Canonical Transformations

For the transformation to be canonical, the Poisson bracket of

Q

and

P

must satisfy:

%5C%7B%20Q%2C%20P%20%5C%7D%20%3D%201.

The Poisson bracket is given by:

%5C%7B%20Q%2C%20P%20%5C%7D%20%3D%20%5Cfrac%7B%5Cpartial%20Q%7D%7B%5Cpartial%20r%7D%20%5Cfrac%7B%5Cpartial%20P%7D%7B%5Cpartial%20p%7D%20-%20%5Cfrac%7B%5Cpartial%20Q%7D%7B%5Cpartial%20p%7D%20%5Cfrac%7B%5Cpartial%20P%7D%7B%5Cpartial%20r%7D.

Substituting the given forms of

Q

and

P

Q%20%3D%20%5Cvec%7Ba%7D%20%5Ccdot%20%5Cvec%7Br%7D%2C%20P%20%3D%20%5Cvec%7Bb%7D%20%5Ccdot%20%5Cvec%7Br%7D.

The Poisson bracket simplifies to:

%5C%7B%20Q%2C%20P%20%5C%7D%20%3D%20%5Cvec%7Ba%7D%20%5Ccdot%20%5Cvec%7Bb%7D.

For the transformation to be canonical, we require:

%5Cvec%7Ba%7D%20%5Ccdot%20%5Cvec%7Bb%7D%20%3D%201.

Thus, the value of

%5Cvec%7Ba%7D%20%5Ccdot%20%5Cvec%7Bb%7D

-1

Chapter Questions
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