Physics: Most Fundamental Equations

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coco
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Physics: Most Fundamental Equations

I saw this guy that had the most fundamental equations and constants in physics displayed in his office in a set of picture frames. One frame had Newton's laws, another Maxwell's equations, another 34 constants, etc., and the final one had a big question mark. He would say to his students, "This is everything we know about the universe. Yet, there is more to be known, represented by the question mark. You have to fill that one in."

I wish to do something similar in my classroom. I realize, of course, that the most fundamental equations are a matter of opinion. In your opinion, what are the most fundamental equations?
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Re: Physics: Most Fundamental Equations

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Re: Physics: Most Fundamental Equations

I took my two required physics courses during summer semesters, while taking no other courses. For each final we were allowed one 3x5 card with whatever we could fit on it. I saved one of mine, if I can find it, I'll take a pic. I remember using the finest lead I could find, and completely filled the card front and back.
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gaining_age
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Re: Physics: Most Fundamental Equations

Is this for high school students? I'd find a blend of the principles expressed in advanced math as well as expressed in math they may recognized.

Example:
Euler Lagrange/Hamilton's principle

$S[\matbf{q}]\equiv \int_{t_1}^{t_2}L(\mathbf{q}(t),\dot{\mathbf{q}}(t),t)dt$
and
$\delta S=0$

But the more common expression would be Newton's 2nd Law
$\sum_{i=1}^{N} \mathbf{F_i}=m\mathbf{a}$

Maxwell's equations in differential form are sleek but seen in integral form may be closer to understanding
\begin{align} \oint \mathbf{B} \cdot d \mathbf{s} &= -\frac{d\Phi_m}{dt} \\ \oint \mathbf{B} \cdot d \mathbf{s} &= \mu_0 I +\epsilon_0 \mu_0 \frac{d \Phi_e}{dt} \\ \oint \mathbf{E} \cdot d\mathbf{A} &= \frac{Q}{\epsilon_o} \\ \oint\mathbf{B}\cdot d\mathbf{A} &= 0 \end{align}

From them and vector/scalar potential you can make a sleek 4-potential and d'alembertian for fancy yet simple look

$\Box\Phi=J$

Simple harmonic oscillator is fantastically useful.
$\ddot s +\omega^2s=0$

Quantum mechanics-- Schrodinger equation is key

$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t)=\Big[-\frac{\hbar^2}{2\mu}\nabla^2+V(\mathbf{r},t) \Big]\Psi(\mathbf{r},t)$
and Heisenberg's uncertainty principle
$\Delta x\Delta y\geq\hbar/2$

Energy momentum relation (relativistic)
$E^2=m^2c^4+p^2c^2$

These are a few to consider.....
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Re: Physics: Most Fundamental Equations

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Re: Physics: Most Fundamental Equations

S#@/ runs downhill.
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Re: Physics: Most Fundamental Equations

coco wrote:
Sat Aug 05, 2017 9:45 am
I saw this guy that had the most fundamental equations and constants in physics displayed in his office in a set of picture frames. One frame had Newton's laws, another Maxwell's equations, another 34 constants, etc., and the final one had a big question mark. He would say to his students, "This is everything we know about the universe. Yet, there is more to be known, represented by the question mark. You have to fill that one in."

I wish to do something similar in my classroom. I realize, of course, that the most fundamental equations are a matter of opinion. In your opinion, what are the most fundamental equations?
Ah! One equation. This is a laundry list of the most fundamental stuff known. Everything else comes out of this. It's the Standard Model of particle Physics plus General Relativity. I think there is not any debate about the last statement. This is everything known about the ingredients of the universe in physics. It's not a matter of opinion. But of course there is lots that happens that can't practically be specified or derived from this in this context eg all of biology. So there is certainly more to the world but it is all consistent with this underlying all of it at the micro level and at the large scale with gravity. And whether the two are really fundamental is certainly a debate. GA's list are just motivational examples of nice stuff rather than anything fundamental and it's incomplete against "everything known about the universe". I think he acks that in saying "these are a few". All the other equations are in the same category against the goal you stated. It's all subsumed in this. Even "S#@/ runs downhill" is subsumed in this, where the hill can be anywhere under any circumstances in the universe.

