Introduction to the symplectic group Sp(2)

C. J. McKinstrie; M. V. Kozlov

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arXiv · 2508.17001

Introduction to the symplectic group Sp(2)

C. J. McKinstrie·M. V. Kozlov

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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.

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C. J. McKinstrie, M. V. Kozlov. 2025-08-23. Introduction to the symplectic group Sp(2). https://arxiv.org/abs/2508.17001

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