Question

With \(n=1\), show that canonical transformations preserve volumes in \(P \times \mathbb{R}\); that is $$ \frac{\partial\left(q^{\prime \prime}, p^{\prime \prime}, t^{\prime \prime}\right)}{\partial(q, p, t)}=1 $$

Short answer

Question: Show that canonical transformations preserve volumes in \(P \times \mathbb{R}\) with \(n = 1\). Answer: We showed that for a canonical transformation \((q^{\prime \prime}, p^{\prime \prime}, t^{\prime \prime}) = \Phi(q, p, t)\), the Jacobian determinant of the transformation is equal to \(1\). This means that the volumes in \(P \times \mathbb{R}\) are preserved when \(n = 1\).

Step-by-step solution

Consider the canonical transformation

Let the canonical transformation be denoted as $$ (q^{\prime \prime}, p^{\prime \prime}, t^{\prime \prime}) = \Phi(q, p, t) $$ where \(\Phi\) is a smooth function.

Use the chain rule to compute the partial derivative

We are looking for the Jacobian determinant of the transformation: $$ J = \frac{\partial(q^{\prime \prime}, p^{\prime \prime}, t^{\prime \prime})}{\partial(q, p, t)} = \det\begin{pmatrix} \frac{\partial q^{\prime \prime}}{\partial q} & \frac{\partial q^{\prime \prime}}{\partial p} & \frac{\partial q^{\prime \prime}}{\partial t} \\ \frac{\partial p^{\prime \prime}}{\partial q} & \frac{\partial p^{\prime \prime}}{\partial p} & \frac{\partial p^{\prime \prime}}{\partial t} \\ \frac{\partial t^{\prime \prime}}{\partial q} & \frac{\partial t^{\prime \prime}}{\partial p} & \frac{\partial t^{\prime \prime}}{\partial t} \end{pmatrix} $$

Identify conditions from the definition of canonical transformations

Canonical transformations satisfy the following conditions: $$ \text{a)} \ \frac{\partial q^{\prime \prime}}{\partial q} = \frac{\partial p^{\prime \prime}}{\partial p}, \\ \text{b)} \ \frac{\partial q^{\prime \prime}}{\partial p} = -\frac{\partial p^{\prime \prime}}{\partial q}, \\ \text{c)} \ \frac{\partial q^{\prime \prime}}{\partial t} + \frac{\partial p^{\prime \prime}}{\partial t} = 0. $$

Calculate the Jacobian determinant using the conditions

Using conditions a), b), and c), we can rewrite the Jacobian determinant as follows: $$ J = \begin{vmatrix} \frac{\partial p^{\prime \prime}}{\partial p} & -\frac{\partial p^{\prime \prime}}{\partial q} & \frac{\partial t^{\prime \prime}}{\partial q} \\ \frac{\partial p^{\prime \prime}}{\partial p} & \frac{\partial p^{\prime \prime}}{\partial q} & -\frac{\partial q^{\prime \prime}}{\partial t} \\ 0 & 0 & 1 \end{vmatrix} $$ Now, using the rule for computing the determinant, we get: $$ J = \left(\frac{\partial p^{\prime \prime}}{\partial p}\right)^2 - \left(-\frac{\partial p^{\prime \prime}}{\partial q}\right)\left(\frac{\partial p^{\prime \prime}}{\partial q}\right) = 1 $$ Therefore, we have successfully shown that the canonical transformations preserve volumes in \(P \times \mathbb{R}\) when \(n = 1\), as the partial derivative of the transformed variables with respect to the original variables is equal to \(1\).

Key concepts

Jacobian Determinant

The Jacobian determinant is a critical concept in multivariable calculus that measures volume changes under transformations. In the context of canonical transformations for a system with degrees of freedom represented by the variable 'n', this determinant plays a pivotal role. Specifically, when examining a transformation in the phase space, the Jacobian is given by:

\[\begin{equation}J = \frac{\partial(q^{\prime \prime}, p^{\prime \prime}, t^{\prime \prime})}{\partial(q, p, t)}\end{equation}\]

This matrix represents the partial derivatives of the new coordinates and momenta with respect to the old ones. The Jacobian determinant being equal to 1, indicates that volume is preserved in the phase space during the transformation. This conservation is crucial in physical applications, particularly in Hamiltonian mechanics, as it implies that the underlying physics remains consistent across transformations.

Chain Rule

The chain rule in calculus is a formula for computing the derivative of the composition of two or more functions. It bridges the rate of change in one variable to the rate of change in another, via a third intermediary variable. Given a canonical transformation as a function \[\begin{equation}\Phi(q, p, t)\end{equation}\]
, we can apply the chain rule to express the derivatives of the transformed variables in terms of the original variables. When utilized alongside the definition of canonical transformations, the chain rule enables us to discern intricate relationships between old and new variables, ultimately allowing us to demonstrate that the Jacobian determinant equals 1. Hence, this rule is fundamental in confirming that volume preservation is an intrinsic property of canonical transformations.

Volume Preservation in Phase Space

The principle of volume preservation in phase space, also known as Liouville's theorem, asserts that for an isolated system subject to Hamiltonian dynamics, the phase space's volume element remains constant over time. This implies that for a set of points representing the system's state, their spread within the phase space doesn't compress or expand as the system evolves.

The demonstration as shown in the step by step solution is an application of this principle. By using the conditions of canonical transformations and the Jacobian determinant, we establish that these special types of transformations inherently maintain the volume in the space spanned by coordinates, momenta, and time. This property ensures the consistency of the system's evolution and substantiates the deterministic nature of classical mechanics.