  This paper provides a theoretical background for Lagrangian Descriptors
(LDs). The goal of achieving rigourous proofs that justify the ability of LDs
to detect invariant manifolds is simplified by introducing an alternative
definition for LDs. The definition is stated for $n$-dimensional systems with
general time dependence, however we rigorously prove that this method reveals
the stable and unstable manifolds of hyperbolic points in four particular 2D
cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic
saddle point for nonlinear autonomous systems, a hyperbolic saddle point for
linear nonautonomous systems and a hyperbolic saddle point for nonlinear
nonautonomous systems. We also discuss further rigorous results which show the
ability of LDs to highlight additional invariants sets, such as $n$-tori. These
results are just a simple extension of the ergodic partition theory which we
illustrate by applying this methodology to well-known examples, such as the
planar field of the harmonic oscillator and the 3D ABC flow. Finally, we
provide a thorough discussion on the requirement of the objectivity
(frame-invariance) property for tools designed to reveal phase space structures
and their implications for Lagrangian descriptors.
