1.3 Understanding the Methods of Applied Mathematics
1.3.2 Modeling Simple Pendulums
Before considering the ideal double pendulum, we will begin by considering a simpler case. This serves both as a warm up to the double pendulum case, since we will see that even this simple case the detailed method of application is quite complex, and as a recurring example that we will use throughout the study. The simple case we begin with, then, is point-particle newtonian mechanics4 and the construction of a model
of the simple pendulum within it. We will use newtonian mechanics again when we turn to the consideration of near-Earth object modeling in chapter 3.
The basic equation we may take to define the theory of point-particle newtonian
4I am following the convention of Arnold(1989) in using lower case letters for names that have
mechanics is Newton’s second law,5
F=md 2x
dt2 (1.1)
which is taken to determine the motion of each particle in ann-particle system. The symbols in this equation have both a mathematical and a physical interpretation. Physically, x(t) is a function specifying the position of a given particle as a function of timet,mis the mass of that particle, andF(x(t), t)is the total vector force acting on the particle at time t. The mathematical meaning of this equation is made more clear by representing it in a slightly different form.
A system of n bodies in three dimensional space is represented by a system of
n copies of (1.2), one for each of n particles in a system. This determines a 3n- dimensional configuration space, a point of which determines the state of the system, with the dynamics being governed by Newton’s second law
d2x
dt2 =f, (1.2)
where f is a function on this 3n dimensional space that specifies the total force to mass ratio Fi/mi for each of the n particles. We will refer to this as the state space of the system,6 and we will use the same symbolx(t) to refer to the 3n-dimensional
position vector function. The 3n-dimensional position vector x(t) then, away from collisions, lies in a space of twice differentiable functions from an interval I of the real line, picking out the time domain, into the state space. The mass m is simply some real number. And the total force-to-mass function f(x(t), t) for the n-particle system lies in a space of continuous functions from the state space to itself, which may or may not vary with the value in I, the time.7 Such a function from the
state space to itself is called avector field.8 Physically the total force-to-mass on each
5There is also Newton’s third law, viz., F
ij = −Fji, where Fij is the force on particlei exerted by particlej, but we will not consider this explicitly here.
6This is distinguished from aphase space, which specifies 6nvariables, the position state and the
state of motion (velocity or momentum) of thenparticles.
7The case where the forces do not depend explicitly on time is called theautonomous case, and
where the forces do depend on time the non-autonomous case. We will be restricting attention to the autonomous case, but mathematically this implies no loss of generality, as will be explained below.
particle determines its acceleration at a positionxi and timet, which is what equation (1.2) says. Mathematically, the vector field (inversely scaled by the parameter m) determines the second order derivative of the position function x(t).
θ `
m
Figure 1.1: An ideal simple pendulum.
So much for the theory equation (1.2); what about methods used to construct models? Well, this requires the adding of constraints that determine the dimension of the state space (number of degrees of freedom). The simple case we consider here is the modeling a simple pendulum (see figure1.1). We will suppose that the physical pendulum is such that the pivot has very little friction, the rod is very light and effectively rigid and the weight is small and dense. Under these conditions, we can construct an ideal model of the simple pendulum that has only one degree of freedom, specified by the angle θ that the rod makes with the vertical (see figure 1.1).9 To
do this we suppose that the pivot is frictionless, that the rod is perfectly rigid and massless, and that the weight is a point mass. We then represent these conditions as mathematical constraints.
determines the direction of the vector field at that point and its length determines the magnitude of the vector field at that point. For the circle that the ideal simple pendulum travels on, the vector field always points tangent to the circle and toward the ground. It reaches its maximum value when the pendulum rod is horizontal and the force is directed straight down and its minimum value of zero when the rod is vertical.
9Strictly speaking,θ is ageneralized position variable and not a newtonian position variable. θ
parameterizes a one dimensional circular manifold that defines the state space of the pendulum. The mathematical framework of newtonian mechanics uses only positions of particles in a three di- mensional cartesian space. Accordingly, geometrically this model fits in the framework oflagrangian mechanics, which treats dynamics in terms of an n-dimensional configuration manifold parameter- ized byngeneralized variables, together with theirntime rates of change parameterizing the tangent bundle of configuration space. Nevertheless, by imposing the rigid body constraint on the rod in a cartesian newtonian setting, this still restricts the motion of the pendulum to a circle, which makes it natural to choose a parameterization of the circle that the motion is constrained to. Thus, in this way generalized variables arise naturally in newtonian mechanics as well. Furthermore, we have not used the apparatus of a lagrangian function and the euler-lagrange equations to construct the model, we have indeed used the framework of newtonian mechanics in the sense of computing the sum of vector forces on bodies and using Newton’s second law to find the equation of motion. In this case, then, the angleθcan simply be regarded as a change of variable.
