− f i n a l 2 ( 2 ) ) ;
dx=norm ( f i n a l P 2 − f i n a l P 1 ) ;
dx0=norm ( i n i t i a l P 2 − i n i t i a l P 1 ) ;
dx0
K=norm ( i n i t i a l 2 − i n i t i a l 1 )
t=t+dt ;
Li=l o g ( dx/dx0 ) ;
L=L+Li ;
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end
end
L=(1/T) ∗ L ;
LYAPS( i +1 ,1)=t h e t a 0 ;
LYAPS( i +1 ,2)=L ;
i % P r i n t i to CLI
p l o t (LYAPS( : , 1 ) ,LYAPS( : , 2 ) )
x l a b e l ( ' I n i t i a l a n g l e ( t h e t a 2=t h e t a 1 ) ' ) ;
y l a b e l ( ' Lyapunov exponent ' ) ;
../Matlab/laypps/lyap.m
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clear all ;
% Total time o f s i m u l a t i o n
T=10
n p o i n t s =100000 % Number o f s i m u l a t e d p o i n t e
A = importdata ( ' 2552 − t r a c k . csv ' , ' , ' ) ; % Experimental motion data .
expt=A( : , 1 ) ;
exptheta=A( : , 2 ) ;
p l o t ( expt , exptheta )
l e g e n d ( ' Experimental ' )
i n i t i a l ( 1 )=exptheta ( 1 ) ;
i n i t i a l ( 2 ) =3.467633769E− 1; % From Tracker
i n i t i a l ( 3 ) =0;
i n i t i a l ( 4 ) =0;
[ t , y]= ode113 ( @odesample , 0 :T/ n p o i n t s : T, [ i n i t i a l ] ) ;
hold on
p l o t ( t , y ( : , 1 ) , ' red ' )
l e g e n d ( ' Experimental ' , ' Simulated ' )
x l a b e l ( ' Time [ s ] ' )
y l a b e l ( ' Angle t h e t a 1 [ rad ] ' )
../Matlab/Trackcompare/comptrajectory.m
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% Parametric p l o t o f double pendulum ( parametrized by e x p e r i m e n t a l l y
% determined a n g l e s ) . P l o t s ( x , y ) t r a j e c t o r i e s o f pendula .
L2 = 0 . 3 2 5 ; % Length o f pendulum 2
h1 = 0 . 4 0 3 5 ; % Distance between p o i n t s o f s u s p e n s i o n
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A = importdata ( ' t r a c k . csv ' , ' , ' ) ;
n=1;
t=z e r o s ( l e n g t h (A) , 1 ) ;
X1=z e r o s ( l e n g t h (A) , 2 ) ; % Will c o n t a i n c o o r d i n a t e s o f pendulum 1 .
X2=X1 ; % Will c o n t a i n c o o r d i n a t e s o f pendulum 2 .
N=l e n g t h (A)
w h i l e n <= l e n g t h (A)
t ( n , 1 )=A( n , 1 ) / 1 0 ;
th1=A( n , 2 ) ;
th2=A( n , 3 ) ;
th3=A( n , 4 ) ;
X1( n , 1 )=h1 ∗ s i n ( th1 ) ;
X1( n , 2 )=−h1 ∗ cos ( th1 ) ;
X2( n , 1 )=h1 ∗ s i n ( th1 )+L2 ∗ s i n ( th3 ) ;
X2( n , 2 )=−h1 ∗ cos ( th1 )−L2 ∗ cos ( th3 ) ;
n=n+1;
end
p l o t (X1 ( : , 1 ) ,X1 ( : , 2 ) )
hold on
p l o t (X2 ( 1 :N, 1 ) ,X2 ( 1 :N, 2 ) , ' red ' )
l e g e n d ( ' Upper pendulum ' , ' Lower pendulum ' )
x l a b e l ( ' x [m] ' )
y l a b e l ( ' y [m] ' )
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t i t l e ( ' Double pendulum , e x p e r i m e n t a l data , runtime : 10 s ' )
a x i s equal
../Matlab/Paramplot/param.m
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clear all ;
I1 =0.08623;
I2 =0.017681;
m1=0.955;
m2=0.63;
g =9.82;
h1 =0.4035;
d1 =14.8/100;
d2 =8/100;
% Plot time
t f i n a l =30;
n p o i n t s =10000; % Number o f s i m u l a t e d p o i n t s
t i n t e r v a l=t f i n a l / n p o i n t s ; % I n t e r v a l between p o i n t s .
i n i t i a l ( 1 ) =1.3;
i n i t i a l ( 2 ) =1.2;
i n i t i a l ( 3 ) =0.3;
i n i t i a l ( 4 ) = − 0.1;
opts=o d e s e t ( ' r e l t o l ' , 1e − 9, ' a b s t o l ' ,1 e − 12) ;
[ t , y]= ode113 ( @odesample , 0 : t i n t e r v a l : t f i n a l , i n i t i a l , opts ) ;
n=1;
E=z e r o s ( l e n g t h ( t ) , 2 ) ;
w h i l e n<=l e n g t h ( t )
E( n , 1 )=t ( n , 1 ) ;
E( n , 2 ) =0.5 ∗ I1 ∗ y ( n , 3 ) ^2+0.5 ∗ I2 ∗ y ( n , 4 ) ^2+0.5 ∗m2∗ h1^2 ∗ y ( n , 3 ) ^2+0.5 ∗m2∗ d2^2 ∗ y ( n , 4 ) ^2+m2∗ h1 ∗ d2 ∗ y ( n
, 3 ) ∗ y ( n , 4 ) ∗ cos ( y ( n , 1 ) −y ( n , 2 ) )−m1∗ g ∗ d1 ∗ cos ( y ( n , 1 ) )−m2∗ g ∗ h1 ∗ cos ( y ( n , 1 ) )−m2∗ g ∗ d2 ∗ cos ( y ( n , 2 ) )
;
n=n+1;
end
p l o t (E ( : , 1 ) ,E ( : , 2 ) ) ;
x l a b e l ( ' Time [ s ] ' ) ;
y l a b e l ( ' Energy [ J ] ' ) ;
E0 = E( 1 , 2 ) ;
E30=E( l e n g t h ( t ) , 2 ) ;
E0str = num2str (E0) ;
E30str = num2str ( E30 ) ;
p l o t t i t l e = s t r c a t ( ' Energy / time . I n i t i a l energy : ' , E0str , ' F i n a l energy : ' , E30str ) ;
title ( plottitle ) ;
../Matlab/Energy/energy.m
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References
[1] Arnold, V. I.
Graduate Texts in Mathematics: Mathematical Methods of Classical Mechanics, 2. ed.
Springer, 2010.
[2] Aruldhas, G.
Classical Mechanics.
[3] Gardiner, S. A.
Prentice-Hall Of India Pvt. Limited.
Quantum Measurement, Quantum Chaos, and Bose-Einstein Condensates.
Uni-
versität Innsbruck, 2000.
[4] Hobbie, R., and Roth, B.
Intermediate Physics for Medicine and Biology.
Biological and Medical
Physics, Biomedical Engineering Series. Springer, 2007.
[5] Lanczos, C.
[6] Morin, D.
The Variational Principles of Mechanics.
Dover Publications, 1970.
Introduction to Classical Mechanics: With Problems and Solutions.
Cambridge University
Press, 2008.
[7] Webber, B., and Barnes, C.
Lecture Notes: Theoretical Physics 1.
Cambridge University, 2008.
[8] Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A. Determining Lyapunov Exponents
from a Time Series.
Physica
(1985), 285317.