How do canonical transformations relate to Hamiltonians?

[dyn]

Hi
The Hamiltonian for a harmonic oscillator is H = 1/(2m) ( p²+m²ω²q²). A canonical transformation is then made to a new Hamiltonian K( P , Q )

It is said that K ( P , Q ) = H ( p , q ) but K ( P , Q ) = ωP ( cos²Q +sin²Q ) = ωP

I don't understand how K ( P , Q ) = H ( p , q ) when they have different forms ? I thought if K = H then they must have the same form but H is a sum of 2 squares but K just equals ωP

Thanks

 

[anuttarasammyak]

I don't understand how K ( P , Q ) = H ( p , q ) when they have different forms ?

They have different forms for different parameters, i.e. (P,Q) and (p,q). One uses Cartesian coordinates. Another uses polar coordinates.

 

[dyn]

Thank you. I think i might be getting confused with symmetries , so let me see if i have got this right.

K ( P , Q ) = H ( p , q ) means that if i evaluate H at a certain value of p and q and then evaluate K at the transformed values of P and Q i get the same numerical answer ? There is no implication that K and H have the same functional form ?

If K and H had the exact same functional form then i could write H ( P , Q ) = H ( p , q ) and this occurs when the canonical transformation is a symmetry ?

Is that right ? Thanks

 

[anuttarasammyak]

K and H does not have the same function form. Q does not appear in Hamiltonian K, which is called cyclic coordinate. Hamilton equation of motion for conjugate momentum is
$$\dot{P}=0$$
The generating function of transformation is
$$\displaystyle W_{1}(q,Q)={\frac {1}{2}}m\omega q^{2}\operatorname {cot} {Q}$$

 

[dyn]

Thank you. I think i might be getting confused with symmetries , so let me see if i have got this right.

K ( P , Q ) = H ( p , q ) means that if i evaluate H at a certain value of p and q and then evaluate K at the transformed values of P and Q i get the same numerical answer ? There is no implication that K and H have the same functional form ?

If K and H had the exact same functional form then i could write H ( P , Q ) = H ( p , q ) and this occurs when the canonical transformation is a symmetry ?

Is that right ? Thanks

This post is a general question. It is not specific to the harmonic oscillator. I am just trying to find out if i understand the concept in general terms ?