Symplectic manifold

A symplectic manifold is a special kind of space studied in mathematics, especially in a field called differential geometry. Imagine it as a smooth, bendable surface with extra rules. These rules help mathematicians understand how things move and change in a precise way.

The key part of a symplectic manifold is called a "symplectic form." This is a special way to measure pairs of directions on the surface. It always gives a clear answer that is not zero, so it never becomes meaningless. This form also has a property called "closed," which keeps the measurements the same across the whole space.

Symplectic manifolds are very useful in physics. They help us study how objects move without forces like friction. In classical mechanics, which explains the motion of things from planets to pendulums, these manifolds describe all the possible states of a system. They connect to something called the "cotangent bundle," which is like a collection of all possible positions and speeds a system can have. Because of this, symplectic geometry is a powerful tool for understanding the rules that control motion in the real world.

Motivation

Symplectic manifolds come from classical mechanics. They help us describe the space where we track a system's condition, called phase space. In mechanics, there are special equations, called the Hamilton equations, that show how a system changes over time. Symplectic manifolds give us a way to turn these ideas into math using differential equations.

The main idea is that symplectic manifolds connect changes in the system to special math tools. This helps 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 measurement. This helps us describe how things move on the space.

These spaces are important in physics. They help us study how objects move naturally, especially when no forces are acting on them. They often appear around smooth objects, helping us understand motion in a simple 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 symplectic geometry. They follow a rule called the "symplectic creed," which says that everything in this area of math can be best understood using these special submanifolds.

A Lagrangian fibration happens when all the pieces, or fibers, of a symplectic manifold are Lagrangian submanifolds. These 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 the same under certain changes.

One important type of symplectomorphism is called a symplectic flow, which comes from a vector field on the manifold. When these flows happen, they keep a key property unchanged, which makes them useful for studying the geometry of these special spaces. Properties that stay the same under all symplectomorphisms are called symplectic invariants, and they are important for 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 with a unique structure called a symplectic form. One simple example is when we look at pairs of numbers, like points on a grid. Here, the symplectic form helps us see how these pairs are balanced.

Another example is the cotangent bundle, which naturally has a symplectic form. This is useful for studying 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