Most General form of Canonical Transformation

In summary, to find the most general form of the canonical transformation for the given equations, one must evaluate the Poisson brackets and determine constraints on the functions f, g, and h. These constraints must satisfy the conditions of having a canonical transformation, such as $\{Q,Q\}=\{P,P \}=0$ and $\{Q,P \}=1$. However, it may be difficult to determine the most general form without a specific relationship between (Q,P) and (q,p).
  • #1
kolawoletech
4
0
How do I go about finding the most general form of the canonical transformation of the form
Q = f(q) + g(p)
P = c[f(q) + h(p)]
where f,g and h are differential functions and c is a constant not equal to zero. Where (Q,P) and (q,p) represent the generalised cordinates and conjugate momentum in the new and old system
 
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  • #2
Isn't this "homework" of some kind? You should post it in the Homework and Coursework section! Anyway, here's some hint:

I'd try to determine constraints on the functions by evaluating the Poisson brackets which must be
$$\{Q,Q\}=\{P,P \}=0, \quad \{Q,P \}=1$$
in order to have a canonical transformation.
 
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Likes AlphaCentaury
  • #3
I am done all that with certain kinds of relationship between (Q,P) and (q,p) but I am unable to do so with this general formula that does not give the function itself
 

Related to Most General form of Canonical Transformation

1. What is the purpose of a Canonical Transformation?

A Canonical Transformation is a mathematical method used in classical mechanics to transform a set of coordinates and momenta into a new set that preserves the Hamiltonian structure of the system. This allows for simplification of the equations of motion and the identification of conserved quantities, making it easier to analyze and solve physical problems.

2. What is the difference between a Canonical Transformation and a Symplectic Transformation?

A Canonical Transformation is a special case of a Symplectic Transformation, which is a transformation that preserves the symplectic structure of a system. The symplectic structure is a mathematical structure that describes the evolution of a system over time. A Canonical Transformation is also a one-to-one map, meaning that there is a unique correspondence between the old and new coordinates and momenta.

3. What are the most general form of Canonical Transformation equations?

The most general form of Canonical Transformation equations are the Hamilton's Equations, which describe the evolution of the system in terms of the Hamiltonian function. They are:

dQ/dt = ∂H/∂P
dP/dt = -∂H/∂Q

where Q and P are the generalized coordinates and momenta, and H is the Hamiltonian function.

4. How do you determine if a Canonical Transformation is canonical?

A Canonical Transformation is canonical if it preserves the Poisson bracket structure of the system. This means that the Poisson bracket of any two variables in the old coordinate system should be equal to the Poisson bracket of the corresponding variables in the new coordinate system. In other words, the transformed Hamilton's Equations should have the same form as the original Hamilton's Equations.

5. What are some common examples of Canonical Transformations?

Some common examples of Canonical Transformations include the transformation from Cartesian to polar coordinates, the transformation from position and momentum to action-angle coordinates, and the transformation from the Lagrangian to the Hamiltonian formalism. Other examples include transformations that simplify the equations of motion for systems with symmetries, such as the Kepler problem or the harmonic oscillator.

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