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*From*: John Denker <jsd@av8n.com>*Date*: Sat, 12 Dec 2020 19:07:41 -0700

Hi --

Another couple of thoughts on the recalcitrant differential equation.

Last time I said "make a graph of what you know" at the start of the

exercise. That's always good advice. I also said beware of the

proximity of other solutions. We are about to see that concept return

in a big way.

This time let's plot something that we /didn't/ know at the outset.

That is, we use 20-20 hindsight to work backwards from the solution.

That's sometimes (not often, but sometimes) a good way to solve the

problem, and it is very often a way to /understand/ the problem after

it has been solved. In particular, we can work backwards all the way

to the starting point, and then identify the fewmets that should have

told us the situation was problematic.

Here's what we know:

1) Using 20-20 hindsight, we know the solution is h = ½ g t².

2) At the beginning of the exercise we wrote h' = √(2 g h). The RHS

is clearly a function of h. Everything we told the computer was

expressed as a function of h.

3) We are only interested in solutions for positive t and positive h,

but the computer doesn't know this. It can't read minds. Also, we

must consider the possibility that the calculation of h is not quite

exact. In particular, suppose at the point where h is supposed to be

zero, our best estimate of h comes out slightly negative. Then the

square root becomes imaginary. This is a huge fewmet!

Whenever you see a variable that might go imaginary behind your back,

you should say hmmm, the last ten times this happened I got badly

burned; maybe I should be careful here, or start over and reformulate

the whole problem so this doesn't happen. At the very least, it's

time to haul out some additional machinery, starting with a

root-locus plot, like this:

https://www.av8n.com/physics/img48/dt-dh-root-locus.png

4) You can see that as the points get closer to the origin, points

that are equally spaced in terms of h become disproportionately far

apart in terms of t. In fact dt/dh becomes infinite. This is

already something to worry about.

5) The equation of motion was derived from an energy principle.

That's always asking for trouble. The momentum will tell you the

kinetic energy, but the kinetic energy will not (in general) tell you

the momentum. (Given the *Lagrangian* you can figure out the

momentum, but that's the answer to a different question.)

Given a certain h such as h=1, there are two possible times when that

could occur, namely t=-1 (inbound, shown in red in the diagram) and

t=+1 (outbound, shown in blue in the diagram. This double-valuedness

is a huge worry. This is a /proximity/ issue. The outbound

trajectory is awfully close to the inbound trajectory.

6) The trajectory exists only for h≥0, but the universe as a whole

still exists even for h<0. For example, negative h could arise due

to a miscalculation. There are always uncertainties. We can revisit

item 4, because a tiny amount of negative h creates a huge amount of

imaginary t.

We can also revisit item 5, because the solution for positive t is in

proximity to negative t *and also* to imaginary t, as we now see. So

the "proximity" issue is even worse than you might have imagined when

reading my previous note.

You know and I know that we're not interested in imaginary-time

solutions, but as far as the computer knows, the system could have

arrived at h=0 from below, via imaginary time.

===================

We needed to know the solution to construct the root-locus plot, but

we have now learned from it a bunch of ideas that we can apply to the

original unsolved problem, and in particular to a wide range of other

problems that we may encounter.

a) Phase space is a thing. Phase space is your friend. It does not

get nearly enough emphasis in most classes nowadays. If you know

what's going on in phase space you can figure out the energy, but not

vice versa. Trying to derive the equation of motion from the energy

is asking for trouble. It's better to try to formulate things in

terms of phase space starting from Day One. It's worth taking some

time to reformulate them if necessary. Symplectic integrators are

your friend.

b) Square roots may seem familiar and innocuous, but near the origin

they are asking for trouble, and they are representative of a wider

class of troublesome things. For starters, √h is not differentiable

at h=0. Trying to solve differential equations at places where

things are not differentiable is no fun.

c) If h = t² that does *not* mean that t=√h. The correct solution is

t=±√h. That ± sometimes matters a lot. That may sound like "duh"

... but I have seen physics professors at Big Name universities fool

themselves this way.

d) Near h=0, not only do you need to worry about negative-t solutions

(which may or may not be physically significant), you also need to

worry about imaginary-t solutions (which may or may not be physically

significant). The /equation/ doesn't know what's significant and

what's not. It can't read minds. "Beware the proximity of other

solutions."

When in doubt, draw the root-locus plot. You might discover some

new physics. Sometimes the analytic continuation of your equation

has some hitherto-unappreciated meaning.

Here's a famous example: Suppose d/dt d/dt y = y. Suppose you know

you are looking for a decaying exponential solution. The differential

equation is notoriously impossible to solve numerically, because

there is also an exponentially *increasing* solution, and no amount

of fiddling with the initial conditions will keep that solution from

sneaking in and wiping out your answer.

e) The fact that dh/dt goes to zero may seem innocuous, but the fact

that dt/dh goes to infinity should make your hair catch on fire.

Using good old physics intuition you may think of the situation in

terms of functions of t, but the computer can't read minds. The

equation that the computer sees is a function of h. Yesterday's graph

https://www.av8n.com/physics/img48/dt-dh-integrator.png

as well as the dot-spacing in today's graph

https://www.av8n.com/physics/img48/dt-dh-root-locus.png

should raise the alarm.

**Follow-Ups**:**Re: [Phys-L] Mathematica question; stiff differential equation***From:*Carl Mungan <mungan@usna.edu>

**References**:**[Phys-L] simple Mathematica question (one more typo fixed)***From:*Carl Mungan <mungan@usna.edu>

**Re: [Phys-L] simple Mathematica question (one more typo fixed)***From:*Francois Primeau <fprimeau@gmail.com>

**Re: [Phys-L] Mathematica question; stiff differential equation***From:*John Denker <jsd@av8n.com>

**Re: [Phys-L] Mathematica question; stiff differential equation***From:*Carl Mungan <mungan@usna.edu>

**Re: [Phys-L] Mathematica question; stiff differential equation***From:*John Denker <jsd@av8n.com>

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