In 1980’s, Lee proposed an energy-preserving discrete mechanics with variable time steps by taking (discrete) time as dynamical variable [2, 3, 4]. On the other hand, motivated by the symplectic property of Lagrangian mechanics, a version of discrete Lagrangian mechanics has been devoloped and variational integrators that preserve discrete symplectic two form have been obtained [11, 12, 15, 16,...

We also use soft methods from symplectic geometry (the relative version of the h–principle for Lagrangian immersions) and tools from algebraic topology to prove (both positive and negative) results about Lagrangian desingularizations of Lagrangian submanifolds with isolated singularities; we view the (Maslov-zero) Lagrangian desingularization problem as the natural soft analogue of the special ...

This article presents a novel direct Jacobian total Lagrangian explicit dynamics (DJ-TLED) finite element algorithm for real-time nonlinear mechanics simulation. The nodal force contributions are expressed using only the operator, instead of deformation gradient tensor and tensor, fewer computational operations at run-time. Owing to this proposed formulation, expressions developed strain invari...

We consider deformations of singular Lagrangian varieties in symplectic spaces. We prove that a Lagrangian deformation of a Lagrangian complete intersection is analytically rigid provided that this is the case infinitesimally. This result solves a problem posed by Colin de Verdière concerning Lagrangian curves. Finally, we prove the coherence of the direct image sheaves of relative infinitesima...

We verify in an elementary way a result of Peterson for the maximal orthogonal and Lagrangian Grassmannians, and then find Vafa-Intriligator type formulas which compute their 3point, genus zero Gromov-Witten invariants. Additionally, we study total positivity of the related Peterson varieties and investigate its relationship with the positivity of Schubert basis elements.

Lagrangian //-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian //-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean «-space and each unit speed curve F in the complex plane, we introduce the notion of the co...