It is interesting and fruitful to study differential geometry of Lagrangian submanifolds in specific Kähler manifolds and related Hamiltonian variational problems. In this talk, we focus on Lagrangian submanifolds in complex hyperquadrics, obtained from Gauss images of isoparametric hypersurfaces in unit spheres. We shall discuss our recent results on properties of such Lagrangian submanifolds and their Hamiltonian stability and Hamiltonian instability in the homogenous case. This talk is based on the joint work with Prof. Yoshihiro Ohnita.

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