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Double Pendulum

A double pendulum is undoubtedly a true miracle of nature. The jump in complexity, which is observed at the transition from

a simple pendulum to a double pendulum is amazing. The oscillations of a simple pendulum are regular. For small deviations

from equilibrium, these oscillations are harmonic and can be described by sine or cosine function. In the case of nonlinearoscillations, the period depends on the amplitude, but the regularity of the motion holds. In other words, in the case ofa simple pendulum, the approximation of small oscillations fully reflects the essential properties of the system.

Double pendulum "behaves" quite differently. In the regime of small oscillations, the double pendulum demonstratesthe phenomenon of beats. The character of oscillations of the pendulums changes radically with increasing energy -the oscillations become chaotic. Despite the fact that the double pendulum can be described by a system of several ordinary

differential equations, i.e. by a completely deterministic model, the appearance of chaos looks very unusual. This situation is

reminiscent of the Lorenz system where a deterministic model of three equations also shows chaotic behavior. Try toexperiment with the application below and watch the movement of the double pendulum at different mass ratios and initialangles.

Next, we will build a mathematical model of the double pendulum in the form of a system of nonlinear differential equations.

Let's start with the derivation of the Lagrange equations.

Lagrange Equations

In Lagrangian mechanics, evolution of a system is described in terms of the generalized coordinates and generalizedvelocities. In our case, the deflection angles of the pendulums α1, α2 and the angular velocities can be taken as

the generalized variables. Using these variables, we construct the Lagrangian for the double pendulum and writethe Lagrange differential equations.

A simplified model of the double pendulum is shown in Figure 1. We assume that the rods are massless. Their lengths are l1

and l2. The point masses (they are represented by the balls of finite radius) are m1 and m2. All pivots are assumed to be

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frictionless.

Fig.1 Fig.2

We introduce the coordinate system Oxy, the origin of which coincides with suspension point of the upper pendulum.The coordinates of the pendulums are defined by the following relations:

The kinetic and potential energy of the pendulums (respectively, T and V) are expressed by the formulas

Then the Lagrangian can be written as

Take into account that

Hence,

As a result, the Lagrangian of the system takes the following form:

Now we can write the Lagrange equations (sometimes they are called as the Euler-Lagrange equations):

The partial derivatives in these equations are expressed by the following formulas:

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Hence, the first Lagrange equation can be written as

Cancelling l1 ≠ 0, we obtain:

Similarly, we derive the second differential equation:

After canceling m2l1 ≠ 0, the equation takes the form

Thus, the nonlinear system of two Lagrange differential equations can be written as

Small Oscillations of the Double Pendulum

Assuming that the angles α1(t), α2(t) are small, the oscillations of the pendulums near the zero equilibrium point can be

described by a linear system of equations. To get such a system, let's get back to the original Lagrangian of the system:

We write this Lagrangian in a simpler form, expanding it in a Maclaurin series and retaining the linear and quadratic terms.

The trigonometric functions can be replaced by the following approximate expressions:

Here we have taken into account that the term with cos cos(α1 − α2) contains the product of small quantities and has

the second order of smallness. Therefore, we can leave only the linear term in the cosine expansion.

Substituting this in the original Lagrangian and considering that the potential energy is defined up to a constant, we obtain

the quadratic Lagrangian for the double pendulum in the form:

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We derive the Lagrange differential equations for the given Lagrangian. They are written as

Find the partial derivatives:

We get the system of two differential equations

or

This system of equations can be written in a compact matrix form. We introduce the matrices

Then the system of differential equations can be represented as

Ii the case of one body, this equation describes the free undamped oscillations with a certain frequency. In the case ofthe double pendulum, the solution (as you will see below) will contain oscillations with two characteristic frequencies, which

are called normal modes. The normal modes are the real part of the complex-valued vector function

where H1, H2 are the eigenvectors, ω is the real frequency. The values of the normal frequencies ω1, 2 are determined by

solving the characteristic equation

We derive general formulas for the angular frequencies ω1, 2 in the case of arbitrary masses m1, m2 and lengths l1, l2:

