Lagrangian pairs.

The idea that Lagrangians come in pairs can be understood; that it is a consequence of the fact of obsever-dependent measurements.

We simply need to understand that the presence of one Lagrangian immediately implies another. We can suggest that Lagrangians come in j= 2n multiples where n takes on specific values and n>0 and j is the number of Lagrangians in the system.

There is a reasoning that we can use to come to this point of view.

Consider that from Newton...we can imagine that a particle in motion continues in its state of motion unless otherwise disturbed by an external stimulus.

From Einstein...we can understand that a particle in freefall does not sense its own motion through the gravitational field.

These facts, if we consider them in tandem, lead us to the notion that a Lagrangian cannot exist in isolation. There must always be at least two Lagrangians or more but never just one. For example, a system with three distinct energy packets has n =3/2 and j = 3. The reason for this is trivial, an isolated energy packet of any size cannot observe its own motion, there must be an observer present who can define motion. Of-course, implicit in this argument, the fact that ordinary spacetime can be defined as flat within an infinitesimally small volume or at some appropriate scale so that at such scales the background geometry does not become an 'observer'.

Notice also that the assertion that there always exists more than one Lagrangian is independent of the background geometry. However, I shall show that even though such an assertion is made independently of the background (curved spacetime or flat spacetime) it is possible to construct a geometry that identifies the same said assertion.