Nonsqueezing theorem - a proof with the action

In summary, the conversation discusses the proof of Gromov's nonsqueezing theorem using the fact that the action is a symplectic invariant. The proof involves expressing the action in terms of the radius of a sphere and a cylinder, and using these expressions to show that the sphere cannot fit into the cylinder by symplectic transformation. The speaker has read the proof superficially and is now looking for a more detailed explanation, but is unable to find it. However, they later find it in Gosson's book on page 100.
  • #1
mma
245
1
I've read somewhere a proof of Gromov's nonsqueezing theorem using the fact that the action is a symplectic invariant (just like the Hamiltonian). As far as I remember, the action was expressed in terms of the radius of a sphere and also in terms of the radius of a cylinder, and the proof of the statement that the sphere doesn't fit by symplectic transormation into the cylinder was based on these expressions. But I 've read this poof only very superficially and just now wanted to read it in details, but now I don't find it, and don't remember where it was. Does somebody know this proof? Coul'd somebody tell me where can I find it?
Thanks in advance.
 
Physics news on Phys.org
  • #2
Don't trouble, I found it in Gosson's book on page 100. (perhaps this isn't that I saw originally, but it doesn't matter)
 

1. What is the Nonsqueezing theorem?

The Nonsqueezing theorem is a mathematical result that states that certain types of symplectic manifolds cannot be "squeezed" into smaller dimensions without changing the topology of the manifold. In other words, there are certain geometric shapes that cannot be deformed into smaller shapes without creating holes or changing the number of dimensions.

2. What is the significance of the Nonsqueezing theorem?

The Nonsqueezing theorem is significant because it has applications in many areas of mathematics and physics, such as in the study of Hamiltonian dynamics and symplectic geometry. It also has implications for the behavior of physical systems, as it shows that certain types of energy cannot be squeezed into smaller volumes.

3. How does the Nonsqueezing theorem relate to the concept of "action"?

The Nonsqueezing theorem is often proved using the concept of action, which is a mathematical quantity that describes the motion of a system in classical mechanics. In particular, the theorem can be proved by showing that the action is conserved under certain transformations, which is a fundamental principle in classical mechanics.

4. Can you provide an example of how the Nonsqueezing theorem is used in practice?

One example of the Nonsqueezing theorem being used in practice is in the study of fluid dynamics. This theorem can be applied to show that certain types of fluid cannot be compressed into smaller volumes without changing the underlying structure of the fluid. This has important implications for understanding the behavior of fluids in various physical systems.

5. Are there any limitations to the Nonsqueezing theorem?

Like any mathematical theorem, the Nonsqueezing theorem has certain limitations. For example, it only applies to certain types of symplectic manifolds and does not hold in all cases. Additionally, it is a purely mathematical result and does not take into account other factors that may affect the behavior of physical systems, such as friction or external forces.

Similar threads

I Contrapositive of theorem and issue with proof
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
237
I Noether's second theorem: two questions
    • Special and General Relativity
Replies
7
Views
1K
MHB Checking Proof of Theorem 6.2.8 Part (ii)
    • Topology and Analysis
Replies
2
Views
2K
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
    • Linear and Abstract Algebra
Replies
3
Views
1K
Surface area of a sphere ##= \pi * (a^2+b^2+c^2+d^2)##
    • Precalculus Mathematics Homework Help
Replies
18
Views
1K
I Spectral theorem for Hermitian matrices-- special cases
    • Linear and Abstract Algebra
Replies
2
Views
607
Virial Theorem for an expanding Star Cluster
    • New Member Introductions
Replies
1
Views
50
I Is the proof of the KRK endgame theorem rigorous and original?
    • General Math
Replies
13
Views
1K
Proof: Gromov's short basis, volume comparison
    • Differential Geometry
Replies
1
Views
2K
B What is the 'proof' of the no-hair theorem of black holes?
    • Cosmology
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top