Question

13.13 \star\star Consider a particle of mass \(m\) constrained to move on a frictionless cylinder of radius \(R\), given by the equation \(\rho=R\) in cylindrical polar coordinates \((\rho, \phi, z) .\) The mass is subject to just one external force, \(\mathbf{F}=-k r \hat{\mathbf{r}},\) where \(k\) is a positive constant, \(r\) is its distance from the origin, and \(\hat{\mathbf{r}}\) is the unit vector pointing away from the origin, as usual. Using \(z\) and \(\phi\) as generalized coordinates, find the Hamiltonian \(\mathcal{H}\). Write down and solve Hamilton's equations and describe the motion.

Short answer

The particle exhibits simple harmonic motion in the \(z\) direction and uniform circular motion in \(\phi\).

Step-by-step solution

Express the Potential Energy

The given force is \ \( \mathbf{F} = -k r \hat{\mathbf{r}} \). The potential energy \( U \) associated with a central force is given by integrating the force: \\[U(r) = - \int \mathbf{F} \cdot d\mathbf{r} = \frac{k}{2} r^2.\]Since \( \rho = R \), we have \( r^2 = R^2 + z^2 \). Therefore, the effective potential energy is \\[U(z) = \frac{k}{2} (R^2 + z^2).\]

Determine the Kinetic Energy

The kinetic energy \( T \) in cylindrical coordinates for a particle constrained to the surface of the cylinder is \\[T = \frac{1}{2} m (\dot{z}^2 + R^2 \dot{\phi}^2).\]Here, \( \dot{z} \) and \( \dot{\phi} \) are the time derivatives of \( z \) and \( \phi \).

Form the Lagrangian

The Lagrangian \( \mathcal{L} \) is given by \ \[\mathcal{L} = T - U = \frac{1}{2} m (\dot{z}^2 + R^2 \dot{\phi}^2) - \frac{k}{2} (R^2 + z^2).\]

Obtain Generalized Momenta

For \( z \), the momentum \( p_z \) is given by \[p_z = \frac{\partial \mathcal{L}}{\partial \dot{z}} = m \dot{z}.\]For \( \phi \), the momentum \( p_\phi \) is given by \[p_\phi = \frac{\partial \mathcal{L}}{\partial \dot{\phi}} = m R^2 \dot{\phi}.\]

Construct the Hamiltonian

The Hamiltonian \( \mathcal{H} \) is \[\mathcal{H} = \dot{z} p_z + \dot{\phi} p_\phi - \mathcal{L}.\]Substitute \( \dot{z} = p_z/m \) and \( \dot{\phi} = p_\phi/(mR^2) \) to get\[\mathcal{H} = \frac{p_z^2}{2m} + \frac{p_\phi^2}{2mR^2} + \frac{k}{2} (R^2 + z^2).\]

Write Hamilton's Equations

Hamilton's equations are \[\dot{z} = \frac{\partial \mathcal{H}}{\partial p_z} = \frac{p_z}{m}, \quad \dot{p_z} = -\frac{\partial \mathcal{H}}{\partial z} = -k z,\]\[\dot{\phi} = \frac{\partial \mathcal{H}}{\partial p_\phi} = \frac{p_\phi}{mR^2}, \quad \dot{p_\phi} = -\frac{\partial \mathcal{H}}{\partial \phi} = 0.\]

Solve Hamilton's Equations for Motion

The equation \( \dot{p_z} = -k z \) gives \( p_z = m \dot{z} = -k z t + C \).This is a simple harmonic equation that can be solved to find the motion in the \( z \) direction: \ \[ z(t) = A \cos(\omega t + \phi_0), \quad \omega = \sqrt{\frac{k}{m}}. \]For \( \phi \), since \( \dot{p_\phi} = 0 \), \( p_\phi \) is constant, leading to uniform circular motion: \[ \phi(t) = \phi(0) + \frac{p_\phi}{mR^2} t.\]

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a natural way to describe the position of a point in a three-dimensional space. Instead of using Cartesian coordinates which rely on (x, y, z), cylindrical coordinates focus on radial and angular positions. In this system, a point is defined by three values: \((\rho, \phi, z)\). Here, \(\rho\) is the radial distance from the z-axis, \(\phi\) is the angular displacement around the z-axis often measured from the positive x-axis, and \(z\) is the height above the xy-plane.

These coordinates are particularly useful when dealing with problems that have some form of rotational symmetry. For example, a particle moving along a cylinder fits perfectly into this system since we can easily describe its position with \(\rho = R\) (a constant) due to the cylinder's radius, alongside the variable \(\phi\) and \(z\). When working with problems involving rotation or circular motion, this system simplifies calculations. So if you picture a particle on the side of a smooth can, cylindrical coordinates easily describe its up-down and rotational motion.

Potential Energy

Potential energy is a measure of stored energy within a system due to its position or arrangement. For a particle subjected to an external force field described in this exercise, the force \( \mathbf{F} = -k r \hat{\mathbf{r}} \) hints at a central force, where potential energy varies with distance \(r\) from the origin.

Calculating the potential energy \(U\) involves integrating this force over space. Here, the integration gives \[U(r) = \frac{k}{2}r^2.\] When a particle is constrained to a cylinder surface, \(r\) remains dependent on \(z\) and radius\(R\), thus, \(U(z) = \frac{k}{2}(R^2 + z^2)\). Understanding potential energy enables us to determine how forces act upon a body and how this translates into movement energy-wise, influencing calculations of dynamic motion in further steps.

Kinetic Energy

Kinetic energy is the energy a particle possesses due to its motion. In cylindrical coordinates, for a particle constrained to a cylinder's surface, kinetic energy \(T\) is defined as the sum of the energies due to movement in each coordinate direction. The precise formula is given by:

\[T = \frac{1}{2} m (\dot{z}^2 + R^2 \dot{\phi}^2). \]

Here, \(\dot{z}\) and \(\dot{\phi}\) represent the time derivatives of \(z\) and \(\phi\), respectively. While \(\dot{z}\) denotes linear motion through vertical displacement, \(\dot{\phi}\) refers to angular motion around the cylinder, scaled by \(R\).
Knowing how kinetic energy behaves allows us to track how energy is spent or preserved as the mass moves - vital for setting up problems and moving on to determine the system's Hamiltonian.

Hamilton's Equations

Hamilton's equations provide a reformulation of classical mechanics, offering a pathway to solving problems in dynamics by providing a set of first-order differential equations. Derived from the Hamiltonian \(\mathcal{H}\), which encapsulates total energy within the system, these equations drive the state of a mechanical system over time.

Here, we derive the Hamiltonian for our constrained particle as: \[\mathcal{H} = \frac{p_z^2}{2m} + \frac{p_\phi^2}{2mR^2} + \frac{k}{2}(R^2 + z^2).\] This equation combines the kinetic and potential energies. The evolution of the system is then unveiled through Hamilton's equations:

• \(\dot{z} = \frac{\partial \mathcal{H}}{\partial p_z} = \frac{p_z}{m}\)
• \(\dot{p_z} = -\frac{\partial \mathcal{H}}{\partial z} = -k z\)
• \(\dot{\phi} = \frac{\partial \mathcal{H}}{\partial p_\phi} = \frac{p_\phi}{mR^2}\)
• \(\dot{p_\phi} = -\frac{\partial \mathcal{H}}{\partial \phi} = 0\)

These provide equations of motion that describe how the system dynamically evolves. Solving these lead us to solutions such as harmonic oscillation in \(z\) and constant angular velocity in \(\phi\), giving us a complete picture of the particle's motion.