Peeter Joot's (OLD) Blog.

Math, physics, perl, and programming obscurity.

Archive for January, 2010

nausea

Posted by peeterjoot on January 28, 2010

Focus is so hard right now.

It’s hard to see through a shitstorm.

My stomach twitches all the time.

Not in a good way.

At least I am sleeping again.

Exercise helps.

I punish my body to clear the head.

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maintainance needs to be timely.

Posted by peeterjoot on January 23, 2010

A man and his wife bought a first car together.

It did not come with an owners manual.

In the beginning they had fun driving and taking care of it.

Before long they also got a nice trailer for it.

A short while later got the roof rack.

Eventually the novelty of the car wore off, and it got neglected more and more.

It got some scratches.  They could have been buffed out.

It got some dents.  They could have been hammered out.

The wife saw the effects of the neglect much earlier, and complained to the husband about them.

“Take the car in for service, or else it will break down.”

He took it in for oil changes occasionally, but didn’t give it the 50000 km service it needed.

The wife complained more and more about all the little issues with the vehicle that were accumulating.

The husband grew to resent the complaining, now about a lot more than the car.

He was dumb, and didn’t understand the root causes for the complaining.

The car did survive a long time, but was no longer pretty.

Eventually the wife saw a new shiny car for herself.

Something exploded under the hood of the old car one day, and it rolled into a ditch.

She was hurt getting out, but didn’t turn back.

She even gave the car a little kick on the way out.

Maybe then the husband would get rid of it.

The husband still loved the car, even if it was beaten up.

He insisted that it could be fixed.

Other owners have the same problems.

The years of skipped maintainance could be made up for.

But by this time it was too late.

She wasn’t interested in the old car any longer.

The shiny new car was so much more appealing, even if she couldn’t buy it right away.

Eventually the husband understood the wife, and said goodbye to the old car.

He’ll get himself a new car too eventually.

They have to figure out what to do with the roof rack and trailer.

Over time those became the best features of the old car.

There is no easy way to fit those nicely onto both replacement vehicles.

Posted in Incoherent ramblings | 3 Comments »

rocked

Posted by peeterjoot on January 18, 2010

I sit bewildered and numb.  Physically shaking.

Such a cruel and painful request.

Resistance.   “Please no.”

Posted in Incoherent ramblings | 4 Comments »

Noise.

Posted by peeterjoot on January 13, 2010

Deafening silence.

A dysfunctional sub woofer pounding my body at near zero hertz.

Noise is eminent, its potential building like entropy.

Time passes.

The switch will be thrown.

Will the fuse blow, or will something absorb the sound?

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darkness.

Posted by peeterjoot on January 12, 2010

Dead cut flowers.

I throw out some more.

Change the stagnant water.

It would not have taken much too keep them looking pretty longer.

But they would die anyways.

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Integrating the equation of motion for a one dimensional problem.

Posted by peeterjoot on January 2, 2010

[Click here for a PDF of this post with nicer formatting].

Motivation.

While linear approximations, such as the small angle approximation for the pendum, are often used to understand the dynamics of non-linear systems, exact solutions may be possible in some cases. Walk through the setup for such an exact solution.

Guts

The equation to consider solutions of has the form

\begin{aligned}\frac{d}{dt} \left( m \frac{dx}{dt} \right) = -\frac{\partial {U(x)}}{\partial {x}}.\end{aligned} \hspace{\stretch{1}}(2.1)

We have an unpleasant mix of space and time derivatives, preventing any sort of antidifferentiation. Assuming constant mass m, and employing the chain rule a refactoring in terms of velocities is possible.

\begin{aligned}\frac{d}{dt} \left( \frac{dx}{dt} \right) &= \frac{dx}{dt} \frac{d}{dx} \left( \frac{dx}{dt} \right) \\ &= \frac{1}{{2}} \frac{d}{dx} \left( \frac{dx}{dt} \right)^2 \\ \end{aligned}

The one dimensional Newton’s law (2.1) now takes the form

\begin{aligned}\frac{d}{dx} \left( \frac{dx}{dt} \right)^2 &= -\frac{2}{m} \frac{\partial {U(x)}}{\partial {x}}.\end{aligned} \hspace{\stretch{1}}(2.2)

This can now be antidifferentiated for

\begin{aligned}\left( \frac{dx}{dt} \right)^2 &= \frac{2}{m} (E - U(x)).\end{aligned} \hspace{\stretch{1}}(2.3)

Taking roots and rearranging produces an implicit differential form x in terms of time

\begin{aligned}dt = \frac{dx}{\sqrt{ \frac{2}{m} (E - U(x)) } }.\end{aligned} \hspace{\stretch{1}}(2.4)

One can concievably integrate this and invert to solve for position as a function of time, but substitution of a more specific potential is required to go further.

\begin{aligned}t(x) = t(x_0) + \int_{y=x_0}^{x} \frac{dy}{\sqrt{ \frac{2}{m} (E - U(y)) } }.\end{aligned} \hspace{\stretch{1}}(2.5)

TODO: doing stuff with this.

EDIT: This was a stupid way to do this. It is nothing more than rearranging the Hamiltonian for \dot{x}^2.

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