Symplectic capacity

Definition

Symplectic capacity is a numerical invariant associated with a symplectic manifold that measures the size of a subset in a way that respects the symplectic structure. It plays a significant role in understanding the geometry and topology of symplectic manifolds, providing insights into their properties and behaviors, especially in the context of embedding problems and dynamical systems.

5 Must Know Facts For Your Next Test

1. Symplectic capacity is defined using the notion of symplectic volumes, which gives a way to quantify how 'large' a subset is within a symplectic manifold.
2. There are several types of symplectic capacities, including the Hofer-Zehnder capacity and the Ekeland-Hofer capacity, each with distinct definitions and applications.
3. In celestial mechanics, symplectic capacities help analyze stability and behavior of orbits and dynamical systems by providing invariants under symplectic transformations.
4. Symplectic capacities are essential in proving Gromov's non-squeezing theorem, illustrating that certain geometric configurations cannot be transformed into others without changing their capacities.
5. The concept of symplectic capacity has implications beyond pure mathematics, impacting fields such as mathematical physics and classical mechanics by linking geometric properties with physical phenomena.

Review Questions

• How does symplectic capacity relate to the properties of symplectic manifolds and their subsets?
  • Symplectic capacity serves as an important invariant for symplectic manifolds, helping to categorize subsets based on their 'size' while respecting the underlying symplectic structure. This relationship allows mathematicians to study how different subsets interact and behave under various transformations. Understanding these properties is key to addressing questions about embedding problems and dynamical behaviors within symplectic geometry.
• What role does symplectic capacity play in Gromov's non-squeezing theorem and its implications in symplectic geometry?
  • Symplectic capacity is central to Gromov's non-squeezing theorem, which asserts that you cannot embed a ball with larger capacity into another with smaller capacity through a symplectomorphism. This result shows that certain geometric configurations are rigid under symplectic transformations, influencing how we understand embeddings and dynamical systems. The theorem's implications highlight fundamental limitations within symplectic geometry, shaping our understanding of spatial properties.
• Evaluate the significance of symplectic capacity in celestial mechanics and its impact on understanding dynamical systems.
  • Symplectic capacity is significant in celestial mechanics because it provides insights into the stability and behavior of celestial bodies' orbits. By quantifying the sizes of phase space regions in terms of their capacities, researchers can better understand how dynamical systems evolve over time. This understanding has far-reaching consequences not just within mathematics but also in areas like astrophysics, where predicting the motion of celestial objects is crucial.

Related terms

Symplectic manifold: A smooth manifold equipped with a closed non-degenerate 2-form, which allows for the definition of the symplectic structure.

Gromov's non-squeezing theorem: A fundamental result in symplectic geometry that states that no symplectic embedding can increase the symplectic capacity of a ball.

Dynamical systems: Mathematical models that describe the evolution of points in a given space over time, often studied within symplectic geometry for understanding Hamiltonian systems.