Finding mathematical patterns in physics text examples

 Posts: 5
 Joined: Tue Nov 11, 2008 6:51 pm
Finding mathematical patterns in physics text examples
Hi,
After pushing my way through an example on the calculus of variations from Marion and Thornton 'Classical Dynamics' it occurred to me, (for about the 100th time), that there must be handy rules of thumb or easily recognizable patterns that experienced physicists recognize. For example, how do you know which 'clever substitution of variables' to perform when evaluating an integral? There are many other examples of such questions. I've came across three more of them at:
http://copaseticflow.blogspot.com/2008/ ... dfew.html
just expanding the aforementioned example from Marion and Thornton.
Does anyone know of a list of such patterns? For example, I know it's handy, (actually kind of crucial), to memorize all the trig identities. What other rules and patterns should physics students memorize to make their lives easier?
Thanks!
Hamilton
After pushing my way through an example on the calculus of variations from Marion and Thornton 'Classical Dynamics' it occurred to me, (for about the 100th time), that there must be handy rules of thumb or easily recognizable patterns that experienced physicists recognize. For example, how do you know which 'clever substitution of variables' to perform when evaluating an integral? There are many other examples of such questions. I've came across three more of them at:
http://copaseticflow.blogspot.com/2008/ ... dfew.html
just expanding the aforementioned example from Marion and Thornton.
Does anyone know of a list of such patterns? For example, I know it's handy, (actually kind of crucial), to memorize all the trig identities. What other rules and patterns should physics students memorize to make their lives easier?
Thanks!
Hamilton
 coreycwgriffin
 Posts: 249
 Joined: Fri Nov 07, 2008 1:28 am
Re: Finding mathematical patterns in physics text examples
I was actually thinking about calculus of variations this morning in the shower (don't ask) and thought it was really strange that I have only ever seen it in my classical class (I used T and M for my text, too).

 Posts: 5
 Joined: Tue Nov 11, 2008 6:51 pm
Re: Finding mathematical patterns in physics text examples
Aaahhh, good! Well, maybe I'll never see it again after T&M. Although it seems like a fairly handy trick for finding the minimums of integrals of diff eqs.
Being able to recognize the patterns in these derivations, examples and homework patterns ahead of time is what's keeping me up. Any ideas?
Hamilton
Being able to recognize the patterns in these derivations, examples and homework patterns ahead of time is what's keeping me up. Any ideas?
Hamilton
Re: Finding mathematical patterns in physics text examples
As you will find in your classical mechanics class in grad school, there was another dude named Hamilton who was pretty fond of the calculus of variations. Not to mention Lagrange and Feynman. No, you're not done with the variational principle yet. In fact, it's crucial to many parts of physics.
 WontonBurritoMeals
 Posts: 102
 Joined: Sat Feb 23, 2008 9:43 pm
Re: Finding mathematical patterns in physics text examples
Don't take this the wrong way Doom, but I think I'm in love with you.As you will find in your classical mechanics class in grad school, there was another dude named Hamilton who was pretty fond of the calculus of variations. Not to mention Lagrange and Feynman. No, you're not done with the variational principle yet. In fact, it's crucial to many parts of physics.
May the wind be always at your back,
Wonton Burrito Meals
 coreycwgriffin
 Posts: 249
 Joined: Fri Nov 07, 2008 1:28 am
Re: Finding mathematical patterns in physics text examples
You do know that Thornton and Marion covers the Hamiltonian and Lagrangian, and are thus included in my statement, right?doom wrote:As you will find in your classical mechanics class in grad school, there was another dude named Hamilton who was pretty fond of the calculus of variations. Not to mention Lagrange and Feynman. No, you're not done with the variational principle yet. In fact, it's crucial to many parts of physics.

 Posts: 5
 Joined: Tue Nov 11, 2008 6:51 pm
Re: Finding mathematical patterns in physics text examples
Hi Doom,
You're right about the Hamiltonian. That's what M&T are working up to in Chapter 6.
So, getting back on the topic, have you ever seen a list of 'clever' physics 'patterns'? Is there a way to know when to multiply a quantity by x/x or to make the substitution x=sin(theta) to make an integral simpler? The books tend to indicate that kind of knowledge can only be gained by experience, but experience is only building up a bank of patterns and memories right?
Hamilton
You're right about the Hamiltonian. That's what M&T are working up to in Chapter 6.
So, getting back on the topic, have you ever seen a list of 'clever' physics 'patterns'? Is there a way to know when to multiply a quantity by x/x or to make the substitution x=sin(theta) to make an integral simpler? The books tend to indicate that kind of knowledge can only be gained by experience, but experience is only building up a bank of patterns and memories right?
Hamilton
Re: Finding mathematical patterns in physics text examples
Yeah, it's called "taking math classes" and "doing problems." That's how you know.
 coreycwgriffin
 Posts: 249
 Joined: Fri Nov 07, 2008 1:28 am
Re: Finding mathematical patterns in physics text examples
Rofl, your comments are hilarious.twistor wrote:Yeah, it's called "taking math classes" and "doing problems." That's how you know.
Re: Finding mathematical patterns in physics text examples
And also true.coreycwgriffin wrote:
Rofl, your comments are hilarious.

