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It's a well established concept in various fields of physics that the action of the field / trajectory that becomes physically real, minimizes / maximizes the action functional. For the calculations, we only require the action to be extremal, which means, in principle if the value of the action would be kind of a "saddlepoint", this would suffice for the equations of motion to hold.
Even before asking such a question: Can there be an equivalent to a "saddlepoint" for functionals definded over a function space? If so, is it a meaningful definition to define a "saddlepoint" as the case when some trajectories in the vicinity of the extremal trajectory do have higher action, while others have lower action?
If so: are there any cases out there in any field of physics, where the extremal trajectory actually is such a "saddlepoint" trajectory?
Is there any case (from any physics field) where the choosen trajectories follow some of those saddlepoint lines? Here, by saddlepoint I mean that some trajectories in the vicinity of the extremal trajectory do have higher action, while others have lower action.