Comment on “A variational theory of hyperbolic Lagrangian coherent structures, Physica D 240 (2011) 574–598”

In this brief note, we state the main result of Haller [G. Haller, A variational theory of hyperbolic Lagrangian coherent structures, Physica D 240 (7) (2011) 574–598] on a variational approach to Lagrangian coherent structures (LCSs) more precisely by adding an extra assumption to ensure the existence of the involved derivatives. Under that assumption we further simplify the positive definiteness condition on the matrix L, which is also referred to as “hyperbolicity test”, to a condition on the second-order directional derivative of the largest eigenvalue. This extends an observation in the two-dimensional case made by Farazmand and Haller [M. Farazmand, G. Haller, Erratum and addendum to “A variational theory of hyperbolic Lagrangian coherent structures [Physica D 240 (2011) 574–598]”, Physica D 241 (4) (2012) 439–441] on the simplified detection of LCS from finite-time Lyapunov exponent ridges.

► We state the main theorem of Haller (2011) [1] more precisely by adding an assumption.
► This specification leads to a major simplification of the LCS criteria for flows in arbitrary dimensions.
► The consequences of this specification are discussed for LCS detection by FTLE ridges.