Cannonical transformation

Transformation: A given system can be described by more than one set of generalized coordinates. We can choose a set which is most convenient for the solution of the problem under consideration. For example to discuss the motion of a particle in a plane, we may use as generalised coordinates, the Cartesian coordinates q1=x, q2=y or plane polar coordinates q1=r, q2= θ.

We note that second set is more convenient because for central force θ is a cyclic. By choosing second set, solution of the problem becomes easy as only one variable r is to appear in the Hamiltonian. Thus we want to discuss here a specific procedure for transforming one set of variables into other set which may be move convenient. If a problem has been formulated in the form of Hamiltonian cannonical equation, the cannonical tranformation can be aimed to put the equations into a more easily soluble form, i.e to make integration of the equation of motion simpler.

Point transformation: Transformation from one set of coordinates qj to a new set of coordinates Qj can be expressed as

Qj = Qj(q,t)

Such transformations are called point transformations. This type of transformation are called point transformation of configuration space because configuration space is adequate only in providing the information about the position coordinate qj and not about the velocities.

Cannonical transformation: The transformation of the type represented by

Qj = Qj(q,p,t)

pj = pj(q,p,t)

are called contact or canonical transformations. Canonical transformation are transformation are the transformations of phase space. They are characterised by the property that they leave the form of Hamiltonian's equation of motion invariant.

Lagrangian equation of motion are invariant in form with respect to the choice of the set any generalized coordinates. Therefore new set Qj, Lagrangian's equation will be

Therefore this transformation is extended to Hamilton's formulation. In Hamiltonian formulation, we admit the existence one more independent variable called momentum and therefore above general form must be widened to accommodate the new variable. Consequently the simultaneous transformation of the independent coordinate and momenta qj,pj to a new set Qj, Pj can be represented in the form

Qj = Qj(q,p,t)

pj = pj(q,p,t)

For Qj, Pj to be cannonical as is demanded in Hamiltonian formulation, they should be able to expressed in Hamiltonian form of equation of motion i.e

Generating Function:

The first bracket of 5 is regarded as a function of qj,pj and t, the second as a function of Qj, Pj and t. F is thus, in general a function of 4n+1 variables qj,pj, Qj,Pj and t. But however the two setd of variables are connected by the 2n transformation equation 1 and thus out of 4n variables, besides t only 2n are independents. Now F is a function of both old and new set of coordinates and therefore out of 2n variables, n should be take from new and n from old set, i.e one variable should be out of qj and pj and other should be from Qj, Pj set. Thus following four forms of function F are possible:

F1(q,Q,t), F2(q,P,t), F3(p,Q,t) and F4(p,P,t)

As f is a function of old and new set of coordinates, it can affect the transformation from old set to new set, i.e. transformation relation can be derived by the knowledge of the function F. It is thus termed as the generating function.

1. First form: F1(q,Q,t)

For this case we can write 5 as

2. Second Form: F2(q,p,t)

To apply 6 for this case, change of basic of description from q, Q to q, P should be affected for which we use Legendre transformation discussed in brief as follows:

Consider a function f(x,y)

df = ∂f/∂x dx + ∂f/∂y dy = udx +vdy...............A

Suppose we want to change the basis of description from (x,y) to (u,y) then let g(u,y) = f(x,y) - ux so that

dg = df - udx - xdu

= udx + vdy - udx - xdu

= vdy - xdu

which is exactly in the form desired so that we can write

dg = ∂g dy/ ∂y + ∂g du / ∂u where v= ∂g/∂y and x = - ∂g/∂u

and is identical to A

Thus if u = ∂f/∂x then the relation g(u,y) = f(x,y) - ux.............B

would be appropriate to effect a change from the basis x,y to u,y. Now we apply it to the present case. Here since

Third Form: F3(p,Q,t)

To connect F3 with F1, we again apply Legendre transformation for this case we see that

This note is taken from classical Mechanics, MSC physics, Nepal