Find the Hamiltonian for a charged particle in electric and magnetic fields in cylindrical polars, starting from the Lagrangian function (10.29). Show that in the case of an axially symmetric, static magnetic field, described by the single component \(A_{\varphi}(\rho, z)\) of the vector potential, it takes the form \(H=\frac{1}{2 m}\left(p_{z}^{2}+p_{\rho}^{2}+\frac{\left(p_{\varphi}-q \rho A_{\varphi}\right)^{2}}{\rho^{2}}\right)\) (Note: Remember that the subscripts \(\varphi\) on the generalized momentum \(p_{\varphi}\) and on the component \(A_{\varphi}\) mean different things.)

In classical mechanics, the Lagrangian function is a fundamental concept that captures the dynamics of a system. It is defined as the difference between the kinetic and potential energies of the system, denoted as L = T - V. In the context of a charged particle moving in electromagnetic fields, the Lagrangian includes not only the mechanical kinetic energy but also an interaction term that incorporates the effects of electric (\textbf{E}) and magnetic (\textbf{B}) fields.

Specifically, the Lagrangian for a charged particle in an electromagnetic field is given by \(L=\frac{1}{2} m(\dot{\rho}^{2}+\rho^{2} \dot{\varphi}^{2}+\dot{z}^{2})+q(\phi-\frac{1}{c} \mathbf{v} \cdot \mathbf{A})\), where \(m\) is the mass of the particle, \(q\) is its charge, \(\phi\) is the electric potential, \(\mathbf{A}\) is the magnetic vector potential, and \(\mathbf{v}\) is the velocity of the particle. The dot over a coordinate denotes its time derivative, and it is essential to express the velocity \(\mathbf{v}\) in the appropriate coordinate system, here being cylindrical coordinates.

Generalized momenta are crucial in constructing the Hamiltonian formalism from Lagrangian mechanics. They are formally defined as the partial derivatives of the Lagrangian with respect to the time derivatives of generalized coordinates: \(p_i = \frac{\partial L}{\partial \dot{q_i}}\) where \(\dot{q_i}\) is the time derivative of the \(i^{th}\) generalized coordinate.

In our specific example, the generalized momenta in cylindrical coordinates are found as follows:

• \(p_{\rho}=\frac{\partial L}{\partial \dot{\rho}} = m\dot{\rho}\)
• \(p_{\varphi}=\frac{\partial L}{\partial \rho \dot{\varphi}} = m \rho^{2} \dot{\varphi} + \frac{q}{c}\rho A_{\varphi}\)
• \(p_{z}=\frac{\partial L}{\partial \dot{z}} = m\dot{z}\)

By computing these, the dynamics of the particle in the direction of each coordinate can be quantitatively described.

The Legendre transformation is a mathematical process that transitions from the Lagrangian to the Hamiltonian formalism. It involves changing the description of the system from one using velocities (time derivatives of positions) to one using momenta. The transformation is given by the equation \(H = \sum_{i}(p_i \dot{q}_i) - L\), where \(H\) is the Hamiltonian, \(p_i\) are the generalized momenta, \(\dot{q}_i\) are the time derivatives of generalized coordinates, and \(L\) is the Lagrangian.

This process results in a new function, the Hamiltonian, that is particularly useful for systems with constraints or when dealing with quantum mechanics. The Hamiltonian represents the total energy of the system and plays a similar role as the Lagrangian in that it can be used to derive equations of motion, but it operates within a different framework.

Cylindrical coordinates \((\rho, \varphi, z)\) are a three-dimensional coordinate system that extends polar coordinates into three dimensions by adding a height dimension, \(z\). Here, \(\rho\) is the radial distance from the z-axis, \(\varphi\) is the angular coordinate, and \(z\) is the height above the xy-plane.

For a particle moving in a 3D space under the influence of forces that have axial symmetry, like those created by an axially symmetric magnetic field, cylindrical coordinates offer a natural way to describe the motion. They simplify many calculations, especially when the Lagrangian or Hamiltonian has terms that are explicitly independent of the angular coordinate \(\varphi\), which signifies conservation of angular momentum.

An axially symmetric magnetic field is one where the magnetic field strength and direction remain constant along the axis of symmetry. In such fields, the vector potential \(\mathbf{A}\) can often be simplified. For example, it may only have a component in the azimuthal direction \(A_{\varphi}(\rho, z)\) that depends solely on the radial distance \(\rho\) and height \(z\).

For a charged particle moving in an axially symmetric magnetic field, its dynamics can be factored into simpler expressions, specifically within the Hamiltonian. The azimuthal component of the velocity \(\rho \dot{\varphi}\) interacts with the magnetic field through the vector potential, leading to a term in the Hamiltonian that resembles the kinetic energy for angular motion but is adjusted by the presence of the magnetic field: \(\frac{(p_{\varphi}-q \rho A_{\varphi})^{2}}{\rho^{2}}\). This represents the modified angular momentum of the charged particle due to the magnetic field interaction.