I haven't dealt with action in a long time, so I'm a bit fuzzy on the
particulars at the moment. Here's what I'm trying to do:
Given a starting point (0,0) and an ending point (5m, 9.08m), I'm trying
to show that the parabolic path beginning with a 70 degree initial angle
from +x (the path given by N's laws, kinematics, etc.) minimizes the action.
Here's how I'm trying to do it (which apparently has one or more flaws):
- I made a spreadsheet, with the columns x, y, v, KE, PE, E, K-U
* x increments in .1 m steps from 0 to 5m
* y is a function of x, defining the path
* v is root(v_i^2-2gy), satisfying cons. of E
* KE and PE are defined in the ordinary way
* E is there to check my formulas, verifying cons. of E
* I average all of the K-U entries to give something similar to the action
Since the x steps are all the same, integrating K-U dx and dividing by
the total delta x should give me the same thing that the average does (I
think). It occurs to me now that there's a problem with paths that go
straight up at any point, but I'm willing to work with that later. My
current issue is that there are other parabolic paths that give a lower
K-U average than the correct path.
Anyone see where I went awry?
Thanks,
Josh
--
Joshua Gates
Physics Faculty
Tatnall School – Wilmington DE
Johns Hopkins Center for Talented Youth