= Symplectic plane
{title2=$(\mathbb R^2,dq\wedge dp)$}
The standard symplectic plane is $\mathbb R^2$ with the closed nondegenerate two-form $\omega=dq\wedge dp$. For tangent vectors $u=(a,b)$ and $v=(c,d)$,
$$
\omega(u,v)=ad-bc,
$$
their signed parallelogram area. This alternating pairing is not a positive-definite inner product: every vector pairs to zero with itself.
A linear map $A$ preserves the form exactly when $\det A=1$, because $\omega(Au,Av)=\det(A)\omega(u,v)$. Thus $\operatorname{Sp}(2,\mathbb R)=\operatorname{SL}_2(\mathbb R)$. A nonlinear example is the shear $(q,p)\mapsto(q,p+q^2)$, whose pullback form is $dq\wedge(dp+2q\,dq)=dq\wedge dp$.
Every smooth embedded curve in the plane is <Lagrangian>: the restricted two-form vanishes on its one-dimensional tangent spaces, which have half the ambient dimension. The symplectic plane is therefore the simplest model of symplectic area preservation and Lagrangian geometry.
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