Symplectic

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.

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.

In his monumental Mathematical methods of classical mechanics, V.I. Arnold states that

Classical mechanics is geometry in phase space. Phase space has the structure of a symplectic manifold

A manifold is a structure that the vicinity of any given point has the structure of a vector space (plus some technicalities). By vicinity of a point I mean, somewhat informally, the set of points that have a distance of less than epsilon, for some value of epsilon. A symplectic manifold is a manifold equipped with a closed nondegenerate skew-symmetric 2-form w(.,.) called the symplectic form, much like a Riemannian manifold is equipped with a closed nondegenerate symmetric 2-form, the metric form that generalizes inner products to the "curved" context of manifolds and gives a notion of length of a curved path, etc.

To present the qualitative features of symplectic structure, I'm going to restrict myself to discussing symplectic vector spaces, which are particular symplectic manifolds where a single coordinate system (identified in linear algebra with the basis). The rest is a bunch of record-keeping that comes useful when trying to generalize to non-intuitive cases. To expand our definition of the symplectic form:

1. 2-form (or bilinear map): a scalar-valued function of two vectors such that the rules of linear algebra apply to each vector (for example: f(u,v)+f(w,v) = f(u+w,v). A well known bilinear map is the inner product. Another bilinear map is the determinant of the two-by-two matrix that has u, v as columns.
2. Skew-symmetry: The property that f(u,v)=-f(v,u)
3. Non-degeneracy: The property that if f(u,v)=0 then either one of u,v is zero or both.

It's not hard to prove that for a vector space to have a symplectic form satisfying all of the above, it must be even-dimensional with a basis {e₁,...e_n,f₁,...f_n} such that w(e_i,e_j) = w(f_i,f_j)=0 for any i,j and w(e_i,f_j)=1 iff i=j. Furthermore, the symplectic vector space admits a decomposition V = span(e₁,f₁)⊕ ... ⊕ span(e_n,f_n)

If the dimension of our vector space is 2, skew-symmetry implies that the symplectic form will be w((u₁,u₂),(v₁,v₂)) = u₁ v₂ - u₂ v₁) -- i.e. the determinant or area spanned by two vectors. Thus the symplectic form is often referred as "symplectic area" to emphasize its geometric nature; but because of the decomposition of vector spaces that admit a symplectic form in disjoint "symplectic subspaces", the symplectic area between 2n vectors will be the sum of 2x2 areas. The "symplectic area" is then counterintuitive in all sorts of wonderful ways.

This poses restrictions on what are the allowable linear transformations that preserve symplectic geometry. A symplectic linear transformation is one that satisfies A^TJA = J where J is given in block matrix form by

   ( 0 | I )
J =( -I| 0 )

By a reasonable argument with infinitesimal limits, a nonlinear transformation is said to be symplectic if its Jacobian matrix preserves symplectic form. As it turns out, the flow maps (which take points in phase space to other points in phase space) of Hamiltonian systems are symplectic. This isn't terribly difficult to see, and advanced physics textbooks will work this out if you really need it.

More abstractly, symplectic geometry is the founding principle of modern classical mechanics (which is by now more or less a field of Pure Mathematics). Hamiltonian mechanics was originally founded on variational principles and to some concrete degree the phase space of a system with n degrees of freedom can be modeled as either R²ⁿ or a subset of it; but the modern theory manages to start from symplectic structure on manifolds, develop vector fields on cotangent bundles and have Hamiltonian structure fall out of it.

On the Applied Mathematics setting, symplectic structure is interesting because it's a fundamental qualitative structure of Hamiltonian systems, which in turn describe many kinds of systems that fall out of variational principles. Most of these are physical in nature, but there are also Hamiltonian problems in control theory (and, from there, in economics) and in MCMC methods of statistical inference. Therefore, computational methods that have as a desiderata to preserve the qualitative Hamiltonian structure can be designed so that the more concrete property of symplecticity is preserved. This is the field of geometric numerical integration.

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.

n.

The symplectic bone.

© Webster 1913.