Confused about symmetries and canonical transformations

In summary, the conversation revolves around a problem in the book Principles of Quantum Mechanics by R. Shankar, specifically in the chapter of Review of Classical Mechanics. The problem pertains to canonical transformations and their relation to Lagrangian and Hamiltonian methods, as well as their significance in the study of quantum mechanics. The conversation also touches on the importance of understanding classical mechanics for further studies in quantum mechanics. The use of exclamation points is cautioned against in the conversation.
  • #1
Sillyboy
27
0
this is a problem confusing me, which is in the book named Principles of Quantum Mechanics by R. Shankar. This problem is not about quantum mechanics, but just in the chapter of Review of Classical Mechanics. (The ******** is just to avoid to be deleted).

The problem is in the attachment, first I have to sorry about my expression in English.
Please make it clear that what is canonical transformation(I just know some kind of comprehension in the book, I want your guy's comprehension about Lagrangian and Hamiltonian mathod, the canonical transformations, the symmetries and their consequences)
I am here expecting your comprehension and reply to my problem.

I didn't study the analytical mechanics, but now we are self-studying the principles of quantum mechanics in the dicussion class! Hence, we do not have a good comprehension about the classical mechanics. But we think it is very important for our further study.
 

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  • #2
Why?
Please answer me!
 
  • #3
Sillyboy said:
Why?
Please answer me!

They're some transformations that allow us to use Hamiltonian formulation in the new coordinates.
http://en.wikipedia.org/wiki/Canonical_transformation#Notation

PS: You should be careful with the usage of exclamation, it looks like yelling.
 

Related to Confused about symmetries and canonical transformations

1. What are symmetries in physics?

Symmetries in physics refer to the invariance or unchanged nature of a physical system under certain transformations. These transformations can include changes in position, orientation, or time. Symmetries play a crucial role in understanding the fundamental laws of physics and can help simplify complex equations.

2. What is the difference between continuous and discrete symmetries?

Continuous symmetries refer to transformations that involve a smooth and continuous change, while discrete symmetries involve a distinct and finite change. For example, rotational symmetry is a continuous symmetry, while mirror reflection is a discrete symmetry.

3. How do symmetries relate to conservation laws?

Symmetries and conservation laws are closely related in physics. In fact, Noether's theorem states that every continuous symmetry in a physical system corresponds to a conserved quantity. For example, the symmetry of translation in time leads to the conservation of energy.

4. What is a canonical transformation?

A canonical transformation is a transformation in the phase space of a physical system that preserves the equations of motion. It is used to transform coordinates and momenta in a way that simplifies the equations of motion and can reveal underlying symmetries in the system.

5. What is the importance of canonical transformations in physics?

Canonical transformations are essential in physics as they allow us to simplify and understand complex systems. They can reveal symmetries, lead to conservation laws, and make calculations more manageable. They are also crucial in the development of Hamiltonian mechanics, which is a powerful tool for studying classical and quantum systems.

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