This article deals with the development of a self-consistent theory of coherent trajectory patterns in dynamical systems defined over a finite time-interval. It is used the term Lagrangian coherent structures (LCSs) [G. Haller and G. Yuan, Physica D 147, No. 3–4, 352–370 (2000; Zbl 0970.76043)] to describe the core surfaces around which such trajectory patterns form. As proposed there repelling LCSs are the core structures generating stretching, attracting LCSs act as centerpieces of folding, and shear LCS delineate swirling and jet-type tracer patterns. In order to act as organizing center for Lagrangian patterns, LCSs are expected to have two key properties:
An LCs should be a material surface, i.e., a codimension-one invariant surface in the extended phase of a dynamical system. This is because (a) an LCS must have sufficiently high dimension to have visible impact and act as a transport barrier and (b) an LCS must move with the flow to act as observable core of evolving Lagrangian patterns.
An LCS should exhibit locally the strongest attraction, repulsion or shearing in the flow. This is essential to distinguish the LCS from all nearby material surfaces that will have the same stability type, as implied by the continuous dependence of the flow on initial conditions over finite times.
The authors develop a mathematical theory that clarifies the relationship between observable LCSs and invariants of the Cauchy-Green strain tensor field. Motivated by physical observations of trajectory patterns hyperbolic LCSs are defined as material surfaces (i.e., codimension-one invariant manifolds in the extendend phase space) that extremize an appropriate finite-time normal repulsion or attraction measure over all nearby material surfaces. Weak LCss (WLCSs) are defined as stationary solutions of the above variational problem. Solving these variational problems computable sufficient and necessary criteria are obtained for WLCSs and LCSs that link them rigorously to the Cauchy-Green strain tensor field. The notion of a Constrained LCS (CLCS) is introduced that extremizes normal repulsion or attractor under constraints. This construction allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.