As an example, consider the length of the rod of the pendulum. A more detailed model would allow the length to vary depending on the applied forces, perhaps by modeling the rod as a very stiff spring. To explicitly formulate the rigid body as- sumption in a newtonian framework, then, we could add the constraint that the time rate of change of the length ` of the pendulum is zero, viz., ˙` = 0, where the dot indicates a time derivative.
Under the applied constraints, and given that the force exerted by gravity on a mass is mg, where g is a vector pointing toward the centre of the Earth and has magnitude g=9.81 m/s, simple trigonometry allows one to show that the total force exerted on the weight is f = −mg` sinθ (in a polar coordinate system (r, θ) centred at the pivot, see figure 1.2).10 Thus, (1.2) becomes
¨
θ= −ω2sinθ, (1.3)
where the dots indicate time derivatives and ω=√g/`, and the mass m has dropped out of the model, indicating the motion of the pendulum is independent of the mass of the weight.
θ
F
mg
Figure 1.2: Forces on an ideal simple pendulum. The tension in the rod and the component of the gravitational force acting on the rod is not shown because these two forces cancel no matter what the massm is because the rod is assumed to be rigid.
The equation (1.3) specifies the equation of motion of the ideal model pendulum, which when supplemented with initial conditions, will determine the form of the motion θ(t) of the model pendulum over time. Physically, this will describe the motion of the target physical pendulum for as long as the assumptions (no friction,
10r does not appear in the model because the constraint that the rod is rigid means that ˙r=0,
rigid bodies, point mass, etc.) are effectively true of the system, which occurs when the behaviour of the components of the system is not significantly distinguishable from the behaviour of their idealized counterparts.11 The most significant factor in this
case is the friction, which will quickly dissipate energy from the physical pendulum system, causing it to slow to a stop. But, for a short period of time the model pendulum will accurately describe the motion of the physical pendulum.
Mathematically, quite a lot has happened in the transition from the theory equa- tion (1.2) to the model equation (1.3). The dimension of the state space, which could have been any finite dimension n, is fixed to be one. This implies that the functions
x(t) and f(x) are in the function spaces C2(
R → R) and C(R → R), respectively.
Before these constraints were applied, however, n was not determined and so they could have been in any of the spacesC2(
R→Rn)andC(Rn→Rn), respectively. Ge-
ometrically, however, the model is representing a pendulum that is located in three dimensional space R3. The model constraints imply that the state of the system is
restricted to a circle of radius ` embedded in R3. Thus, the geometric presentation
of the state of the model is as a (twice differentiable) function from an interval I⊆R
into a one-dimensional circular manifold M embedded inR3.
There is also another significant shift from equation (1.2) to equation (1.3). In the theory equation (1.2) the symbols x and f have no distinct reference. Rather, they pick out a vast plurality of function spaces in which the interpretation of these symbols could lie (C2(
R→Rn) and C(Rn →Rn), respectively, for any n). Already, m is sufficiently meaningful because it is required to be a positive real number. Once the dimension of the state space was restricted to one, however, the symbols x and
f are specified to lie in C(R →R). This makes them sufficiently meaningful, since we know that they are functions of a specific type. And this also defines a distinct
space of models, indexed by the possible values of f in C(Rn →
Rn). The model
construction process did not stop here, however, since we also specified the total force on the weight of the pendulum, which specified a particular value for f, viz.,
−g
`sinθ, which determines the dynamics on the state space. This now provides us with adistinctly meaningful equation, in the sense that it is talking about a particular
11Note that what counts assignificantly distinguishabledepends on tolerances for error in structure
state space, and a specific dynamics on that state space. Moreover, we represented this mathematical form of the model in three dimensional space, in such a way that mirrors the motion of the physical pendulum. This sort of equation, which has a specific geometric interpretation (state space, vector field, with or without projection into 3D space), is what applied mathematicians often mean by a model. This model can then be studied subject to a variety of different initial and or boundary conditions in order to understand its behaviour.
Determining how the model will behave under particular circumstances requires imposing particular conditions: initial conditions or boundary conditions. This re- quires finding the solution to the model equation (1.3) under specific conditions. Suppose that here we are interested in starting the pendulum in motion from a state of rest. In this case we impose the initial conditions
θ(t0) =θ0, θ˙(t0) =0, (1.4)
where t0 is the initial time, the initial angle is θ0 and the initial angular velocity is
zero. These conditions together with the model equation (1.3) define a particular mathematical problem, called an initial value problem. The solution to this problem will tell us how the model behaves under the imposed conditions.
