1. MAT 303 1.0 Classical Mechanics
Canonical
Transformations

2. Canonical Transformations
For a given system of particles, we can define different sets of generalized coordinates.
A transformation between two sets of generalized coordinates is called a canonical transformations if it leaves the Hamilton’s Equations invariant.
Suppose {qi , pi }is one set of generalized coordinates and generalized momenta. H is the Hamiltonian of the system then
and .

3. Suppose {Qi , Pi } is another set of generalized coordinates and generalized momenta.
The transformation
{qi, pi }
{Qi , Pi }
Is called a canonical transformation if there exists a function Ho , called Hamiltonian of the system with the coordinates {Qi , Pi }, satisfying the equations
and .

4. Theorem The transformation {qi , pi } {Qi , Pi } is canonical if
is an exact differential.
E.g.
Show that the following transformation of one degree of freedom is canonical.
Solution :
Since
We get dQ=

5. So transformation is canonical.

6. E.g. Show that the transformation is canonical. Solution : Since
So transformation is canonical.

7. Generating Functions qi pi Qi T U Pi S V The transformation {qi , pi }
{Qi , Pi } is canonical
provided that there is a function
called generating function, satisfying
, here L’s are the
Lagrangians with respect to the old and new variables.
We can express the generating function G using one of the two old variables and one of the two new variables. qi pi Qi T U Pi S V

8. Result I
When the generating function
is expressed using
and t
It is denoted by T and we get
and
Proof of the result :
Eq. 1
Further
Eq. 2

9. Eq. 1
and
Eq. 2
Give us
and
E.g.
Show that the following transformation is canonical, and find the generating function in terms of p and q.
Solution :
Canonical part is easy.
We rewrite the given transformation as
Above first equations becomes

10. E.g.
Show that the following transformation is canonical, and find the generating function in terms of p and q.
Solution :
Canonical part is easy.
By rewriting the given transformation becomes
The equation
becomes

11. By integrating partially with respect to q, we get
The other equation
becomes
, and hence
, neglecting the constant.
So the generating function is

12. Result II
When the generating function
is expressed using
and t
It is denoted by S and we get
and
Proof of the result :
Eq. 1
Further

13. Eq. 2
Eq. 1
and
Eq. 2
Give us
and
E.g.
, find the
For the canonical transformation
generating function in terms
of p and q. 1
Solution :
Here

14. Above first equations becomes
By integrating partially with respect to q, we get
, and hence
The other equation
becomes

15. So the generating function is
, neglecting the constant.

16. Result III
When the generating function
is expressed using
and t
It is denoted by U and we get
and
Proof of the result :
Eq. 1

17. Eq. 2
Eq. 1
and
Eq. 2
give us
and
E.g. Does there exist a generating
of the form G=G( p, Q, t ) for the transformation

18. Result IV
When the generating function
is expressed using
and t
It is denoted by V and we get
and
Proof of the result :
Eq. 1

19. Eq. 2
Eq. 1
and
Eq. 2
give us
and
E.g. Find the generating function for the transformation