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.

Sym*plec"tic (?), a. [Gr. plaiting together, fr. to plait together.] Anat.

Plaiting or joining together; -- said of a bone next above the quadrate in the mandibular suspensorium of many fishes, which unites together the other bones of the suspensorium.



The symplectic bone.


© Webster 1913.

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