Canonical Transformations

Chapter 6

Module 1

Aspects of canonical transformation

Hamiltonian formulation usually does not decrease the difficulty of solving problems of mechanics if the formulation is applied in a straight forward way. It provides almost the same differential equations like the Lagrangian formulation. The advantages of Hamiltonian formulation do not lie on its use as a tool for calculation rather it provides deeper insights into the structure of mechanics. In Hamiltonian mechanics, the generalized coordinates and the generalized momenta, used as the independent variables, have equal status which provides a greater freedom in choosing the quantities which are to be treated as ‘coordinates’.

Sometimes it is very useful to switch over from original set of coordinates in which the problem is expressed to another different set of coordinates in which the problem can be expressed in a simpler form. For example, in Lagrangian mechanics if we are allowed to change

/ the coordinates qii q( q , t ), i 1,2,..., n and if we redefine the Lagrangian of the system as

/// L(,,)(,,) qi q i t L q i q i t then the form of Euler-Lagrange equation remain the same in terms of both LL,./

It is well known that cyclic coordinates have advantages since they can be determined by integration only. So, it is desirable to have as many cyclic coordinates as possible in order to get advantage in the solution of a dynamical system within the frame work of Hamiltonian Mechanics. Generally, a given system of generalized coordinates reflect all the cyclic coordinates in Lagrangian or Hamiltonian expressed in terms of those generalized coordinates, generalized velocities or generalized momenta. Thus, a transformation in coordinates and momenta is needed which can transform normal looking Hamiltonian to a simple desired form of our choice. In such cases, solving the Hamilton’s equations of motion becomes very easy to solve.

6.1 Point transformation

A transformation from a given set of generalized coordinates ( q12, q ,..., qn ) (say), to another set of generalized coordinates ( QQQ12, ,..., n ) (say), is called a point transformation and is expressed as Qi Q i( q12 , q ,..., q n ), i 1,2,..., n . (6.1)

In order that inverse transformation

qi q i( Q12 , Q ,..., Q n ), i 1,2,..., n (6.2) also exists, the Jacobian of this transformation should not vanish. This type of transformation is called point transformation as the set ( q12, q ,..., qn ) represents a definite point in the configuration space spanned by these n-number of q-coordinates and will lead to another definite point expressed by the transformation (6.1) in the configuration space spanned by the n-number of Q- coordinates. Such a transformation preserves the form of Lagrange’s equation of motion.

Point transformations (6.1) and (6.2) are defined over the configuration space. But Hamilton’s equations of motion describe the evolution of the state of a system in its phase space.

Therefore, for Hamilton’s equations of motion, we have to use some suitable phase space transformation, namely q,,,, p t   Q P t defined by

Qi Q i q1, q 2 ,..., q n ; p 1 , p 2 ,..., p n ; t and PPqqi i 1, 2 ,..., qpp n ; 1 , 2 ,..., pti n ; , 1,2,..., n . (6.3)

In order to have a transformation which are physically meaningful phase space transformation, it would be quite logical to demand that new QP, are canonical coordinates i.e. the transformation (6.3) should keep the form of Hamilton’s equations of motion invariant.

We may demand the invariance of Hamilton’s principle under the transformation (6.3) i.e.

tt11nn    pq  Hqptdt( , , )  PQ  KQPtdt ( , , )  0 (6.4) i i   i i  ii11 tt00    where K(,,) Q P t is the transformed Hamiltonian under the transformation (6.3). The condition (6.4) immediately gives Hamilton’s canonical equations of motion

 KK Pii , Q  , i  1,2,..., n QPii (6.5)

which must be satisfied by the new coordinates and momenta (,)QPiishowing that QP, are canonical coordinates.

6.2 Canonical variables

Two variables xt()and y(t) are said to be canonical (in the Hamiltonian sense) if K K x  and y  where K(x, y, t ) is a function defined over the space spanned by x and y y x at time t.

6.3 Canonical transformation

The phase space transformations (6.3) preserve the forms of the Hamilton’s canonical equations of motion and are called canonical transformations (C.T.)

The phase space transformations (6.3) are called canonical if

 K  K Qi  and Pi  , i  1,2,..., n Pi Qi

where K(x, y, t) is a function defined over the space spanned by Q and P at a time t. The function K plays the role of Hamiltonian.

Thus, to look for a set of generalized coordinates in terms of which the system simplifies as much as possible, the idea of canonical transformations developed.

If the canonical transformation does not include the time t explicitly, the transformation is known as restricted canonical transformation.

Remarks:

1. Identity transformation is canonical. 2. If a transformation is canonical, its inverse is also so. 3. Two successive canonical transformations constitute a canonical transformation.

6.4 Properties of canonical transformation

1. The Jacobian determinant of the univalent canonical transformation is unity. 2. All univalent transformations form a group with respect to the composition defined

by c1* c 2 c 1 c 2 where cc12, are any two canonical transformations and cc12denotes the transformations that are performed in sequence.

6.5 Poincare Theorem

Under canonical transformation, the integral J dq dp remains invariant where S is  ii S i a two dimensional surface in phase space.

Proof: The position of any point on a two-dimensional surface can be specified uniquely by

two parameters. Let u, be the parameters. Then qi q i( u , ), p i p i ( u , ) . Now,

qpii (,)qp uu dq dp iidud dud . (6.27)  ii  ii(,)u  qp S i SSii 

Let the canonical transformation be Qk Q k( q , p , t ), P k P k ( q , p , t ) .

(,)QP dQ dPkk dud . (6.28) kk  SSkk(,)u 

J will be invariant under the canonical transformation, if we have

dq dp dQ dP . i i  k k SSik

(,)(,)q p Q P Or, i idud  k k dud . SSik(,)(,)uu

(,)(,)q p Q P Or, i i k k (since S is arbitrary). (6.29) ik(,)(,)uu

Now we have to prove the relation given by (6.29).

Let F2 (,,) q P t be the generating function for this canonical transformation. Then,

F2 (,,) q P t pi  . qi

p22FF  q  P i 22 k  k . uk   qi  q k  u  q i  P k  u

p22FF  q  P & i22 k k . k  qi  q k    q i  P k  

22 qp qiFF22  q k  P k ii  (,)qp uu u  qi  q k  u  q i  P k  u ii      22 i(,)u  iqpii i, k qFF  q  P i22 k k    qi  q k    q i  P k  

 qi  q k  q i  P k  22FFu  u  u  u   22. (6.30) ik, qi  q kqi  q k  q i  P k  q i  P k          

Now,

qi  q k  q k  q i 22FFu  u  u  u 22 i,, kqi  q kqi  q k i k  q k  q i  q k  q i        (interchanging the indices i, k) qqik 2 F uu  2 . ik, qqikqqik 

qqik 2 F uu So,  2  0. (6.31a) ik, qqikqqik 

PPik 2 F uu Replacing q by P in (6.31a) we have  2  0. (6.31b) ik, PPikPPik 

 Pi  P k  q i  P k  (,)qp 22FFu  u  u  u ii22 i(,)u i, k   Pi  P kPi  P k  q i  P k  q i  P k        22FFFP   q   P     P 2i 2 i  k  2  k Pk  P i  u  P k  q i  u   u  u   P k  u   22  ik, FFP  q  P k F P 22i i k 2 k Pk  P i   P k  q i      Pk 

QPkk uu (,)QP kk kkQPkk (,)u  

(6.32)

F2 where Qk  . Pk

Hence the theorem is proved.