Show that the map \(A: \mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}\) sending \((\mathbf{p}, \mathbf{q}) \rightarrow(\mathbf{P}(\mathbf{p}, \mathbf{q}), \mathbf{Q}(\mathbf{p}, \mathbf{q}))\) is canonical if and only if the Poisson brackets of any two functions in the variables \((\mathbf{p}, \mathbf{q})\) and \((\mathbf{P}, \mathbf{Q})\) coincide: $$ (F, H)_{\mathbf{p}, \mathbf{q}}=\frac{\partial H}{\partial \mathbf{p}} \frac{\partial F}{\partial \mathbf{q}}-\frac{\partial H}{\partial \mathbf{q}} \frac{\partial F}{\partial \mathbf{p}}=\frac{\partial H}{\partial \mathbf{P}} \frac{\partial F}{\partial \mathbf{Q}}-\frac{\partial H}{\partial \mathbf{Q}} \frac{\partial F}{\partial \mathbf{P}}=(F, H)_{\mathbf{p}, \mathbf{q}} $$ Solution. Let \(A\) be canonical. Then the symplectic structures \(d \mathbf{p} \wedge d \mathbf{q}\) and \(d \mathbf{P} \wedge d \mathbf{Q}\) coincide. But the definition of the Poisson bracket \((F, H)\) was given invariantly in terms of the symplectic structure; it did not involve the coordinates. Therefore, $$ (F, H)_{\mathbf{p}, \mathbf{q}}=(F, H)=(F, H)_{\mathbf{P}, \mathbf{Q}} . $$ Conversely, suppose that the Poisson brackets \(\left(P_{i}, Q_{j}\right)_{\mathbf{p}, \mathbf{q}}\) have the standard form of Problem \(2 .\) Then, clearly, \(d \mathbf{P} \wedge d \mathbf{Q}=d \mathbf{p} \wedge d \mathbf{q}\), i.e., the map \(A\) is canonical.

Step by step solution

01

Assume that the map \(A: \mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}\) sending \((\mathbf{p}, \mathbf{q}) \rightarrow(\mathbf{P}(\mathbf{p}, \mathbf{q}), \mathbf{Q}(\mathbf{p}, \mathbf{q}))\) is canonical. We need to show that \((F, H)_{\mathbf{p}, \mathbf{q}}=(F, H)_{\mathbf{P}, \mathbf{Q}}\). #Step 2: Show that Poisson brackets coincide when A is canonical#

Since A is canonical, then the symplectic structures \(d \mathbf{p} \wedge d \mathbf{q}\) and \(d \mathbf{P} \wedge d \mathbf{Q}\) coincide by definition. The Poisson bracket \((F, H)\) is given by the symplectic structure, which does not involve the coordinates. Therefore, $$ (F, H)_{\mathbf{p}, \mathbf{q}}=(F, H)=(F, H)_{\mathbf{P}, \mathbf{Q}}. $$ #Step 3: Assume that Poisson brackets coincide for both variables#

02

Now we assume that the Poisson brackets \((F, H)_{\mathbf{p}, \mathbf{q}}=(F, H)_{\mathbf{P}, \mathbf{Q}}\) where $$ \frac{\partial H}{\partial \mathbf{p}}\frac{\partial F}{\partial \mathbf{q}}-\frac{\partial H}{\partial \mathbf{q}}\frac{\partial F}{\partial \mathbf{p}}=\frac{\partial H}{\partial \mathbf{P}}\frac{\partial F}{\partial \mathbf{Q}}-\frac{\partial H}{\partial \mathbf{Q}}\frac{\partial F}{\partial \mathbf{P}}. $$ We need to show that the map A is canonical. #Step 4: Show that map A is canonical when Poisson brackets coincide#

Suppose that the Poisson brackets \((P_{i}, Q_{j})_{\mathbf{p}, \mathbf{q}}\) have the standard form. Then, from our assumption, the symplectic structures are equal: \(d \mathbf{P} \wedge d \mathbf{Q}=d \mathbf{p} \wedge d \mathbf{q}\). Hence, the map A is canonical. Thus, we have shown that the map A is canonical if and only if the Poisson brackets of any two functions in the variables \((\mathbf{p}, \mathbf{q})\) and \((\mathbf{P}, \mathbf{Q})\) coincide.

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Key Concepts

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

Poisson Brackets

The Poisson Bracket is a critical concept in classical mechanics, particularly when studying Hamiltonian systems. It is a mathematical operator that measures the infinitesimal change of one observable in a direction of another. For two functions, F and H, their Poisson bracket is defined by:

$$ (F, H)_{\textbf{p}, \textbf{q}} = \frac{abla H}{abla \textbf{p}} \frac{abla F}{abla \textbf{q}} - \frac{abla H}{abla \textbf{q}} \frac{abla F}{abla \textbf{p}} $$

In the context of canonical maps, the coincidence of Poisson brackets between the old variables \textbf{p}, \textbf{q} and transformed variables \textbf{P}, \textbf{Q} indicates the preservation of the Hamiltonian structure of the system upon transformation. This leads to the classical result that dynamics are invariant under canonical transformations, ensuring that the equations of motion are form-invariant.

Symplectic Structure

Symplectic structure is a geometric framework essential for the mathematical formulation of Hamiltonian mechanics. It can be expressed using differential forms, specifically the symplectic form, which is a non-degenerate, closed differential two-form. In coordinates, it's often written as:

$$ d \textbf{p} \text{\textbackslash}wedge d \textbf{q} $$

Symplectic structure provides the language to speak about how volumes in phase space are conserved and how they can encode the physics of the system. In the case of our problem, the coincidence of symplectic structures under transformation is a signature of the map being canonical—preserving the intrinsic geometric and physical properties of the phase space.

Classical Mechanics

In classical mechanics, the physical laws governing the motion of bodies are based on two major formulations: Newtonian and Hamiltonian mechanics. While the Newtonian approach is more commonly taught initially, Hamiltonian mechanics offers a powerful toolset for more complex and abstract investigations with the use of Poisson brackets, symplectic structures, and differential forms. It allows for the analysis of systems with many degrees of freedom and is intimately related to symmetries and conservation laws through Noether's theorem. Canonical maps maintain the structure of Hamiltonian mechanics and therefore are critical in preserving the equations of motion in a transformed coordinate system.

Differential Forms

Differential forms play an important role in modern theoretical physics, particularly in the context of symplectic geometry. A differential form is a geometrical object that is integrable over a manifold, enabling calculations of quantities like flux or oriented volume. They generalize the concepts of vectors and scalars by allowing for the integration over arbitrary smooth manifolds. In the language of classical mechanics, differential forms can represent the symplectic structure of the phase space, encoding conserved properties across transformations.

For instance, the conservation of the symplectic structure in a canonical map, which can be written using differential forms as a two-form $$ d \textbf{p} \text{\textbackslash}wedge d \textbf{q} = d \textbf{P} \text{\textbackslash}wedge d \textbf{Q} $$

is vital in ensuring that the physics represented by this geometry do not change as we move between different descriptions or coordinate systems of our system.