We obtain a biquadratic equation for the frequencies ω. Compute the discriminant:

Thus, the squares of the normal frequencies ω1, 2 are given by

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or

This expression is somewhat cumbersome. Therefore we consider the case when the lengths of the rods of both pendulums

are equal: l1 = l2 = l. Then the normal frequencies will be determined by the more compact formula:

As can be seen, the eigenfrequencies ω1, 2 depend only on the mass ratio μ = m2/m1. The dependencies of the frequencies

ω1, ω2 on the parameter μ (provided g/l = 1) are shown in Figure 2 above. In particular, at equal masses m1 = m2 = m, i.e.

when μ = 1, the frequencies are equal to:

Now, after the eigenfrequencies ω1, 2 are known, we still have to determine the eigenvectors H1, 2 to describe the normal

modes. They can be found by solving the vector-matrix equation

Let the eigenvector H1 = (H11, H21) T (the superscript T denotes transposition) be corresponded to the normal frequency ω1.

Then we have the following equation for H1:

The coordinates of the eigenvector H1 satisfy the equation

Thus, the eigenvector H1 is given by

Similarly, we find the coordinates of the second eigenvector H2 = (H12, H22) T :

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Hence, the eigenvector H2 has the coordinates:

The general solution of the matrix equation can be written as

where the constants C1, C2, φ1, φ2 depend on the initial positions and velocities of the pendulums.

Consider the character of small oscillations for a specific set of initial data. Suppose, for example, that the initial positions and

velocities of the pendulums have the following values:

In this case, the initial phases are zero: φ1 = φ2 = 0. Determine the constants C1 and C2:

Then the law of oscillations of the pendulums is expressed by the formulas

where the angular frequencies ω1, 2 are given by

Here the angles α1(t), α2(t) are expressed in radians, and the time t in seconds. Figures 3-5 show plots of small oscillations

for three values of μ: μ1 = 0.2, μ2 = 1, μ3 = 5, provided l = l1 = l2 = 0.25 ì, g = 9.8 ì/c2. For convenience, the deflection

angles of the pendulums are given in degrees. As can be seen from the graphs, the oscillations in the system occur in the formof beats, in which energy is cyclically transferred from one pendulum to the other. When one of the pendulums almost stops,

the other swings with maximum amplitude. After some time, the pendulums "switch roles" and so on. Oscillations with higherfrequency ω1 are modulated by low-frequency oscillations with the frequency ω2. This is particularly apparent in Figure 5 for

the large value of μ (μ3 = 5) when the difference between the frequencies ω1 and ω2 is great.

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Fig.3 Fig.4

Fig.5 Fig.6

So, the small oscillations of the double pendulum are periodic and described by the sum of two harmonics with frequencies

ω1, ω2 depending on the system parameters. The characteristic property of small oscillations of the double pendulum is

the phenomenon of beats.

Legendre Transformation and Hamilton Equations

We now turn back to the original nonlinear system of equations and examine the character of oscillations of arbitrary

amplitude. This system of equations can not be solved analytically. Therefore, we consider a numerical model of the doublependulum.

The Lagrange equations given above are second order differential equations. It is more conveniently to convert them intothe form of Hamilton's canonical equations. As a result, instead of the two second-order equations, we obtain a system offour differential equations of the first order.

In Hamiltonian mechanics, the state of a system is determined by the generalized coordinates and generalized momenta. Inour case, we can use again as in the Lagrange equations the angles α1, α2 as the generalized coordinates. Instead of

the generalized velocities (in the Lagrangian), we now introduce the generalized momenta p1, p2 related to

the velocities by the formulas

or in the brief description:

The transition from the Lagrangian to the Hamiltonian form of the equations is performed using the Legendre transformation,which is defined as follows.

