Introduction to the symplectic group Sp(2)
Optics
2025-08-26 v1
Abstract
In this article, we derive and discuss the properties of the symplectic group Sp(2), which arises in Hamiltonian dynamics and ray optics. We show that a symplectic matrix can be written as the product of a symmetric dilation matrix and a rotation matrix, in either order. A symplectic matrix can be written as the exponential of a generating matrix, and there is a one-to-one relation between the coefficients of the symplectic and generating matrices. We also discuss the adjoint and Schmidt decompositions of a symplectic matrix, and the product of two symplectic matrices. The results of this article have applications in many subfields of physics.
Cite
@article{arxiv.2508.17001,
title = {Introduction to the symplectic group Sp(2)},
author = {C. J. McKinstrie and M. V. Kozlov},
journal= {arXiv preprint arXiv:2508.17001},
year = {2025}
}