Question

Classical Mechanics

Working with Poisson bracket we consider the set of variables q and p to be canonical variables if the variables satisfy the following:

{??,??}=0 (1) whatever i and j

{??,??}=0 (2) whatever i and j

{??,??}=??,? (3) ( 0 for i ≠ j; 1 for i = j )

which are derived with the use of Hamilton’s equations. A canonical transformation is a change of canonical variables (q,p) to (Q,P) that preserves the form of Hamilton’s equations. That is, Q and P satisfy the relations (1-3) above.

Let us consider the following transformation equations between two sets of coordinates (q,p) to (Q,P) are:

? = ln [1+?^(1/2) cos(?)]

? = 2 [1+ ? ^ (1/2) cos(?)] ? ^ (1/2) sin (?)

Show that these transformation equations yield Q and P to be canonical variables if q and p are canonical.

Answer #1

Classical Mechanics -
Working with Poisson bracket we consider the set of variables q
and p to be canonical variables if the variables satisfy the
following:
{??,??}=0 (1) whatever i and j
{??,??}=0 (2) whatever i and j
{??,??}=??,? (3) ( 0 for i ≠ j; 1 for i = j )
which are derived with the use of Hamilton’s equations. A
canonical transformation is a change of canonical variables (q,p)
to (Q,P) that preserves the form of Hamilton’s equations....

Classical Mechanics -
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