Symplectic Matrix

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

A real square matrix satisfying M^T J M = J, where J is the canonical symplectic structure matrix. Symplectic matrix eigenvalues come in reciprocal pairs (lambda and 1/lambda). In Hamiltonian mechanics, the phase flow map of a Hamiltonian system preserves the symplectic structure, so its Jacobi matrix (linearized state transition matrix) is symplectic. The paper notes that the differential D_z P(z) of the Poincaré map is a symplectic matrix, which is the mathematical basis for determining Halo orbit stability.

Application Value

This concept provides fundamental support in cislunar space research and mission design. In practical applications, the appropriate analysis method should be selected based on specific mission scenarios and constraints.

Related Concepts

• On-Board Software
• Barycentric Coordinate Time, TCB
• Horizontal Coordinate System

References

• 徐明和徐世杰 - 2008 - Halo轨道维持的线性周期控制策略
• Meyer和Offin - 2017