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In this article we study the fragility of Lagrangian periodic tori forsymplectic twist maps of the $2d$-dimensional annulus and prove a rigidityresult for completely integrable ones. More specifically, we consider$1$-parameter families of symplectic twist maps$(f_\varepsilon)_{\varepsilon\in \mathbb{R}}$, obtained by perturbing thegenerating function of an analytic map $f$ by a family of potentials$\{\varepsilon G\}_{\varepsilon\in \mathbb{R}}$. Firstly, for an analytic $G$and for $(m,n)\in \mathbb{Z}\times \mathbb{N}^*$ with $m$ and $n$ coprime, weinvestigate the topological structure of the set of $\varepsilon\in \mathbb{R}$for which $f_\varepsilon$ admits a Lagrangian periodic torus of rotation vector$(m,n)$. In particular we prove that, under a suitable non-degeneracy conditionon $f$, this set consists of at most finitely many points. Then, we exploitthis to deduce a rigidity result for integrable symplectic twist maps, in thecase of deformations produced by a $C^2$ potential. Our analysis, which holdsin any dimension, is based on a thorough investigation of the geometric anddynamical properties of Lagrangian periodic tori, which we believe is of itsown interest.

On the fragility of periodic tori for families of symplectic twist maps