What is the definition of canonical transformation?

In summary, a canonical transformation is a mathematical transformation that preserves the canonical form of Hamilton's equations of motion. Its importance lies in simplifying the equations of motion for a physical system and finding more elegant expressions for its dynamics. Unlike a coordinate transformation, a canonical transformation also preserves the physical dynamics of the system. However, not every transformation can be considered canonical as it must satisfy certain conditions. These transformations are closely related to symmetries in physics, with every symmetry corresponding to a canonical transformation that leaves the equations of motion unchanged, known as Noether's theorem.
  • #1
zheng89120
149
0
Why is it that only Canonical transformations preserve the Hamilton's equations? Or what makes non-canonical transformations not preserve the Hamilton's equations?
 
Physics news on Phys.org
  • #2
But that is the definition of canonical transformation that they preserve the Hamilton's equation. Whatever transformation of coordinates you find that preserves Hamilton's equation is a canonical transformation.
 

What is a canonical transformation?

A canonical transformation is a mathematical transformation that preserves the canonical form of Hamilton's equations of motion. It is used to transform one set of coordinates and momenta into a new set of variables that describe the same physical system.

What is the importance of canonical transformations?

Canonical transformations are important because they allow us to simplify the equations of motion for a physical system. By transforming to new coordinates and momenta, we can often find simpler and more elegant expressions for the dynamics of the system.

What is the difference between a canonical transformation and a coordinate transformation?

A coordinate transformation only changes the way we describe a system mathematically, while a canonical transformation also preserves the physical dynamics of the system. This means that the equations of motion will remain the same, even after the transformation.

Can any transformation be considered a canonical transformation?

No, not every transformation is a canonical transformation. For a transformation to be canonical, it must satisfy certain conditions, such as preserving the Poisson brackets between the coordinates and momenta of the system.

How are canonical transformations related to symmetries in physics?

Canonical transformations are closely related to symmetries in physics. In fact, every symmetry in a physical system corresponds to a canonical transformation that leaves the equations of motion unchanged. This is known as Noether's theorem.

Similar threads

Replies
4
Views
561
  • Classical Physics
Replies
6
Views
1K
Replies
3
Views
550
Replies
25
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
761
  • Introductory Physics Homework Help
Replies
3
Views
891
  • Introductory Physics Homework Help
Replies
5
Views
212
  • Advanced Physics Homework Help
Replies
1
Views
653
  • Classical Physics
Replies
3
Views
4K
Replies
2
Views
990
Back
Top