# Symplectic Condition For Canonical Transformation

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• CassiopeiaA
In summary, the symplectic condition for a transformation to be canonical is given as MJM' = J, where M' is transpose of M. This condition is true for any matrix M and can be checked for the 2D case by setting M as a 2x2 matrix and checking if it satisfies the condition ad-bc=1. However, not all real 2x2 matrices fulfill this condition, as symplectic matrices must have a determinant of 1.
CassiopeiaA
I am reading Chapter 9 of Classical Mech by Goldstein.The symplectic condition for a transformation to be canonical is given as MJM' = J, where M' is transpose of M. I understood the derivation given in the book. But my question is : isn't this condition true for any matrix M? That is it doesn't matter whether we are talking about canonical transformation or not, any real matrix will satisfy the condition with the given J. And if so then how do I check if the given transformation is canonical without using the Poisson Bracket condition?

How do you come to that conclusion? Why don't you check the condition for the most simple case of a 2D phase space?

vanhees71 said:
How do you come to that conclusion? Why don't you check the condition for the most simple case of a 2D phase space?
I took 2X2, 3X3 and 4X4 matrices with unknown variables and did the operation on them. All of them satisfied the condition. I also did a small proof for NXN matrix, that the above condition is true for any matrix.

Then there must be some misunderstanding on your part. Let's check the 2D case. By definition
$$J=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$$
Now let ##M \in \text{Sp}(2)##, i.e.,
$$M J M^T=J.$$
Set
$$M=\begin{pmatrix} a & b \\ c & d \end{pmatrix}.$$
Then the condition for symplecticity is
$$ad-bc=\det{M} \stackrel{!}{=}1.$$
This means for the 2D case the symplectic matrices are the ##\mathrm{SL}(2,\mathbb{R})## matrices, i.e., the matrices with determinant 1, and not all real ##2 \times 2## matrices fulfill this condition.

For more details see

https://en.wikipedia.org/wiki/Symplectic_group

## What is the symplectic condition for canonical transformation?

The symplectic condition for canonical transformation is a fundamental principle in classical mechanics that ensures the preservation of Hamilton's equations of motion. It states that the transformation between two sets of canonical coordinates must preserve the symplectic structure of the system, which is described by the Poisson bracket.

## Why is the symplectic condition important?

The symplectic condition is important because it guarantees that the equations of motion in one set of canonical coordinates can be transformed to another set of coordinates without changing the dynamics of the system. This allows for a more convenient and efficient way of solving complex mechanical systems.

## What are the implications of violating the symplectic condition?

If the symplectic condition is violated, the transformed equations of motion will not accurately describe the dynamics of the system. This can lead to incorrect predictions and inconsistencies in the behavior of the system.

## How is the symplectic condition related to the conservation of energy?

The symplectic condition is closely related to the conservation of energy in classical mechanics. If the symplectic condition is satisfied, then the Hamiltonian, which represents the total energy of the system, is conserved throughout the transformation.

## Can the symplectic condition be generalized to other physical theories?

Yes, the symplectic condition has been generalized to other physical theories, such as quantum mechanics and field theory. In these theories, the symplectic structure is described by the commutator instead of the Poisson bracket, but the fundamental principle remains the same.

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