Problem 9 - Classical Mechanics

Question

A particle of mass \(m\) and charge \(q\) is moving around a fixed point charge \(-q^{\prime}\left(q q^{\prime}>0\right)\), and in a uniform magnetic field \(\boldsymbol{B}\). The motion is confined to the plane perpendicular to \(\boldsymbol{B}\). Write down the Lagrangian function in polar co-ordinates rotating with the Larmor angular velocity \(\omega_{\mathrm{L}}=-q B / 2 m\) (see \(\left.\S 5.5\right) .\) Hence find the Hamiltonian function. Show that \(\varphi\) is ignorable, and interpret the conservation law. (Note that \(J_{z}\) is not a constant of the motion.)

Step-by-Step Solution

Answer

In conclusion, we have written the Lagrangian function for a charged particle in a magnetic field in rotating polar coordinates, found the Hamiltonian function, and showed that the azimuthal angle \(\varphi\) is ignorable. The conservation law associated with the ignorable coordinate indicates that the angular momentum of the particle projected onto the \(z\)-axis (\(p_\theta\)) is conserved during its motion in the presence of a magnetic field, even though the total angular momentum is not conserved due to the torque from the magnetic field.

Step 1: Write down the Lagrangian function in rotating polar coordinates

First, we need to express the position and velocity of the particle in rotating polar coordinates. Let \(\boldsymbol{r}\) and \(\boldsymbol{v}\) be the position and velocity of the particle in the lab frame. In polar coordinates, they are given by: \begin{align*} \boldsymbol{r} &= r\hat{r}\\ \boldsymbol{v} &= \dot{r}\hat{r} + r\dot{\theta}\hat{\theta} \end{align*} In the rotating frame with angular velocity \(\omega_{L}\), the position remains the same but the velocity vector is modified by the Coriolis effect. The velocity of a point in the rotating frame is given by: \(\boldsymbol{v'} = \boldsymbol{v} - \boldsymbol{\omega}_{L} \times \boldsymbol{r}\) So, we have: \(\boldsymbol{v'} = \dot{r}\hat{r} + r\dot{\theta}\hat{\theta} + \frac{qB}{2m}r\hat{\theta}\) Now, the kinetic energy of the particle is given by: \(T = \frac{1}{2}m\boldsymbol{v'}^2 = \frac{1}{2}m(\dot{r}^2 + r^2(\dot{\theta} + \frac{qB}{2m})^2)\) The potential energy is given by the electrostatic interaction between the charges: \(U = \frac{qq'}{4\pi\epsilon_{0}r}\) The Lagrangian function is then: \(L = T - U = \frac{1}{2}m(\dot{r}^2 + r^2(\dot{\theta} + \frac{qB}{2m})^2) - \frac{qq'}{4\pi\epsilon_{0}r}\)

Step 2: Find the Hamiltonian function

We will use the following formula to find the Hamiltonian function from the Lagrangian: \(H = p_{r}\dot{r} + p_{\theta}\dot{\theta} - L\) In order to find the conjugate momenta, we take the derivatives of the Lagrangian with respect to \(\dot{r}\) and \(\dot{\theta}\): \begin{align*} p_{r} &= \frac{\partial L}{\partial\dot{r}} = m\dot{r}\\ p_{\theta} &= \frac{\partial L}{\partial\dot{\theta}} = mr^2(\dot{\theta} + \frac{qB}{2m}) \end{align*} Next, we solve for \(\dot{r}\) and \(\dot{\theta}\): \begin{align*} \dot{r} &= \frac{p_r}{m}\\ \dot{\theta} &= \frac{p_{\theta}}{mr^2} - \frac{qB}{2m} \end{align*} Substitute these expressions back into the formula for the Hamiltonian: \(H = p_{r}\frac{p_{r}}{m} + p_{\theta}\left(\frac{p_{\theta}}{mr^2} - \frac{qB}{2m}\right) - L\) After simplification we get: \(H = \frac{p_{r}^2}{2m} + \frac{p_{\theta}^2}{2mr^2} - \frac{qBp_{\theta}}{2m}- \frac{1}{2}m r^{2}\left(\frac{qB}{2m}\right)^{2} +\frac{qq'}{4\pi\epsilon_{0} r}\)

