= Solution
Fix a primitive $\lambda$ of the <standard symplectic form> on $\mathbb R^{2n}$. With the coordinates of Question 2 one convenient choice is
$$
\lambda=\frac12\sum_j(q_j\,dp_j-p_j\,dq_j),\qquad d\lambda=\omega_0.
$$
For a <Lagrangian submanifold> with inclusion $i:L\hookrightarrow\mathbb R^{2n}$, $d(i^*\lambda)=i^*\omega_0=0$. Its <Liouville class of a Lagrangian submanifold> is
$$
\boxed{\ell_L=[i^*\lambda]\in H^1_{\mathrm{dR}}(L;\mathbb R)}.
$$
Any two primitives differ by a closed one-form on the contractible ambient space, hence by an exact form, so the class is independent of the primitive. The cotangent primitive convention $\theta=\sum_jp_jdq_j$, with $\omega_0=-d\theta$, gives the negative of this class; rationality, positive period size and nonvanishing do not depend on that sign.
The period homomorphism sends an integral one-cycle $c$ to $\int_c i^*\lambda$. Its image $\operatorname{Per}(L)\subset\mathbb R$ is an additive subgroup. A <rational Lagrangian submanifold> is one for which this subgroup is discrete. In the nonzero case it has a unique positive generator:
$$
\boxed{\operatorname{Per}(L)=\gamma(L)\mathbb Z,\qquad\gamma(L)>0}.
$$
Equivalently $\ell_L$ is a real multiple of an integral cohomology class. The invariant $\gamma(L)$ is its <least positive Liouville period>, also expressible as $\inf\{a>0:a\in\operatorname{Per}(L)\}$. If the period group is zero, that infimum is $+\infty$ by the empty-set convention; some treatments instead reserve “rational” for the nonzero discrete case. Because the ambient space is contractible, the <Stokes theorem> identifies these periods with signed symplectic areas of disks whose boundaries lie on $L$.
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