This model is not entirely straightforward to solve as it stands, however, since the vector field f is a nonlinear function of θ.12 If we are content to consider only the
behaviour for small angles, then we can use the small angle approximation for sinθ,
viz., θ, which simplifies (1.3) to
¨
θ= −ω2θ, (1.5)
which is just the equation for simple harmonic motion, which has under the conditions (1.4) the solution
θ(t) =θ0cos(ωt).
This approximative technique, called linearization, is not strictly deductive. It pre- serves the structure of the equation, however, changing only the form of the vector
12It does have a general solution, however, and one that can be fairly straightforwardly found
using Jacobi elliptic functions (see, e.g., Lawden, 1989). More typical cases of nonlinear models, however, are not so amenable to closed-form analytic solution, if they can be solved analytically at all.
field, buying us an equation that is easily solved analytically at the expense of its only being valid under conditions that ensure small angles.
Models are used for a variety of different purposes in applied mathematics, includ- ing description, prediction, explanation and control. Let us suppose that in this case we are interested in the classic case of prediction. This simple model makes a number of predictions. It tells us that for small angles and over a short time interval the motion should be described by a cosine function. It also tells us that the frequency of oscillation ω=√g/` is inversely proportional to the length` of the pendulum rod in a particular way,viz., that quadrupling the length of the pendulum would reduce the frequency by a factor of 12. Or, equivalently, that the period T =2π/ω,i.e., the time of one oscillation, of the pendulum is directly proportional to the the square root of the length. To test these consequences requires observation and measurement.
The square root dependency of the period on the rod length could be studied simply by constructing a series of pendulums with a low friction pivot, rods of different lengths made of a light, rigid material, and a small dense weight, much heavier than the rods. Using a measuring tape and a stop watch to measure the lengths of the rods and the time of one period, respectively, it would be possible to determine if the `1/2 dependency is replicated by real pendulums.13 A strategy for measurement
here could be to run several trials of period measurement for each rod length, each trial being a time average of several oscillations, and then averaging the results of the trials for each rod. The result would be a measurement of the (short time) period of each of the pendulums. To see if the square root dependency obtains, several methods could be used. An example would be to compute the logarithms of the period and length measurements, plotting these on a log-log plot of the period versus the length, performing a linear regression analysis and then seeing if the slope of the regression line is equal to 12 within experimental error. Determining the experimental error is another matter, which would require estimating the uncertainty of the devices
13The`1/2 dependency could actually be tested without the use of a time measuring device, by
using a measuring tape to measure the lengths of the rods and using rod lengths that differ by a factor of 4. It would then be possible to determine if the `1/2 dependency is replicated by real
pendulums by determining whether there are two oscillations of a given pendulum for every one of a pendulum with a rod four times as long. This method avoids the potential circularity of using a time measuring device that itself uses vibrations of a quartz crystal, an essential equivalent of a pendulum. For our purposes here, however, we will require some time measurement device, so it is convenient to work with the simpler case of a common time measuring device.
themselves, calculating the standard error of the trial measurements, and then tracing the propagation of error through the calculation done on these quantities. At the end of this process we would be in a position to determine whether the square root dependency prediction of the model is correct.
The measurement devices for the procedure just described (a tape measure and a stop watch) would not be sufficient to determine whether the form of the function describing the evolution of the angleθ over time is, to a high degree of approximation, a cosine function for small angles. First of all, this means of measuring time will probably not be sufficiently accurate to precisely determine the form of the function. But we also need a device to measure the angle accurately at particular times. For example, this could be done with a digital camera and a strobe light or with a video camera with a precise time stamp. However this is accomplished, the result will be a number of pairs of measurements of angle and time. These could be compared point-wise with the prediction of the models by calculatingθ(t)for each measurement time ti. If the difference is small, then this is one measure of whether the pendulum and model agree. Or the measurements could be interpolated to obtain a twice continuously differentiable function that minimizes the deviation from the measured angles.14 This would provide us with a differentiable function ˜θ(t), which is a model
of the raw angle-time data.
There are two basic ways that we could assess whether the model and the measure- ments agree. One approach is to compute the difference between the two functions
θ(t)and ˜θ(t). This difference
θ(t) =θ(t) −θ˜(t)
is called the forward error. If for any given time, θ(t) is small, then we know that the form of the motion of the physical pendulum is approximately a cosine function. This is a continuous version of the discrete calculation mentioned above where we compute θ(t) from the model solution at each measurement timeti and calculate the
14This can be done, for example, using a combination of forward differences and Hermite interpo-
lation (for an example of this approach to interpolation seeMoir,2010). Forward differences could be used to estimate the angular velocity at each point in time, and Hermite interpolation could then give aC2interpolant that matches the measured angles and the estimated angular velocities at each