Question

Here is a simple example of a canonical transformation that illustrates how the Hamiltonian formalism lets one mix up the \(q^{\prime}\) s and the \(p\) 's. Consider a system with one degree of freedom and Hamiltonian \(\mathcal{H}=\mathcal{H}(q, p)\). The equations of motion are, of course, the usual Hamiltonian equations \(\dot{q}=\partial \mathcal{H} / \partial p\) and \(\dot{p}=-\partial \mathcal{H} / \partial q .\) Now consider new coordinates in phase space defined as \(Q=p\) and \(P=-q .\) Show that the equations of motion for the new coordinates \(Q\) and \(P\) are \(\dot{Q}=\partial \mathcal{H} / \partial P\) and \(\dot{P}=-\partial \mathscr{H} / \partial Q ;\) that is, the Hamiltonian formalism applies equally to the new choice of coordinates where we have exchanged the roles of position and momentum.

Short answer

The transformation is canonical, preserving the Hamiltonian structure.

Step-by-step solution

Identify Original Equations

For the original system with coordinates \(q\) and \(p\), the equations of motion are determined by the Hamiltonian \(\mathcal{H}\): \[ \dot{q} = \frac{\partial \mathcal{H}}{\partial p} \] and \[ \dot{p} = -\frac{\partial \mathcal{H}}{\partial q}. \] These are the standard Hamiltonian equations describing evolution in the original coordinates.

Define New Coordinates

We introduce a new set of coordinates \(Q = p\) and \(P = -q\). This means that in the new coordinate system, the position \(Q\) is actually the momentum \(p\) of the original system, and \(P\) is the negative of the original position \(q\). The role of \(q\) and \(p\) has been interchanged in terms of the assignment of new coordinates \(Q\) and \(P\).

Determine New Equations of Motion

Using the definition of the new coordinates, substitute into the original equations of motion. We have \(q = -P\) and \(p = Q\). The equations become: \[ \dot{Q} = \dot{p} = -\frac{\partial \mathcal{H}}{\partial (-P)} = \frac{\partial \mathcal{H}}{\partial P}, \] and \[ \dot{P} = -\dot{q} = -\frac{\partial \mathcal{H}}{\partial Q}. \] These equations follow the same form as the Hamiltonian equations when interpreted with the new variables.

Verify Hamiltonian Structure

In the new coordinate system, the transformed equations \(\dot{Q} = \frac{\partial \mathcal{H}}{\partial P}\) and \(\dot{P} = -\frac{\partial \mathcal{H}}{\partial Q}\) have the same structure as the original Hamilton's equations. This confirms that the Hamiltonian formalism is preserved under the change of variables, demonstrating the canonical nature of the transformation.

Key concepts

Hamiltonian Mechanics

Hamiltonian Mechanics is a powerful framework used in classical mechanics to describe the evolution of a physical system. It provides a different perspective compared to Newtonian mechanics by focusing on energy rather than force. The core idea revolves around the Hamiltonian function, denoted as \( \mathcal{H}(q, p) \), which represents the total energy of the system, combining both kinetic and potential energy.
In Hamiltonian mechanics:

• We exploit the position \(q\) and momentum \(p\) of the system as variables.
• The equations of motion are derived based on partial derivatives of the Hamiltonian function.

These equations of motion are expressed as:\[\dot{q} = \frac{\partial \mathcal{H}}{\partial p}, \quad \dot{p} = -\frac{\partial \mathcal{H}}{\partial q}. \]The beauty of Hamiltonian mechanics lies in its ability to simplify complex problems, especially when dealing with higher dimensions and when transformations, such as canonical transformations, are involved. This approach lays the foundation for more advanced theories like Quantum Mechanics, making it an essential concept in the study of dynamics.

Phase Space

Phase Space is an essential concept in Hamiltonian mechanics and a powerful tool used to visualize and analyze the state of a mechanical system. Phase space is a multidimensional space where each state of the system is represented as a point.

• Each point in phase space corresponds to a unique combination of position \(q\) and momentum \(p\).
• The evolution of the system is depicted by the trajectory of these points as time progresses.

Phase space provides a complete description of the system's dynamics, as knowing both the position and momentum of each degree of freedom equips us with the information needed to determine its future and past states.
In practical applications, analyzing phase space helps in:

• Understanding the behavior and stability of a system.
• Identifying conserved quantities and symmetries.
• Facilitating the analysis of complex systems where other methods would be cumbersome.

Phase space is also integral to statistical mechanics where it aids in describing systems with a large number of particles, forming the backbone for concepts like the Gibbs ensemble.

Equations of Motion

In the context of Hamiltonian mechanics, the Equations of Motion are fundamental as they describe how the state of a physical system evolves over time. These are expressed as Hamilton's Equations, which link the change in system coordinates to its Hamiltonian. They play a crucial role in understanding the dynamics of the system in phase space.
The standard form of Hamilton's Equations is:\[\begin{align*}\dot{q} &= \frac{\partial \mathcal{H}}{\partial p}, \\dot{p} &= -\frac{\partial \mathcal{H}}{\partial q}.\end{align*}\]These equations reveal how the position \(q\) and momentum \(p\) are influenced by the Hamiltonian function:

• \( \dot{q} \) represents the rate at which position changes, determined by the partial derivative of the Hamiltonian with respect to momentum.
• \( \dot{p} \) indicates the rate of change in momentum, corresponding to the negative derivative of the Hamiltonian with respect to position.

Understanding these equations provide insights into the conservation of energy and symplectic structure—key components in classical mechanics. Moreover, when systems undergo transformations, such as the canonical transformations addressed in the exercise, Hamilton's Equations ensure that the integrity and dynamics of the system remain intact.