- #1
Nusc
- 760
- 2
What's a really good resources with numerous easy-to-follow proofs to theorems on symplectic manifolds?
Arnold is too difficult.
Arnold is too difficult.
A symplectic manifold is a mathematical object that represents a smooth space with a specific geometric structure called a symplectic structure. This structure allows for the definition of a special kind of vector field, called a symplectic vector field, which preserves the symplectic structure.
Examples of symplectic manifolds include the phase space of a mechanical system, the cotangent bundle of a smooth manifold, and the space of orbits of a group action.
Symplectic manifolds play a crucial role in Hamiltonian mechanics, as they provide a geometric framework for studying systems with conserved quantities and Hamiltonian dynamics. In fact, every symplectic manifold can be viewed as a phase space for a Hamiltonian system.
A symplectomorphism is a diffeomorphism (a smooth bijective map) between two symplectic manifolds that preserves the symplectic structure. In other words, it is a map that preserves the geometric properties of the symplectic manifold.
Symplectic manifolds are different from other types of manifolds, such as Riemannian manifolds, in that they have a non-degenerate, closed 2-form, known as the symplectic form, which defines the symplectic structure. This structure is important in studying Hamiltonian systems and has many applications in physics and mathematics.