Classical Mechanics: Canonical transformation problem

[genius2687]

Homework Statement

Show directly that the transformation; Q=log(1/q*sinp), P=q*cotp is canonical.

Homework Equations

Since these equations have no time dependence, the equations are canonical if (with d denoting a partial derivative)

dQ_i/dq_j = dp_j/dP_i, and dQ_i/dp_j = -dq_j/dP_i

The Attempt at a Solution

With

Q=log(1/q*sinp), dQ/dq = -1/q

P=q*cotp => p=tan^-1(q/P), dp/dP = -q/(p^2+q^2).

The first problem I encounter is that -1/q not= -q/(p^2+q^2).

With dQ/dp = cotp, and -dq/dP = -1/(dP/dq) = -cotp

so, cotp not= -cotp.

 

[OlderDan]

It worked for me. I solved for

cosp = Pe^Q

q = sinp/e^Q = sqrt[1 - (Pe^Q)²]/e^Q

and took the derivatives.

 

[genius2687]

I know what I did now. For partial derivatives, dx/dy not= 1/(dy/dx). I falsely made that assumption.