Simple Pendulum Dynamics in Three Dimensions Cole Graham February 3, 2012 The simple pendulum is a classic system of study in introductory physics courses. It offers a tangible glimpse of the simple harmonic oscillator, and is easy to reproduce in lab and demonstration settings. However the analysis performed in such courses is usually restricted to the motion of the pendulum in a vertical two-dimensional plane through the pivot point. Taking the vertical line through the pivot as the z-axis of a three dimensional spherical system of coordinates, the standard presentation of the simple pendulum assumes that the weight has no θ component of angular momentum. Here, the full motion of a pendulum in three dimensions will be considered, with an emphasis on the trajectory of a pendulum with nonzero θ angular momentum. To begin, take the z-axis of the spherical coordinate system as described above, oriented downwards, so that the force of gravity acts in the positive z direction. Take the pivot of the pendulum as the origin of coordinates. Let the pendulum have length l and mass m in a gravitational field of constant acceleration g. Let φ and θ have their usual spherical definitions, so that φ is the angle between the z-axis and the pendulum measured at the pivot, and θ corresponds to rotation about the z-axis. Let the connecting rod of the pendulum be massless but rigid, so that the system remains well behaved for all φ between 0 and π. In this situation, the radial coordinate has a constant value l, leaving two degrees of freedom for the system, which are best described by the coordinates φ and θ. The evolution of the system is determined by the principle of least action applied to the Lagrangian L = T − U : ˙ θ) ˙ = 1 ml2 φ˙ 2 + 1 ml2 sin2 φ θ˙2 − mgl(1 − cos φ) L(φ, θ, φ, 2 2

(1)

∂L From the Lagrangian alone it is apparent that ∂L ∂θ = 0, implying that pθ ≡ ∂ θ˙ is conserved. Let M = pθ . Changing variables from φ˙ and θ˙ to pφ and pθ by applying the appropriate Legendre transformation, we obtain an expression for the Hamiltonian H, where pθ has been replaced by its constant value M : U

eff }|

{ M + + mgl(1 − cos φ) (2) H= | {z } 2ml2 2ml2 sin2 φ | {z } True Potential Kinetic Potential This expression illustrates the presence of a kinetic potential in H, which is added to the true potential to form the effective potential Ueff . Note especially that ∂H ∂t = 0, so in this system the energy E = T + U is conserved. Furthermore, for M 6= 0, limφ→0+ Ueff = limφ→π− Ueff = ∞. Thus, any state with nonzero M results in a bound orbit in which φ is permanently constrained between a maximal and minimal value. Indeed, it is clear that for systems with M 6= 0, φ cannot attain 0 or π for any finite energy E. p2φ

z

2

1

y Typical Orbit

U 25

Equilibrium Orbit 0.7

Effective Potential 20 Stable Equilibrium

15

0.7

-0.7

x

10 True Potential 5

-0.7 Kinetic Potential Î 4

0

Î 2

ÎŚ

3Î 4

Î

Figure 2: A typical and equilibrium orbit for the same physical parameters as Fig. 1, projected onto the xy-plane.

Figure 1: The three potentials for m = l = M = 1 and g = 9.8

Substituting M for pÎ¸ , we have eliminated Î¸ from the Hamiltonian altogether, leaving an expression involving only one coordinate Ď† and its canonical momentum pĎ† . The equations of motion of the system are thus best expressed solely in terms of the Ď† coordinate and its momentum, and a simple conversion may be included to relate Î¸ to these variables. From the Hamiltonian form of the Euler-Lagrange equations, we have the following equations of motion: âˆ‚H âˆ‚H M 2 cos Ď† âˆ‚H pĎ† M Ď†Ë™ = , p Ë™ = âˆ’ = âˆ’ mgl sin Ď†, and Î¸Ë™ = = = Ď† 3 2 2 âˆ‚pĎ† ml2 âˆ‚Ď† âˆ‚p ml sin Ď† ml sin2 Ď† Î¸

(3)

Ë™ is derived by differentiating (2) before M is substituted The third expression, determining Î¸, for pÎ¸ . To begin with an analysis of the trajectories followed by such a pendulum, let us first consider the special case in which pË™Ď† = 0. Let Ď†0 be a solution to the equation pË™Ď† = 0. Substituting Ď†0 into the second part of (3), setting the expression equal to zero, and simplifying, we reach the following transcendental equation for Ď†0 : M2 cos Ď†0 = sin4 Ď†0 m2 gl3

(4)

To further analyze this equilibrium state, consider the effective potential energy as a function âˆ‚U eff = âˆ‚H , so âˆ‚Ueff |Ď†=Ď† = 0. Let U0 = of Ď†: Ueff (Ď†). Differentiating Ueff , we find that âˆ‚Ď† 0 âˆ‚Ď† âˆ‚Ď† Ueff (Ď†0 ). Expanding Ueff (Ď†) about Ď†0 , we obtain the following second order approximation of Ueff (Ď†): âˆ‚ 2 Ueff 1 M2 4m3 g 2 l4 Ueff (Ď†) â‰ˆ U0 + k(Ď† âˆ’ Ď†0 )2 , where k = |Ď†=Ď†0 = + sin4 Ď†0 (5) 2 2 2 âˆ‚Ď† M2 ml2 sin Ď†0 Now, (4) has no solutions with Ď†0 â‰Ľ âˆ‚U

eff |Ď†= < 0. Clearly,

âˆ‚Ď†

Ď€ 2,

since

âˆ‚U

eff |Ď†= Ď€ > 0, and 2

âˆ‚Ď†

2

M2 m2 gl3 âˆ‚2U

> 0. Furthermore, âˆƒ > 0 such that

eff > 0 for 0 < Ď† < Ď€/2. Considering

âˆ‚Ď†2

the geometry of Fig. 1, this implies that in any given system there is only one solution to (4). Thus φ0 is unique, and at φ = φ0 the system attains its only stable equilibrium. At this equilibrium k 6= 0, so the system is approximated by a simple harmonic oscillator for φ ≈ φ0 . That is, the deviation of φ from φ0 varies sinusoidally in time. By substituting the approximationqin (5) into (3), and solving for φ(t), we find the frequency of this variation is

k given by ω = ml 2 . In Fig. 2 a typical trajectory of the pendulum has been superimposed over the equilibrium trajectory. φmax = 1 for the typical trajectory, and φ0 = .572 at equilibrium. For such a large displacement from φ0 , the higher order terms of Ueff (φ) visibly alter the trajectory, but the underlying sinusoidal displacement is apparent. Clearly, the motion of a simple pendulum extended to 3 dimensions neatly exhibits a multitude of classical mechanical principles, which are not limited to those discussed here. In fact, some of the essential results regarding the limits on motion, the conservation of energy, and the equilibrium state may be safely introduced in an introductory course without resorting to the Hamiltonian or the calculus of variations.

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