Symplectic according to some sources is the state of a function which has symplecticness. More usefully it means that a map (which in algebra were called functions) conserves the Hamiltonian.
The Hamiltonian of a system is the Kinetic plus Potential energy of that system. So a symplectic mapping is a mathematical function which upholds energy conservation in the system as far as energy is defined for the purpose of the system. So complex and large systems such as the solar system can be simulated in a computer using a Symplectic Integrator in million year projections.
The way the formulation works is thus:
Where H = T + U, (Hamil. = KE + PE)
dq/dt = ∇pT = ∂T/∂p
dp/dt = ∇qU = ∂U/∂q
The Symplectic Integrator does not need to be extremely complex itself, an Explicit Euler's Method will do, which is striking given Euler Method's typical inaccuracy.