Cannonical transformation
Transformation: More than one set of generalized coordinates can be used to describe a given system. We can decide on a set that will make solving the problem at hand the easiest. The Cartesian coordinates q1=x, q2=y or the plane polar coordinates q1=r, q2= θ, for instance, can be used as generalized coordinates to explain the motion of a particle in a plane.
Because the central force is cyclic, we observe that the second set is more practical. By selecting the second set, the problem is easier to solve because the Hamiltonian only needs one variable for r to exist. Therefore, we want to talk about a specific method for changing one set of variables into another set that might be more useful in the future. If a problem has been expressed as a Hamiltonian canonical equation, the cannonical transformation can be used to simplify the integration of the equation of motion by converting the equations into a more easily solvable form.
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.
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
As a result, Hamilton's formulation is included in the transformation. The above generic form must be enlarged to provide room for the additional independent variable, momentum, which is acknowledged in the Hamiltonian formulation. Therefore, the simultaneous translation of the independent coordinate and momenta qj,pj to a new set Qj, Pj may be expressed as
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
3. 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 a part of the Physics Repository.
This note is taken from classical Mechanics, MSC physics, Nepal