Question

If \( \vec{a} \) and \( \vec{b} \) are constant vectors, \( \vec{r} \) and \( \vec{p} \) are generalized positions and conjugate momenta, respectively, then for the transformation \( Q = \vec{a} \cdot \vec{r} \) and \( P = \vec{b} \cdot \vec{r} \) to be canonical, the value of \( \vec{a} \cdot \vec{b} \) (in integer) is \(\underline{\hspace{2cm}}\).

Hint:

For canonical transformations, ensure the Poisson bracket condition \( \{ Q, P \} = 1 \) is satisfied.

Answer 1

Correct Answer: -1

For the transformation to be canonical, the Poisson bracket of \( Q \) and \( P \) must satisfy: \[ \{ Q, P \} = 1. \] The Poisson bracket is given by: \[ \{ Q, P \} = \frac{\partial Q}{\partial r} \frac{\partial P}{\partial p} - \frac{\partial Q}{\partial p} \frac{\partial P}{\partial r}. \] Substituting the given forms of \( Q \) and \( P \): \[ Q = \vec{a} \cdot \vec{r}, P = \vec{b} \cdot \vec{r}. \] The Poisson bracket simplifies to: \[ \{ Q, P \} = \vec{a} \cdot \vec{b}. \] For the transformation to be canonical, we require: \[ \vec{a} \cdot \vec{b} = 1. \] Thus, the value of \( \vec{a} \cdot \vec{b} \) is \( -1 \).