Question

The "spherical pendulum" is just a simple pendulum that is free to move in any sideways direction. (By contrast a "simple pendulum"- unqualified - is confined to a single vertical plane.) The bob of a spherical pendulum moves on a sphere, centered on the point of support with radius \(r=R\) the length of the pendulum. A convenient choice of coordinates is spherical polars, \(r, \theta, \phi,\) with the origin at the point of support and the polar axis pointing straight down. The two variables \(\theta\) and \(\phi\) make a good choice of generalized coordinates. (a) Find the Lagrangian and the two Lagrange equations. (b) Explain what the \(\phi\) equation tells us about the \(z\) component of angular momentum \(\ell_{z^{*}}\) (c) For the special case that \(\phi=\) const, describe what the \(\theta\) equation tells us. (d) Use the \(\phi\) equation to replace \(\dot{\phi}\) by \(\ell_{z}\) in the \(\theta\) equation and discuss the existence of an angle \(\theta_{\mathrm{o}}\) at which \(\theta\) can remain constant. Why is this motion called a conical pendulum? (e) Show that if \(\theta=\theta_{0}+\epsilon,\) with \(\epsilon\) small, then \(\theta\) oscillates about \(\theta_{\mathrm{o}}\) in harmonic motion. Describe the motion of the pendulum's bob.

Short answer

The pendulum's motion is governed by the Lagrangian mechanics, resulting in conical, harmonic oscillations about a stable angle.

Step-by-step solution

Define the Lagrangian

The Lagrangian, \( L \), is defined as the difference between the kinetic \( T \) and potential \( V \) energies of the system: \( L = T - V \). For a spherical pendulum, the kinetic energy in spherical coordinates is \( T = \frac{1}{2} m R^2 (\dot{\theta}^2 + \sin^2 \theta \dot{\phi}^2) \) and the potential energy is \( V = mgR(1 - \cos \theta) \). Thus, the Lagrangian is \( L = \frac{1}{2} m R^2 (\dot{\theta}^2 + \sin^2 \theta \dot{\phi}^2) - mgR(1 - \cos \theta) \).

Find the Euler-Lagrange Equations

The Lagrange equations for \( \theta \) and \( \phi \) are derived from the Euler-Lagrange equation \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 \), where \( q \) is the generalized coordinate. Applying this to \( \theta \) and \( \phi \), we get:1. \[ \frac{d}{dt} (mR^2 \dot{\theta}) = mR^2 \sin \theta \cos \theta \dot{\phi}^2 - mgR \sin \theta \]2. \[ \frac{d}{dt} (mR^2 \sin^2 \theta \dot{\phi}) = 0 \]

Analyze the \( \phi \) Equation

The equation \( \frac{d}{dt} (mR^2 \sin^2 \theta \dot{\phi}) = 0 \) implies that the quantity \( mR^2 \sin^2 \theta \dot{\phi} \) is conserved. This represents the conservation of the \( z \) component of angular momentum \( \ell_{z^{*}} \).

Evaluate Special Case \( \phi = \) const

For the special case where \( \phi \) is constant, \( \dot{\phi} = 0 \) and consequently, the system simplifies to one-dimensional motion for \( \theta \). The \( \theta \) equation reduces to simple harmonic motion in \( \theta \). The pendulum oscillates in a single plane like a simple pendulum.

Replace \( \dot{\phi} \) Using \( \ell_{z} \)

Using the conservation of angular momentum, \( \dot{\phi} = \frac{\ell_{z}}{mR^2 \sin^2 \theta} \) can replace \( \dot{\phi} \) in the \( \theta \) equation. This substitution helps to find a stable equilibrium angle \( \theta_o \) where \( \theta \) can remain constant, leading to the concept of a conical pendulum where the pendulum swings in a conical motion.

Prove Harmonic Oscillation for Small \( \epsilon \)

Consider \( \theta = \theta_{0} + \epsilon \) with \( \epsilon \) small. Linearize the \( \theta \) equation around \( \theta_0 \) to find that \( \epsilon \) satisfies the equation for harmonic oscillation. The pendulum oscillates about \( \theta_0 \) with small oscillations, indicating it undergoes simple harmonic motion about the cone's axis.

Key concepts

Lagrangian mechanics

Understanding the movement of systems with Lagrangian mechanics is a fundamental aspect of physics studying motion. The Lagrangian, denoted by \( L \), plays a crucial role in formulating the equations of motion. In the context of the spherical pendulum, the Lagrangian is expressed as the difference between the system's kinetic and potential energies: \( L = T - V \). Here, \( T \) represents the kinetic energy, given by \( T = \frac{1}{2} m R^2 (\dot{\theta}^2 + \sin^2 \theta \dot{\phi}^2) \), while \( V \) denotes the potential energy, \( V = mgR(1 - \cos \theta) \). By setting up these equations, one can analyze the motion dynamics of the pendulum.
In simpler terms, the Lagrangian encompasses all the energy aspects of a system: how it moves, how fast it moves, and how high it might swing. This formulation allows us to derive important equations that govern the dynamics, specifically the Euler-Lagrange equations, bringing us a step closer to understanding the complex movements of a spherical pendulum.

Angular momentum

The conservation of angular momentum is a pivotal concept in understanding spherical pendulum motion. Angular momentum, particularly its component \( \ell_{z^{*}} \) (momentum related to the \( z \) direction), is conserved in isolated systems like our pendulum. This means that \( mR^2 \sin^2 \theta \dot{\phi} \), which represents the \( z \) component of angular momentum, remains constant over time. By analyzing this, we can deduce key characteristics of motion.

• Angular momentum conservation tells us that as a pendulum swings and changes its angle \( \theta \), the speed at which it swings around the vertical axis \( \dot{\phi} \) varies to balance any changes in \( \theta \).
• This conservation directly impacts how rotational motion is maintained, thus defining consistent, predictable motion patterns for the pendulum.
• This is why angular dynamics are often more predictable than other types, given the lack of external forces disrupting this balance.

Euler-Lagrange equations

Euler-Lagrange equations are foundational tools in mechanics, especially for analyzing dynamic systems like the spherical pendulum. These equations are formulated as \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 \) for a generalized coordinate \( q \). They allow us to derive the necessary equations of motion for the pendulum.
For a spherical pendulum, applying this form:

• If \( q = \theta \), you derive an equation capturing linear dynamics affected by angles.
• If \( q = \phi \), you gain insight into the rotational components, reflected in the conservation of angular momentum.

These insights enable physicists to explore the deeper mechanics involved in pendulum motion. They are essential for understanding momentum variations and rotational influences. By translating the Lagrangian into the Euler-Lagrange formalism, we distill its essence down to precise, solvable mathematical equations.

Harmonic motion

Harmonic motion becomes particularly significant in the analysis of a spherical pendulum. When the angle \( \theta \) slightly deviates from a mean position \( \theta_0 \), we see this motion come into play. Hence, if \( \theta = \theta_0 + \epsilon \) with \( \epsilon \) being small, the resulting motion can be approximated as simple harmonic. This means:

• The pendulum bob undergoes minor oscillations about \( \theta_0 \), behaving as a harmonic oscillator.
• This behavior is akin to how springs and waves operate, with periodic back-and-forth movements centered around a stable point.
• In the context of the spherical pendulum, such oscillations mean gentle, predictable swagging within the conical path—a fundamental harmonic trait.

Thus, recognizing harmonic motion in these oscillations simplifies the complexity of pendulum motion, giving us clear insights into its predictable patterns of repeated movement.