The primary objective of this paper is to demonstrate the utility of symplectic numerical techniques for ray tracing within gradient-index media. The relevant mathematics are explained in brief, deriving the optical Hamiltonian independently of the Lagrangian optical formalism before constructing a symplectic ray tracing algorithm. Numerical experiments with the Luneburg and Maxwell fish-eye lenses compare the effectiveness of symplectic methods with standard numerical integration techniques, challenging the idea that the increased accuracy of higher-order numerical methods justifies their elevated computational cost. Further uses for symplectic ray tracing are also discussed.