Lagrangian submanifold

A Lagrangian submanifold is a special kind of geometric shape within a larger space that has twice as many dimensions as a familiar notion of position and momentum, like in physics. It is characterized by the property that, when measured with a specific mathematical structure called a symplectic form—which encodes information about how dynamics unfold—these measurements vanish or cancel out. This property makes Lagrangian submanifolds fundamental in classical and quantum mechanics, as they often represent "states" or configurations where the system's energy or action is minimized or conserved, capturing geometric aspects of physical systems in a mathematical framework.