 Posts: 5
 Joined: Tue Nov 11, 2008 6:51 pm
Re: Finding mathematical patterns in physics text examples
Your comments are funny. But seriously, wouldn't you have liked it if you had a list of patterns, (think software design patterns), you could study to have a clue which direction the professor's head was pointed? I took the math classes too. I'm looking for ways to make things simpler.
So, I'm guessing no one has ever seen anything like that.
Hamilton
So, I'm guessing no one has ever seen anything like that.
Hamilton
Re: Finding mathematical patterns in physics text examples
I do feel your pain, hcarter333. I know exactly what you're talking about. Physics books are constantly pulling random tricks, approximations, assumptions, often with little or no justification or explanation, in derivations or solutions to problems. It seems like every problem has something << something else.
If an approximation/assumption is stated, but not justified mathemactically or intuitively, we could still follow the derivation, but we could never fully understand it, nor could we have produced the result ourselves because it involves a step that we can't take without already knowing something beyond the problem.
Lots of problems are like this. In fact, in one of the derivations in a plasma text I saw, they expanded exp(x) to (1+x) in an ODE expression just so they could get an analytical answer, even though the assumption seemed questionable. Later in an extention of this derivation, they integrated the resultant function from 0 to infinity. BUT the original assumption was not valid near zero... How could they do that? They did not address this concern at all.
Sadly, there's no book of tricks that I know of that you can buy to elucidate these magic steps. Having majored in math and physics, I've come to think that it's not something we're missing as students, but that physics books and instructors are just not as good as those in math. The fact that physics models the real world leads more often to intractable problems and so the writers often make simplifications. But they usually don't do the extra work needed to justify the simplifications in detail. Also, any steps that are purely mathematical (like the trick you shared in your first post), are merely stated without motivation because they only care about physics.
If an approximation/assumption is stated, but not justified mathemactically or intuitively, we could still follow the derivation, but we could never fully understand it, nor could we have produced the result ourselves because it involves a step that we can't take without already knowing something beyond the problem.
Lots of problems are like this. In fact, in one of the derivations in a plasma text I saw, they expanded exp(x) to (1+x) in an ODE expression just so they could get an analytical answer, even though the assumption seemed questionable. Later in an extention of this derivation, they integrated the resultant function from 0 to infinity. BUT the original assumption was not valid near zero... How could they do that? They did not address this concern at all.
Sadly, there's no book of tricks that I know of that you can buy to elucidate these magic steps. Having majored in math and physics, I've come to think that it's not something we're missing as students, but that physics books and instructors are just not as good as those in math. The fact that physics models the real world leads more often to intractable problems and so the writers often make simplifications. But they usually don't do the extra work needed to justify the simplifications in detail. Also, any steps that are purely mathematical (like the trick you shared in your first post), are merely stated without motivation because they only care about physics.

 Posts: 5
 Joined: Tue Nov 11, 2008 6:51 pm
Re: Finding mathematical patterns in physics text examples
Thanks quizivex! It sounds like there's a good opportunity as well as pain. If someone, maybe even a team of people, maybe even from this forum put together a compendium of patterns, there should be a market. Not only that, I've got to believe it would help out the field as a whole if more students could become intuitive sooner.
Your point about math instructors is great. I remember sitting in any number of calculus lectures where a trig identity was used to eliminate five steps in a derivation. Eventually it became obvious that the class would be a lot easier for students that memorized their trig identities.
Trig identities are just as important in physics. Beyond those, there are other categories that could be fleshed out like the 'near zero approximations' and 'when to use a Taylor series'.
This definitely provides food for thought!
Hamilton
Your point about math instructors is great. I remember sitting in any number of calculus lectures where a trig identity was used to eliminate five steps in a derivation. Eventually it became obvious that the class would be a lot easier for students that memorized their trig identities.
Trig identities are just as important in physics. Beyond those, there are other categories that could be fleshed out like the 'near zero approximations' and 'when to use a Taylor series'.
This definitely provides food for thought!
Hamilton
Re: Finding mathematical patterns in physics text examples
Learning the mathematical tricks of the trade is like becoming a good lover.
You have to spend of lot of time watching experienced people do it before you finally get it right. Or you can just use Mathematica, Maple, MATLAB, etc. That's the mathematical equivalent of switching yourself with Ron Jeremy while the lights are out and your girlfriend is waiting.
You have to spend of lot of time watching experienced people do it before you finally get it right. Or you can just use Mathematica, Maple, MATLAB, etc. That's the mathematical equivalent of switching yourself with Ron Jeremy while the lights are out and your girlfriend is waiting.
Re: Finding mathematical patterns in physics text examples
Calculus of variations is used throughout FT (Field Theory) and quite a bit in QFT.
As doom said, much of Hamiltonian and Lagrangian mech will come back "to haunt you" ( ).doom wrote:As you will find in your classical mechanics class in grad school, there was another dude named Hamilton who was pretty fond of the calculus of variations. Not to mention Lagrange and Feynman. No, you're not done with the variational principle yet. In fact, it's crucial to many parts of physics.