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Chapter 7: Problem 6

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

Expert verified
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

01

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.
02

Use the chain rule to compute the partial derivative

We are looking for the Jacobian determinant of the transformation: $$ J = \frac{\partial(q^{\prime \prime}, p^{\prime \prime}, t^{\prime \prime})}{\partial(q, p, t)} = \det\begin{pmatrix} \frac{\partial q^{\prime \prime}}{\partial q} & \frac{\partial q^{\prime \prime}}{\partial p} & \frac{\partial q^{\prime \prime}}{\partial t} \\ \frac{\partial p^{\prime \prime}}{\partial q} & \frac{\partial p^{\prime \prime}}{\partial p} & \frac{\partial p^{\prime \prime}}{\partial t} \\ \frac{\partial t^{\prime \prime}}{\partial q} & \frac{\partial t^{\prime \prime}}{\partial p} & \frac{\partial t^{\prime \prime}}{\partial t} \end{pmatrix} $$
03

Identify conditions from the definition of canonical transformations

Canonical transformations satisfy the following conditions: $$ \text{a)} \ \frac{\partial q^{\prime \prime}}{\partial q} = \frac{\partial p^{\prime \prime}}{\partial p}, \\ \text{b)} \ \frac{\partial q^{\prime \prime}}{\partial p} = -\frac{\partial p^{\prime \prime}}{\partial q}, \\ \text{c)} \ \frac{\partial q^{\prime \prime}}{\partial t} + \frac{\partial p^{\prime \prime}}{\partial t} = 0. $$
04

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 \(1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A particle \(P\) of mass \(m\) is moving in the plane under the influence of two inverse-square-law forces: \(\lambda(P A)^{-2}\) directed towards the point \(A\) and \(\lambda(P B)^{-2}\) directed towards the point \(B\), where \(A\) and \(B\) are separated by a distance \(2 b\). Solve the Hamilton-Jacobi equation in the coordinates \(\theta\) and \(\varphi\), where $$ 2 b \cosh \varphi=P A+P B, \quad 2 b \cos \theta=P A-P B . $$

Consider a system with one degree of freedom and Hamiltonian \(h(q, p, t) .\) Show that the dynamical trajectories in \(P T=\mathbb{R}^{3}\) are tangent to the vector field $$ \boldsymbol{x}=\frac{\partial h}{\partial p} i-\frac{\partial h}{\partial q} j+\boldsymbol{k} $$ where \(\boldsymbol{i}, \boldsymbol{j}\), and \(\boldsymbol{k}\) are unit vectors along the \(q, p\), and \(t\) axes. Let \(\Sigma\) be a surface in \(\mathbb{R}^{3}\) given by $$ p=\frac{\partial S}{\partial q} $$ where \(S=S(q, t)\). Show that if \(S\) is a solution of the HamiltonJacobi equation, then \(\boldsymbol{x}\) is tangent to \(\Sigma\). Show conversely that if \(\boldsymbol{x}\) is tangent to \(\Sigma\), then $$ \frac{\partial}{\partial q}\left[\frac{\partial S}{\partial t}+h\left(q, \frac{\partial S}{\partial q}, t\right)\right]=0 $$

Show that if \(q_{a}^{\prime \prime}=q_{a}^{\prime \prime}(q)\) and $$ p_{a}=\frac{\partial q_{b}^{\prime \prime}}{\partial q_{a}} p_{b}^{\prime \prime} $$ then the transformation from \(q_{a}, p_{a}, t\) to \(q_{a}^{\prime \prime}, p_{a}^{\prime \prime}, t^{\prime \prime}=t\) is canonical. Show that a generating function is \(F=q_{a}^{\prime \prime} p_{a}^{\prime \prime}\).

Obtain Hamilton's equations for a particle moving in space under an inverse- square-law central force, taking the \(q_{a}\) s to be spherical polar coordinates.

\({ }^{\dagger}\) The motion of a particle in three dimensions under an inversesquare-law force is governed by the Lagrangian $$ L=\frac{1}{2} m \dot{\boldsymbol{r}} . \dot{\boldsymbol{r}}+\frac{k}{r}, $$ where \(r=|\boldsymbol{r}|\). Show by vector methods, or otherwise, that the components \(J_{i}, R_{i}\) of the two vectors $$ J=m \boldsymbol{r} \wedge \dot{\boldsymbol{r}} \quad \text { and } \quad \boldsymbol{R}=\dot{\boldsymbol{r}} \wedge \boldsymbol{J}-\frac{k \boldsymbol{r}}{r} $$ are constants of the motion. Show that \(\left[J_{1}, r\right]=0\). Find \(\left[J_{1}, J_{2}\right]\) and \(\left[J_{1}, R_{2}\right]\) in terms of components of \(\boldsymbol{J}\) and \(\boldsymbol{R}\).
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