Canonical conversion

Canonical transformations are a fundamental concept in the field of classical mechanics, particularly in the Hamiltonian framework. They provide powerful ways to simplify complex mechanical problems and ensure that transformations do not change the form of the Hamiltonian equations. This preservation of form under transformation makes them particularly valuable for solving problems where a direct solution may not be readily apparent.

Introduction to canonical transform

In classical mechanics, we often deal with configurations of systems that evolve over time. The Lagrangian and Hamiltonian formalisms provide powerful tools for analyzing these dynamics. While the Lagrangian approach uses generalized coordinates and velocities, the Hamiltonian formulation uses generalized coordinates and momenta, making it suitable for applying canonical transformations.

Canonical transformations are transformations in Hamiltonian mechanics that preserve the form of Hamilton's equations. In essence, they are a transformation of variables from the old set of coordinates and their conjugate momenta, denoted as (q, p), to a new set of variables, (Q, P), in such a way that the new coordinates and momenta satisfy the Hamiltonian formalism in the same way as the old ones.

Hamiltonian system

To understand canonical transformations, we must first revisit the Hamiltonian formulation. The Hamiltonian of a system is defined as:

H = ∑(p_i * dq_i/dt) – L

where L is the Lagrangian, q_i are the generalized coordinates, and p_i are the conjugate momenta defined by:

p_i = ∂L/∂(dq_i/dt)

The equations of motion in a Hamiltonian system are given by the Hamiltonian equations:

dq_i/dt = ∂H/∂p_i dp_i/dt = -∂h/∂q_i

Definition of canonical transformation

A transformation from (q, p) to (Q, P) is canonical if it preserves the form of Hamilton's equations. More formally, it preserves the symplectic structure of the phase space.

Mathematically, this means that if the transformation is given by the following function:

q = q(q, p, t) P = P(q, p, t)

Then there exists a generating function that associates the new variable (Q, P) with the old one (q, p). Generating functions can be of different types, and they define the canonical transformation.

Types of generating functions

There are four main types of generating functions used in canonical transformations, classified based on the variables they depend on:

1. Generating function of the first kind, F₁(q, Q, t): This function depends on the old coordinate q and the new coordinate Q.
2. Second kind of generating function, F₂(q, P, t): it depends on the old coordinate q and the new momentum P.
3. Third kind of generating function, F₃(p, Q, t): It depends on the old momentum p and the new coordinate Q.
4. Fourth kind of generating function, F₄(p, P, t): It depends on the old and new momentum p and P respectively.

Expressions to generate functions

Let's look at the expressions for each of these generating functions and see how they combine the old and new variables:

1. F₁(q, Q, t):

    p_i = ∂F₁/∂q_i P_i = -∂F₁/∂Q_i k = h + ∂F₁/∂t 

2. F₂(q, P, t):

    p_i = ∂F₂/∂q_i Q_i = ∂F₂/∂P_i k = h + ∂F₂/∂t 

3. F₃(p, q, t):

    q_i = -∂F₃/∂p_i P_i = -∂F₃/∂Q_i k = h + ∂F₃/∂t 

4. F₄(p, p, t):

    q_i = -∂F₄/∂p_i Q_i = ∂F₄/∂P_i k = h + ∂F₄/∂t 

In each case, K is the new Hamiltonian expressed in terms of the new canonical variables. If ∂F/∂t = 0, then the transformation is called time-independent.

Example of canonical transformation

Let's consider a simple example to show how canonical transformations work. Consider a one-dimensional system where the Hamiltonian is:

h(q, p) = (p²/2m) + v(q)

We want to perform the canonical transformation into new variables (Q, P) using a generating function of the second kind:

F₂(q, P, t) = q * P

Using the relations given for F₂(q, P, t), we have:

P = ∂F₂/∂q = P q = ∂F₂/∂P = q

Thus, the transformation is simple:

q = q P = P

This is a trivial example where the transformation is essentially the identity transformation. However, it shows how the variables change under the transformation, and we can find the new Hamiltonian if necessary.

More complex examples

Consider a harmonic oscillator with the Hamiltonian:

h(q, p) = (p²/2m) + (1/2)mω²q²

Suppose we want to perform a canonical transformation using a generating function:

F₂(q, P) = mωq²/2 * cot(P)

This gives us new variables:

P = ∂F₂/∂q = mωq * cot(P) Q = ∂F₂/∂P = -mωq²/2 * csc²(P)

This example demonstrates how canonical transformations can become more complex and require careful manipulation to achieve the desired simplification or problem-solving benefit.

Applications of the canonical transform

Canonical transformations are incredibly useful in various advanced areas of theoretical physics and classical mechanics:

• Simplifying calculations: By making changes to a set of variables where complex interactions are simplified, calculations can be greatly simplified. For example, the action-angle variables used in solving periodic systems use canonical transformations.
• Identifying constants of motion: A skillful choice of generating functions and canonical transformations can reveal conserved quantities in the system, helping to solve the equations.
• Quantum mechanics transformations: For many quantum mechanical systems the classical background is represented in phase space, where canonical transformations help to understand conversions between different pictures or bases.

Visual representation of canonical transformations

Below is an illustration showing the transformation from the old variable set (q, p) to the new set (Q, P). Note that each transformation will preserve the underlying symplectic structure of the phase space when transitioning between these coordinate systems.

The left block shows the original system in terms of the variables (q, p), while the right block shows the same system represented in the transformed coordinates (Q, P). The arrows indicate the direction of the canonical transformation.

Summary

Canonical transformations are important in Hamiltonian mechanics because they preserve the structure of Hamilton's equations. By using different generating functions, these transformations can be adapted to suit different physics problems. They streamline calculations, lead to simplified interactions, and often reveal conservation laws, making them important in both classical and quantum mechanics studies.

Ultimately, understanding and using canonical transformations enables scientists and engineers to tackle complex physical systems, and provides deeper insights into their behavior and evolution.

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