🔵Symplectic Geometry Unit 5 – Lagrangian Submanifolds in Mechanics

Key Concepts and Definitions

• Symplectic manifold $(M, \omega)$ consists of an even-dimensional smooth manifold $M$ and a closed, non-degenerate 2-form $\omega$ called the symplectic form
• Lagrangian submanifold $L$ is a submanifold of a symplectic manifold $(M, \omega)$ with half the dimension of $M$ on which the symplectic form $\omega$ vanishes
  • Formally, $\dim L = \frac{1}{2} \dim M$ and $\omega|_L = 0$
• Hamiltonian vector field $X_H$ associated with a smooth function $H: M \to \mathbb{R}$ satisfies $\omega(X_H, \cdot) = dH$
• Poisson bracket $\{f, g\}$ of two smooth functions $f, g: M \to \mathbb{R}$ measures the rate of change of $g$ along the Hamiltonian flow of $f$
• Lagrangian intersection $L_1 \cap L_2$ of two Lagrangian submanifolds $L_1, L_2 \subset M$ plays a crucial role in studying the topology of symplectic manifolds
• Maslov index $\mu(u)$ is an integer-valued invariant assigned to each homotopy class $u$ of loops in the Lagrangian Grassmannian $\Lambda(n)$

Historical Context and Development

• Lagrangian mechanics, developed by Joseph-Louis Lagrange in the late 18th century, laid the foundation for the study of Lagrangian submanifolds
• Symplectic geometry emerged in the 1960s and 1970s as a mathematical framework for classical mechanics and geometric optics
  • Pioneering work by Vladimir Arnold, Victor Guillemin, Shlomo Sternberg, and Alan Weinstein
• Lagrangian submanifolds were first studied in the context of geometric quantization and the WKB approximation in quantum mechanics
• The introduction of Floer homology in the 1980s by Andreas Floer revolutionized the study of Lagrangian submanifolds
  • Floer homology provides a powerful tool for investigating the intersection theory of Lagrangian submanifolds
• Recent developments include the study of Fukaya categories, mirror symmetry, and the connection between Lagrangian submanifolds and knot theory

Lagrangian Submanifolds: The Basics

• Lagrangian submanifolds are a fundamental class of objects in symplectic geometry
• Examples of Lagrangian submanifolds include:
  • The graph of a closed 1-form in the cotangent bundle $(T^*M, \omega_{\text{can}})$
  • The fixed point set of an anti-symplectic involution (real locus)
  • The zero section of a cotangent bundle
• Lagrangian submanifolds are locally characterized by the vanishing of the symplectic form
• The Darboux-Weinstein theorem states that a neighborhood of a Lagrangian submanifold is symplectomorphic to a neighborhood of the zero section in the cotangent bundle
• Lagrangian submanifolds are closely related to generating functions and the Hamilton-Jacobi equation in classical mechanics

Mathematical Foundations

• Symplectic linear algebra studies symplectic vector spaces $(V, \omega)$ and linear subspaces, including Lagrangian subspaces
• The Lagrangian Grassmannian $\Lambda(n)$ is the space of Lagrangian subspaces in a symplectic vector space of dimension $2n$
  • $\Lambda(n)$ is a homogeneous space diffeomorphic to $U(n) / O(n)$
• The tangent space to a Lagrangian submanifold $L$ at a point $x \in L$ is a Lagrangian subspace of the symplectic vector space $(T_xM, \omega_x)$
• The Maslov index is a fundamental invariant of loops in the Lagrangian Grassmannian
  • Computed using the intersection theory of Lagrangian subspaces
• The Arnold-Liouville theorem characterizes completely integrable Hamiltonian systems in terms of Lagrangian torus fibrations

Applications in Classical Mechanics

• Lagrangian submanifolds naturally arise in the study of classical mechanical systems
• The configuration space of a mechanical system is often modeled as a Lagrangian submanifold of the phase space (cotangent bundle)
• Hamilton's principle of least action can be formulated in terms of Lagrangian submanifolds
  • Stationary paths correspond to intersections of certain Lagrangian submanifolds
• Lagrangian submanifolds play a crucial role in the study of integrable systems and action-angle coordinates
• The KAM (Kolmogorov-Arnold-Moser) theory describes the persistence of Lagrangian tori under small perturbations of integrable systems
• Lagrangian submanifolds are used in the geometric formulation of optics and wave propagation (WKB approximation)

Symplectic Structures and Lagrangian Submanifolds

• The study of Lagrangian submanifolds is intimately connected to the properties of symplectic structures
• Symplectic reduction allows for the construction of new symplectic manifolds and Lagrangian submanifolds from group actions
  • Marsden-Weinstein reduction, Meyer reduction, and symplectic cutting
• Lagrangian submanifolds can be constructed using symplectic surgeries, such as Lagrangian connected sums and Polterovich surgery
• The Gromov-Lees h-principle provides a flexible method for constructing Lagrangian submanifolds in certain situations
• Lagrangian Floer homology is a powerful invariant that captures the intersection theory of Lagrangian submanifolds
  • Defined using the Maslov index and pseudo-holomorphic curves

Advanced Topics and Current Research

• The Fukaya category is an A-infinity category that encodes the symplectic topology of a manifold through its Lagrangian submanifolds
  • Objects are Lagrangian submanifolds, morphisms are Floer homology groups
• Mirror symmetry relates the symplectic geometry of a Calabi-Yau manifold to the complex geometry of its mirror manifold
  • Homological mirror symmetry conjecture (Kontsevich) states that the Fukaya category is equivalent to the derived category of coherent sheaves on the mirror
• Lagrangian submanifolds play a central role in the study of symplectic knot theory and the symplectic camel problem
• The Lagrangian cobordism group measures the relations between Lagrangian submanifolds under Lagrangian cobordisms
• Lagrangian submanifolds are used in the study of symplectic capacities and displacement energy
• Current research focuses on understanding the topology and geometry of the space of Lagrangian submanifolds and its compactifications

Problem-Solving Techniques

• Identify the symplectic manifold $(M, \omega)$ and the dimension of the Lagrangian submanifold $L$
• Check that $L$ is a submanifold of $M$ and that $\dim L = \frac{1}{2} \dim M$
• Verify that the symplectic form $\omega$ vanishes on $L$, i.e., $\omega|_L = 0$
  • This can often be done by showing that $L$ is locally the graph of a closed 1-form
• Compute the Maslov index of loops in $L$ using the intersection theory of Lagrangian subspaces in the tangent spaces
• Use the Darboux-Weinstein theorem to study the local structure near a Lagrangian submanifold
• Apply symplectic reduction techniques to construct new Lagrangian submanifolds from group actions
• Investigate the intersection properties of Lagrangian submanifolds using Floer homology and pseudo-holomorphic curves
• Utilize the Gromov-Lees h-principle to construct Lagrangian submanifolds in suitable settings