Oscillation Numbers for Continuous Lagrangian Paths and Maslov Index

In this paper we present the theory of oscillation numbers and dual for continuous Lagrangian paths in $${\mathbb {R}}^{2n}$$ . Our main results include a connection given path with Lidskii angles special symplectic orthogonal matrix. We also Sturmian type comparison separation theorems difference two paths. These results, as well definition number itself, are based on comparative index (Elyseeva, 2009). The applications these directed to Maslov derive formula via matrix, hence express certain transformed path. methods generalization recently introduced conjoined bases linear Hamiltonian systems 2019 2020) between matrices (Šepitka Šimon Hilscher, 2021).