Symplectic manifold

A symplectic manifold is a special kind of space studied in mathematics, particularly in a field called differential geometry. Think of it as a smooth, flexible surface that has extra rules built into it. These rules help mathematicians understand how things move and change in a very precise way.

At the heart of a symplectic manifold is something called a "symplectic form." This is a special way of measuring pairs of directions on the surface that always gives a clear, non-zero result — meaning it never collapses or becomes meaningless. This form also has a property called "closed," which makes the measurements consistent over the whole space.

Symplectic manifolds are very important in physics, especially in the study of how objects move without forces like friction. In classical mechanics, which describes the motion of everything from planets to pendulums, these manifolds help describe the complete set of possible states of a system. They are tied to something called the "cotangent bundle," which is like a collection of all possible positions and speeds a system can have. This connection makes symplectic geometry a powerful tool for understanding the laws that govern motion in the physical world.

Motivation

Symplectic manifolds come from classical mechanics. They are a way to describe the space where a system’s condition is tracked, called phase space. In mechanics, there are special equations, called the Hamilton equations, that help us understand how a system changes over time. Symplectic manifolds provide a structure that helps us turn these ideas into mathematical descriptions using differential equations.

The main idea is that symplectic manifolds let us connect changes in the system to a special kind of mathematical object, helping us study how things move and change in a smooth and predictable way.

Definition

A symplectic manifold is a special kind of smooth space used in geometry. It has something called a "symplectic form," which is a special kind of measurement that helps describe how things move on the space.

These manifolds are important in physics, especially in the study of how objects move without forces acting on them. They often appear when looking at the space around a smooth object, helping us understand motion in a natural way.

Submanifolds

A submanifold is a smaller shape inside a bigger shape. In symplectic geometry, we talk about special kinds of submanifolds inside a symplectic manifold.

• Symplectic: The smaller shape keeps the same symplectic form as the bigger shape.
• Isotropic: The symplectic form becomes zero on the smaller shape.
• Coisotropic: The smaller shape fits inside the bigger shape in a special way.
• Lagrangian: This is both isotropic and coisotropic, meaning it is the biggest shape where the symplectic form is zero and the smallest shape that fits in this special way.

Lagrangian submanifolds

Lagrangian submanifolds are very important in the study of symplectic geometry. They follow a rule called the "symplectic creed," which says that everything in this area of math can be best understood using Lagrangian submanifolds.

A Lagrangian fibration happens when all the pieces, or fibers, of a symplectic manifold are Lagrangian submanifolds. These special submanifolds help us understand the structure and behavior of more complex mathematical objects.

Symmetries

Main article: Symplectomorphism

In symplectic geometry, a special kind of map between two symplectic manifolds that keeps the symplectic structure the same is called a symplectomorphism. These maps help us understand how the geometry stays consistent under certain transformations.

One important type of symplectomorphism is called a symplectic flow, which comes from a vector field on the manifold. When these flows happen, they preserve a key property, making them useful in studying the geometry of these special spaces. Properties that stay the same under all symplectomorphisms are called symplectic invariants, and they are central to understanding symplectic geometry, much like how the Erlangen program guides other areas of geometry.

Examples

Main article: Symplectic vector space

Symplectic manifolds are special spaces that have a special kind of structure called a symplectic form. One simple example is when we look at a space made of pairs of numbers, like points on a grid. In this case, the symplectic form helps us understand how these pairs relate to each other in a balanced way.

Another important example is the cotangent bundle, which naturally comes with a symplectic form. This structure is very useful in the study of motion and mechanics, where it helps describe how things move and change over time.

Generalizations

Symplectic manifolds can be extended into different types. Presymplectic manifolds are similar but allow for some degeneracy. Poisson manifolds keep only certain structures from symplectic manifolds. Dirac manifolds are an even broader extension. Multisymplectic manifolds use a special kind of form, while polysymplectic manifolds are used in advanced theories of physics.

Main articles: Presymplectic manifolds, Poisson manifolds, Dirac manifolds, Hamiltonian field theory