Step 3: Show \(\varphi\) is ignorable and interpret the conservation law

Notice that the Lagrangian does not depend on \(\varphi\), so we can say that \(\varphi\) is ignorable. Consequently, the conjugate momentum associated with \(\varphi\), which is \(p_\theta\), is conserved. This conservation law means that the angular momentum of the particle projected onto the \(z\)-axis is conserved during the motion. It is important to note that the total angular momentum \(J_{z}\) is not conserved because the magnetic field \(\boldsymbol{B}\) applies a torque on the particle. However, the projection of the angular momentum onto the \(z\)-axis, given by \(p_{\theta}\), is conserved due to the ignorable \(\varphi\) coordinate.

Key Concepts

Classical Mechanics

Classical mechanics serves as the foundation of physics, describing the motion of bodies under the influence of a system of forces. The central principle of classical mechanics is Newton's second law, which states that the acceleration of a body is directly proportional to the net force acting upon it and inversely proportional to its mass.

However, sometimes it's helpful to use alternative formulations of mechanics, especially when dealing with complex motions or constraints. This is where the Lagrangian and Hamiltonian formulations come into play. They are powerful tools used to derive the equations of motion for any mechanical system. The Lagrangian is a function representing the difference between the kinetic and potential energies, while the Hamiltonian is the total energy of the system, expressed in terms of positions and momenta.

Understanding these concepts is crucial in solving problems like the motion of a charged particle in a magnetic field, particularly in advanced topics such as charged particle dynamics, conservation laws, and rotating reference frames.

Charged Particle Dynamics

The dynamics of charged particlesis a critical topic in classical mechanics, especially relating to electromagnetic fields. Charged particles, such as electrons or protons, experience forces due to electric and magnetic fields, which significantly influences their trajectory and energy states.

In the given exercise, we see a particle of charge q moving in the presence of both a static electric field due to another charge q' and a uniform magnetic field B. When dealing with such a system, it's important to account for the Lorentz force, which combines electric and magnetic forces acting on the particle. However, in the Lagrangian and Hamiltonian mechanics framework, we express dynamics in terms of energy rather than forces.

As part of the solution, the particle's motion in the plane perpendicular to the magnetic field is described using polar coordinates, which simplifies the equations of motion. The Coriolis effect, stemming from a rotating frame of reference, also comes into play and must be handled carefully.

Conservation Laws

Conservation laws are pivotal in physics because they help predict the outcomes of interactions in a system without requiring the detailed knowledge of the forces at play. These laws stem from symmetries in physical laws and are manifest in Lagrangian and Hamiltonian mechanics as conserved quantities.

For instance, the conservation of angular momentum, a principle of paramount importance in mechanics, dictates that the angular momentum of a system remains constant if no external torque acts on it. In the context of our exercise, even though the total angular momentum is not conserved due to the magnetic torque, the component of angular momentum in the direction of the magnetic field remains a constant of motion, showcasing the principle of conservation.

The conservation law associated with an 'ignorable' coordinate -- one that does not appear in the Lagrangian -- implies that the conjugate momentum to that coordinate is conserved. This property offers significant simplifications when analyzing the motion of systems and deriving the Hamiltonian.

Rotating Reference Frames

Rotating reference frames are utilized in mechanics to simplify problems involving rotational motion. When analyzing a system from a rotating frame, additional fictitious forces, including the Coriolis and centrifugal forces, need to be considered. These forces are not ‘real’ in the sense that they do not result from physical interactions, but rather from the acceleration of the frame itself.

In the exercise at hand, the particle's motion is described in polar coordinates that are rotating with a specific angular velocity called the Larmor frequency. This approach simplifies the problem because the Larmor precession is a common phenomenon in charged particle dynamics in magnetic fields. Mathematically, we adjust the velocity and kinetic energy in the Lagrangian to account for the rotating frame effects, leading to accurate descriptions of the system's dynamics. Understanding and correctly applying the principles of rotating reference frames are essential in solving problems related to celestial mechanics, atmospheric sciences, and many other fields of physics.