calculus of variations

i am about to start the topic of variational calculus in classical mechanics.   The idea is that you describe motion as a trajectory (path) in time, and you imagine other paths than the one actually followed.   we call this imagined path the varied path “del-x”.   A key idea is the difference between this type of variation and the derivative from ordinary calculus,  usually called dx:   The derivative compares position at different times, where-as the del-x compares positions at the same time. 

What is he difference between dreaming and hallucinating?  Dreams occur in an imagined time-- events can happen as fast or slow as necessary.  in this sense

they are derivative, the time in which they occur is as imagined as the dream itself.   Things can be imagined to happen fast or slow.  Hallucinations, however occur in real time.  They are analogous to the path variations.  As I experienced them, they could not be sped up or slowed down;  they had to be lived through.    In the course of many long hours, they became more elaborate and ritualized.   For example, with my bed sheet as keyboard and mouse I was typing long paragraphs on the wall and ceiling.   I was seeing all the objects in my room as geometric shapes,  then projecting them onto the walls in two-dimensions and trying to draw lines through all the shapes using my mouse.   this would become more difficult as people would bring supplies into my room and leave them.  the holes in my ceiling tiles needed to be organized into patterns.  yes, pattern recognition needed to be done everywhere.  The hair on the head of the nurse become an elaborate pattern recognition problem, deconstructed into various geometric shapes.  mountains and stars appeared in the window that needed to be catalogued.  Over my shoulder, a man was standing on a platform against the wall praying.  And then one night I was talking in great detail about Egyptian astronomy with a doctor from “the institute”  who was able to dazzle me by spinning his arms.   (There was the clock on the wall above the bathroom door).   But the point is it went on and on, in real time.  Which was two weeks, day and night.   Towards the end, reality started to mix in but is was always hard to tell what was real and what wasn’t.  Meg was there, and our friend Susan.   But so was the little girl in the corner that Meg couldn’t see and the family of young Jewish immigrants that came and camped in my room after Meg left at night.  Please leave and let me get some rest!  

But all this in real-time.  calculus of variations.