Download presentation
Presentation is loading. Please wait.
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.