MAT 303 1.0 Classical Mechanics Canonical Transformations

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 .

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 .

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=

So transformation is canonical.

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

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

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

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

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

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

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

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

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

So the generating function is , neglecting the constant.

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

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

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

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