What is symplectic mathematics?

What is symplectic mathematics as follows:

Symplectic geometry is the frontier of mathematics.

Over the past 40 years of reform and opening-up, China Academy of Sciences has adhered to its position as a national strategic scientific and technological force, and persisted in facing the forefront of world science and technology.

On the basis of systematically sorting out many important scientific and technological achievements made by researchers of China Academy of Sciences in the past 40 years of reform and opening-up, this paper comprehensively summarizes 40 representative and symbolic major scientific and technological achievements with the clue of "three orientations". It has been included in the first aspect (project 15), the thirteenth mathematical mechanization method and symplectic geometry algorithm facing the world's scientific and technological frontier in the "Pioneer Innovation Development Engine of Reform and Opening-up —— Forty Years of Reform and Opening-up of China Academy of Sciences".

Let M be a 2n-dimensional differential manifold, and a quadratic differential form ω is called a symplectic structure or symplectic form on M. If ω satisfies ω, it is a closed form, that is, dω=0. ω is nondegenerate, that is, ω n (n-th outer product of ω) is a 2n-th differential form that is nonzero everywhere. We call (m, ω) a symplectic manifold. Simply put, symplectic geometry is a kind of geometry that studies the properties of symplectic manifolds, which is generally considered to belong to the category of differential geometry.

Symplectic geometry is the mathematical basis of Hamiltonian system. This algorithm has become the mainstream research direction of ordinary differential equations and numerical calculation of dynamic systems, which has brought innovation in scientific and engineering calculation.

It is widely used in celestial orbit evolution, high-energy accelerator design, molecular dynamics simulation, numerical weather prediction, oil and gas exploration, plasma confinement, computational quantum chemistry and other fields.