Lagrangian submanifolds

A Lagrangian submanifold is a special kind of geometric shape within a symplectic space, which is a mathematical setting where concepts like position and momentum coexist. In simple terms, it’s a subspace where the symplectic form, a structure capturing how these quantities relate, "vanishes." This means that on this subspace, certain geometric and physical properties are minimized or balanced. Lagrangian submanifolds are essential in areas like classical mechanics and geometry because they often represent states of equilibrium or optimal configurations, preserving key structure and symmetries of the larger space.