This says that for a bunch of particle in a potential V = a x^k, if you let the system evolve over time T forming a path which has size L in some sense, then there is another path which is a re-scaled version of the original one s.t. if it has size l, then the time taken to form this new path is t.
We can use this trick to create a bunch of “scaling laws” for simple physical systems. For example:
1) Let V = a x ^{-1} i.e. a gravitational potential. Then we have k=-1, so
t/T = (l/L)^{3/2}
(t/T)^{2} = (l/L)^{3}
I.e. Kepler’s third law. In fact, this is a more general result because it also applies to particles falling into a potential well. If you increase the distance the particle takes to fall in, you can use this equation to tell you how much the time increases, too.
2) V = a x^{2} i.e. the potential of an oscillator. Then we have k=2, so
t/T = (l/L)^{0}
That is, we can take an oscillation and increase the amplitude without changing the time it takes to complete the oscillation.
3) V = a x i.e. potential of a constant force. Then we have k=1, so
t/T = (l/L)^{1/2}
(t/T )^{2}= (l/L)
This reflects the fact that the position of a particle experiences constant force instreases like t^2,
So you can see you get a lot of mileage out of this pretty simple equation.
Good job to chatgpt actually, I just dumped your post to it lol. Also you can just ask LLM to typeset the math but keep other content verbatim and im certain it will one shot it
And if you want to convert to HTML/Unicode for places where you don’t have direct LaTeX support, you can also have a LLM do that, albeit there area lot of edge-cases and I don’t think LLMs will usually use more exotic Unicode like FRACTION SLASH for things like ‘3/2’ etc, so I have a big script for that: https://gwern.net/static/build/latex2unicode.py (Github).
Oh, I didn’t realize you could just paste latex into the LW editor and it autoparses, I thought you had to use ctrl+4 to summon a block in which you write math equations.
Perhaps my favourite relation in physics is
t/T = (l/L)^{1-k/2}.
This says that for a bunch of particle in a potential V = a x^k, if you let the system evolve over time T forming a path which has size L in some sense, then there is another path which is a re-scaled version of the original one s.t. if it has size l, then the time taken to form this new path is t.
We can use this trick to create a bunch of “scaling laws” for simple physical systems. For example:
1) Let V = a x ^{-1} i.e. a gravitational potential. Then we have k=-1, so
t/T = (l/L)^{3/2}
(t/T)^{2} = (l/L)^{3}
I.e. Kepler’s third law. In fact, this is a more general result because it also applies to particles falling into a potential well. If you increase the distance the particle takes to fall in, you can use this equation to tell you how much the time increases, too.
2) V = a x^{2} i.e. the potential of an oscillator. Then we have k=2, so
t/T = (l/L)^{0}
That is, we can take an oscillation and increase the amplitude without changing the time it takes to complete the oscillation.
3) V = a x i.e. potential of a constant force. Then we have k=1, so
t/T = (l/L)^{1/2}
(t/T )^{2}= (l/L)
This reflects the fact that the position of a particle experiences constant force instreases like t^2,
So you can see you get a lot of mileage out of this pretty simple equation.
That’s really cool, is there a wiki page for this formula? [Edit: Mechanical similarity but it’s a stub]
btw: LW supports latex
Good job on finding the wikipedia page for this! I didn’t know what it was called, I just read about it in Landau vol. 1 many years ago.
And if I had to typset the equations I probably wouldn’t have written this quick take.
Good job to chatgpt actually, I just dumped your post to it lol. Also you can just ask LLM to typeset the math but keep other content verbatim and im certain it will one shot it
It’ll zero-shot easy cases, yeah.
And if you want to convert to HTML/Unicode for places where you don’t have direct LaTeX support, you can also have a LLM do that, albeit there area lot of edge-cases and I don’t think LLMs will usually use more exotic Unicode like FRACTION SLASH for things like ‘3/2’ etc, so I have a big script for that: https://gwern.net/static/build/latex2unicode.py (Github).
Oh, I didn’t realize you could just paste latex into the LW editor and it autoparses, I thought you had to use ctrl+4 to summon a block in which you write math equations.