Suppose that f(x) is a smooth convex downward function (Figure 6). Consider the line y = px passing through the origin.The distance between the line y = px and the function y = f(x) along the y-axis depends on the coordinate x. This distance

will be maximal at a certain value of x. Clearly, it depends on the slope of the line, i.e., on the parameter p. Thus, weintroduce a new function g(p):

Such transformation of the function f(x) into the conjugate function g(p) is called the Legendre transformation. Note that

the function g(p) reaches a maximum value with respect to the variable x when p = df/dx. Indeed,

Knowing the dependence p(x), one can find the inverse function x(p). Then the Legendre transform is expressed by

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the relation

The Legendre transformation is easily generalized to the case of functions of several variables. In the model of the double

pendulum, the transition from the Lagrangian to the Hamiltonian is described by the Legendre transformation of the form:

In this expression, L is the Lagrangian, and the function H is the Hamiltonian of the system, which depends onthe generalized coordinates α1, α2 and the generalized momenta p1, p2.

As a result of this transformation, each Lagrange equation becomes a system of two Hamilton's canonical equations ofthe form:

We now define the specific form of the Hamilton's equations for the double pendulum. Thre generalized momenta p1, p2 are

expressed in terms of the partial derivatives of the Lagrangian in the form

We solve this system of equations using Cramer's rule and express the angular velocity in terms of the generalizedcoordinates and momenta. Compute the corresponding determinants:

Hence, we obtain the following expression for the velocities:

These formulas are the first two (of four) Hamilton's differential equations. Given these expressions, the Hamiltonian can be

written as follows:

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The last formula can be written as

The numerator N in this expression is very cumbersome. We simplify it:

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Hence, the Hamiltonian takes the form:

Here, the first term is the generalized kinetic energy T, and the other two terms represent the potential energy V,

i.e. the Hamiltonian H is defined as

where

Now we can write two more Hamilton's differential equations for the generalized momenta:

Compute separately the partial derivatives of the generalized kinetic energy:

where the symbols A1 and A2 denote the expressions

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The derivative of the kinetic energy T with respect to α2 will have the same form, but with the opposite sign:

From this we obtain the Hamilton equations in the form:

So, as a result of the cumbersome transformations we finally obtain the system of four Hamilton's canonical equations

describing the motion of the double pendulum. We write them together in the final form:

where

We can now proceed to the numerical analysis of the equations.

Numerical Simulation of Chaotic Oscillations

The most common method of numerical solution of differential equations is the 4th or 5th order Runge-Kutta methodDifferent variations of this method are used in most mathematical packages (MATLAB, Maple, Mathematica, MathCAD),

usually with an automatic error control and adaptive time-stepping.

To model the motion of the double pendulum, we also use the classical 4th order Runge-Kutta method (RK4). We somewhat

simplify the differential equations assuming that the lengths of the pendulums are the same: l1 = l2 = l. By introducing

the parameter μ equal to the mass ratio: μ = m2/m1, we can write the system in the following form:

where

This system can be rewritten in vector form:

The vector Z is composed of 4 canonical variables of the system, and components of the vector f correspond to the right

hand sides of the differential equations.

The Runge-Kutta (RK4) method requires at each step sequential evaluation of the four intermediate vectors:

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The value of the vector Zn+1 in the next node is given by

The total error of the algorithm on a finite interval has the order O(τ4 ), i.e. the computational accuracy increases by 16 times

while reducing the time step τ twice.

The described model is implemented in the animation shown at the beginning of the web page. For simplicity, we assume that

the initial deflection angles of the pendulums are equal: α1 = α2 = α. This application demonstrates the chaotic dynamics of

the double pendulum for different values of μ and α. Interestingly, in some regimes, stable trajectories such as in Figure 7 or

compact regions of attraction as in Figure 8 appear in the system. It seems that the double pendulum is not yet fully studied byphysicists and mathematicians and carries a lot of surprises.

Fig.7 (μ = 2.75, α = 171°) Fig.8 (μ = 1.21, α = 154°)

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