As a participant in the 2012 REU program at Rutgers (Click here for their website.), I will be studying symplectic geometry and its application to analyitcal mechanics, mentored by Professor Shabnam Beheshti (For her website, click here). The eventual goal of this project is to apply analytical mechanics to material science, specifically to the modeling of molecular dynamics.
This May I have completed an introductory analytical mechanics course with Professor Beheshti in which we covered the Lagrangian formulation and introduced Hamilton's technique. The goal of this project is to continue studying Hamilton's formulation with emphasis on the geometry involved, specifically, how the phase space of Hamiltonian systems are naturally symplectic manifolds.
Completed final revision of a write-up of previous semester's work on analyitcal mechanics, gathered relevant resources including Symplectic Geometry and Analytical Mechanics and The Mathematics of Soap Films. Reviewed an excerpt from An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition on alternating tensors and the exterior product in preparation for a discussion on symplectic vector spaces.
Proved that finite dimensional vector spaces endowed with symplectic forms must be of even dimension, as well as other results about symplectic matrices. Read sections 3.1-3.5 in Introduction to Analytical Dynamics on canonical transforms and reviewed notes from Prof. Beheshti on Principal curvatures and regular surfaces.
Proved several basic results about regular surfaces, including uniqueness of the tangent plane and independence of parametrization of a few important quantities defined on surfaces. Read from The Mathematics of Soap Films and Constant Mean Curvature Surfaces, Harmonic Maps, and Integrable Systems about the first and second fundamental forms, mean curvature, and their linear algebraic relationship. Read through Do Carmo's proof that Surfaces with zero mean curvature are minimal with respect to surface area and filled in the details. Read Oprea's proof of the existence of an isothermal parametrization of a minimal surface.
Read from Oprea regarding the basics of complex variable theory and some elementary results from complex analysis. Particularly focused on understranding the geometric properties associated with holomorphic functions. Proved that holomorphic functions with a nowhere vanishing derivative are conformal maps (angle preserving) from C → C. My hope is to use the language of complex analysis to better understand the importance/existence of isothermal coordinates on surfaces.
Looked back at linear symplectic structures, and found an interesting interpretation in introductory notes on symplectic topology, by Professor Sergei Tabachnikov. If one identifies R^2n with C^n, then the standard dot product and standard symplectic form are related in that they are the corresponding real and imaginary parts of the standard hermitian form (Complex dot product) on C^n. Also showed that for every finite dimensional vector space V, the direct sum of V and the dual of V has a natural symplectic structure.