--- title: Symplectic Geometry description: Symplectic geometry is the branch of differential geometry that studies symplectic manifolds. Learners will understand the mathematical structure of phase spaces, classical mechanics, and the geometric formulation of Hamiltonian systems. category: mathematics subcategory: mathematical-physics difficulty: beginner, intermediate, advanced url: /subject/symplectic-geometry --- # Symplectic Geometry Symplectic geometry is the branch of differential geometry that studies symplectic manifolds. Learners will understand the mathematical structure of phase spaces, classical mechanics, and the geometric formulation of Hamiltonian systems. ## Available Resources 4 Books • 1 Courses • 2 Websites ## Websites ### 1. nLab: Symplectic Geometry The nLab wiki entry on symplectic geometry, written from a category-theoretic and mathematical physics viewpoint. Defines symplectic manifolds and related structures, connects them to Hamiltonian mechanics, quantization, and higher geometry, and links to key references for further study. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/symplectic+geometry **Tags:** symplectic-geometry, symplectic-manifolds, hamiltonian-mechanics, geometric-quantization, differential-geometry ### 2. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Symplectic Geometry and Classical Mechanics Learn symplectic geometry and its applications to classical mechanics with Paul Biran's insightful course. Explore fundamental concepts! **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLDfPUNusx1EoVnrQcCRishydtNBYU6A0c **Tags:** symplectic-geometry, classical-mechanics, hamiltonian-systems, symplectic-manifolds, lecture-series ## Books ### 1. Mathematical Methods of Classical Mechanics (2nd Edition) **Author:** V. I. Arnold The book that makes Hamiltonian mechanics a branch of mathematics rather than a physics recipe. Graduate text recasting mechanics in the language of manifolds, differential forms and symplectic geometry, then developing Hamiltonian systems, canonical transformations, integrable systems, action-angle variables and KAM perturbation theory with complete proofs. Springer Graduate Texts in Mathematics 60. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387968903?tag=edmonddante07-20 **Tags:** classical-mechanics, hamiltonian-mechanics, symplectic-geometry, kam-theory, differential-forms ### 2. Introduction to Symplectic Topology **Author:** Dusa McDuff, Dietmar Salamon The standard graduate reference on symplectic topology from Oxford University Press. Develops linear symplectic algebra, symplectic manifolds, Hamiltonian flows, almost complex structures, symplectic capacities, and the Arnold conjecture, giving readers the background needed to read current research in the field. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0198511779?tag=edmonddante07-20 **Tags:** symplectic-topology, symplectic-geometry, hamiltonian-dynamics, symplectic-capacities, differential-topology ### 3. Symplectic Geometry **Author:** Michèle Audin, Ana Cannas da Silva, Eugene Lerman Expanded lecture notes from a 2001 CRM Barcelona summer school, published by Birkhäuser. Introduces symplectic and contact geometry through integrable Hamiltonian systems, covering Lagrangian submanifolds, toric manifolds, moment maps, and contact toric geometry, aimed at graduate students entering the area. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/3764321679?tag=edmonddante07-20 **Tags:** symplectic-geometry, integrable-systems, hamiltonian-systems, toric-geometry, contact-geometry ### 4. Lectures on Symplectic Geometry **Author:** Ana Cannas da Silva Springer Lecture Notes in Mathematics volume based on graduate courses. Moves from symplectic linear algebra and Darboux's theorem through Lagrangian submanifolds, Kähler manifolds, Hamiltonian group actions, moment maps, and symplectic reduction, giving a concise route into modern symplectic geometry. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/3540421955?tag=edmonddante07-20 **Tags:** symplectic-geometry, symplectic-manifolds, moment-maps, symplectic-reduction, kahler-geometry --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/symplectic-geometry