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Re: [Phys-l] Action

A couple quick comments:

1. Why should there be only one parabolic path that minimizes the action for your situation? Backing up a step, we know that there are (in general) for every sufficiently large initial velocity two different firing angles that result in trajectories hitting a given target (if the starting velocity is too small -- i.e., the target is too far away -- there may only be one possible angle or indeed no possible angle). Anyway in general we can aim above or below the 45 degree angle and hit the same spot on a level field. And varying the initial velocity will yield different parabolas. So if even a basic treatment doesn't pin down the initial velocity and angle, we need not expect least action to do better. I think you need to specify not only the end points of the path, but also a starting velocity: speed and angle. (I've done this before by simply locking down the first _two points, as well as the last one.) Then least action will give you the parabolic path that meets these boundary conditions.

2. Why hardcode in conservation of energy? We normally find this as the first integration of the differential equation of motion (after separating variables and/or using other techniques). Least action is supposed to more or less hand us the equations of motion, so I think you're stacking the deck by including a result of least action as an assumption. I would use v^2 = v_x^2 + v_y^2, where the v_x and v_y values are found by simply subtracting neighboring x values and y values, respectively. Since you have a uniform distribution of x values, your v_x values will all be the same, but the v_y's will vary as the y's change.

Other than that, it looks good to me.

Ken Caviness
Physics @ Southern Adventist U

-----Original Message-----
From: [] On Behalf Of Josh Gates
Sent: Thursday, January 07, 2010 7:33 AM
To: Forum for Physics Educators
Subject: [Phys-l] Action

Hi everyone,

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?


Joshua Gates

Physics Faculty
Tatnall School - Wilmington DE
Johns Hopkins Center for Talented Youth

Forum for Physics Educators