This is the amplitude to undergo a transition from one configuration to another in the path-integral formalism of quantum mechanics, within the framework of quantum field theory, with field content and dynamics described by general relativity (for gravity) and the Standard Model of particle physics (for everything else). The notations in red are just meant to be suggestive, don’t take them too seriously. But we see all the parts of known microscopic physics there — all the particles and forces.
But of course your students can't understand this. We can't either. They can't deal with the formalism GA has given you either, not even the SHM differential equation, I suspect. So is any of this motivational for a HS student if they can't understand the language? Probably not.

What do you teach your High School Students about physics that is from 1900 and later?
With mathematics the limit of the HS curriculum is probably back in the 17th century and even all of that is not complete in HS.

There are certainly some things that are taught in HS, and are part of general public knowledge, that were not known in 1900 eg the existence of atoms and molecules. That was a question in 1900. But for the most part HS physics may end in the early to mid 19th C. Is that true?
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Re: Physics: Most Fundamental Equations

The "most fundamental equation" depends on your mischief.

Back when I did a lot of chemical thermodynamics, much of life was devoted to PV=nRT -- the Ideal Gas Law.
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gaining_age
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Re: Physics: Most Fundamental Equations

Physics classes are always a study of history (to a degree). There is formalism and rigorousness to become familiar with the practices of the language but a class is not going to branch into new items. So the equations and relationships that are put forth or are put on posters are about major significant advances in history and a rigor that can be learned in a class so that further input may be created.

Rusty's equation is not very practical. Physics gets that way every so often in trying to piece concepts and fundamentals together. Yet it is approximations and approaches that we an discover new ways to look at things without trying to uncover the impact of "everything". That's part of the beauty of Maxwell's Equations. It pieces relationships together in the equations but you have to have boundary conditions to do an actual problem. Derivations are fun for seeing new relations or proving existing known pieces but you can't do problems without boundaries--- and a description of the boundary itself may lead to some understanding about how the physics works or about the nature of material and how to analyze it.

I had fun in this thread pulling together some of the $LaTeX$ to show off the equations "natively".

G.
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coco
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Re: Physics: Most Fundamental Equations

gaining_age wrote:
Sat Aug 05, 2017 3:20 pm
I had fun in this thread pulling together some of the $LaTeX$ to show off the equations "natively".
Many thanks. I was not expecting the $LaTeX$.
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Re: Physics: Most Fundamental Equations

gaining_age wrote:
Sat Aug 05, 2017 3:20 pm
Physics classes are always a study of history (to a degree). There is formalism and rigorousness to become familiar with the practices of the language but a class is not going to branch into new items. So the equations and relationships that are put forth or are put on posters are about major significant advances in history and a rigor that can be learned in a class so that further input may be created.

Rusty's equation is not very practical. Physics gets that way every so often in trying to piece concepts and fundamentals together. Yet it is approximations and approaches that we an discover new ways to look at things without trying to uncover the impact of "everything". That's part of the beauty of Maxwell's Equations. It pieces relationships together in the equations but you have to have boundary conditions to do an actual problem. Derivations are fun for seeing new relations or proving existing known pieces but you can't do problems without boundaries--- and a description of the boundary itself may lead to some understanding about how the physics works or about the nature of material and how to analyze it.

I had fun in this thread pulling together some of the $LaTeX$ to show off the equations "natively".

G.
It wasn't me that established the goal: "This is everything that is known about the universe." But I lived with it. I think the more interesting question is what can you say to HS students, about this topic, that they can understand? For example, HS students, wouldn't understand why boundary conditions are even required. They don't know anything about solving partial differential equations or why they are even there. This is a serious challenge.

I would like Cocos to weigh in on this and answer the questions I asked too.
Last edited by Rusty on Sat Aug 05, 2017 3:46 pm, edited 1 time in total.
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coco
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Re: Physics: Most Fundamental Equations

JudgeRusty wrote:
Sat Aug 05, 2017 1:03 pm
S#@/ runs downhill.

I should put the three rules of plumbing on the other wall.
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coco
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Re: Physics: Most Fundamental Equations

Rusty wrote:
Sat Aug 05, 2017 3:44 pm
gaining_age wrote:
Sat Aug 05, 2017 3:20 pm
Physics classes are always a study of history (to a degree). There is formalism and rigorousness to become familiar with the practices of the language but a class is not going to branch into new items. So the equations and relationships that are put forth or are put on posters are about major significant advances in history and a rigor that can be learned in a class so that further input may be created.

Rusty's equation is not very practical. Physics gets that way every so often in trying to piece concepts and fundamentals together. Yet it is approximations and approaches that we an discover new ways to look at things without trying to uncover the impact of "everything". That's part of the beauty of Maxwell's Equations. It pieces relationships together in the equations but you have to have boundary conditions to do an actual problem. Derivations are fun for seeing new relations or proving existing known pieces but you can't do problems without boundaries--- and a description of the boundary itself may lead to some understanding about how the physics works or about the nature of material and how to analyze it.

I had fun in this thread pulling together some of the $LaTeX$ to show off the equations "natively".

G.
It wasn't me that established the goal: "This is everything that is known about the universe." But I lived with it. I think the more interesting question is what can you say to HS students, about this topic, that they can understand? For example, HS students, wouldn't understand why boundary conditions are even required. They don't know anything about solving partial differential equations or why they are even there. This is a serious challenge.
The pedagogical goal is just to help them see that physics is useful and that they can be a part of it in the future. "Everything" is hyperbole, and perhaps unjustified.
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Rusty
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Re: Physics: Most Fundamental Equations

coco wrote:
Sat Aug 05, 2017 3:48 pm
Rusty wrote:
Sat Aug 05, 2017 3:44 pm
gaining_age wrote:
Sat Aug 05, 2017 3:20 pm
Physics classes are always a study of history (to a degree). There is formalism and rigorousness to become familiar with the practices of the language but a class is not going to branch into new items. So the equations and relationships that are put forth or are put on posters are about major significant advances in history and a rigor that can be learned in a class so that further input may be created.

Rusty's equation is not very practical. Physics gets that way every so often in trying to piece concepts and fundamentals together. Yet it is approximations and approaches that we an discover new ways to look at things without trying to uncover the impact of "everything". That's part of the beauty of Maxwell's Equations. It pieces relationships together in the equations but you have to have boundary conditions to do an actual problem. Derivations are fun for seeing new relations or proving existing known pieces but you can't do problems without boundaries--- and a description of the boundary itself may lead to some understanding about how the physics works or about the nature of material and how to analyze it.

I had fun in this thread pulling together some of the $LaTeX$ to show off the equations "natively".

G.
It wasn't me that established the goal: "This is everything that is known about the universe." But I lived with it. I think the more interesting question is what can you say to HS students, about this topic, that they can understand? For example, HS students, wouldn't understand why boundary conditions are even required. They don't know anything about solving partial differential equations or why they are even there. This is a serious challenge.
Indeed. The pedagogical goal is just to help them see that physics is useful and that they can be a part of it in the future.
The idea that physicists all work at understanding and advancing "everything known about the universe" is probably wrong. Most of them probably do not work in fundamental physics. There are lots of other areas of physics, including medical physics, bio-physics, material science, <long list> plus there are many people trained in physics that end up in engineering. Some actually go to finance and investment generally. You might remember a little economic crisis that was generally hung on the equation for derivatives. So it may be completely misleading to characterize a career in physics or training in physics as dealing daily with the problem of "everything known in the universe". It may even chase some people away.

There is something else that I think should be explained to HS students too. Having superb technical skills isn't necessarily the only way to success. In industry, especially, combining adequate technical skills or understanding with other skills eg any of: communications (writing, presentation, foreign languages, etc), marketing, sales, administration, project mgmt, managerial skills, entrepreneurial daring, etc is often much more valuable (ie you get paid more and sometimes you even get paid twice) than technical skills alone. Those with superb technical skills often lack the complementary skills mentioned that can allow folks with multiple skills to excel well beyond the pure technical gurus.
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