Timeline for Why do most formulas in physics have integer and rational exponents?
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15 events
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| Jul 3, 2020 at 17:38 | comment | added | Andrew | (6) In statistical physics, there are many examples of non-integer or non-rational scaling behavior ("critical exponents") when the system is close to phase transition. These are not simply an empirical scaling relation, and there is a deep underlying theoretical structure to explain these exponents. | |
| Jul 3, 2020 at 17:01 | comment | added | Andrew | (4) Simple relations between conserved quantities can exist even with non-linear equations. For example, the first law of thermodynamics does not assume the underlying dynamics governing the microstates is linear. (5) The reason physical laws are second order in time has to do with the so-called Ostragradsky instability. GR is 2nd order in time derivatives but not linear. | |
| Jul 3, 2020 at 17:01 | comment | added | Andrew | 5 years too late, but given the number of upvotes I wanted to point out some issues. (1) The equation $dy/dx=a y/x$ is linear and solved by $y=cx^a$, for any real or complex $a$, so linear equations can yield non rational power law. (2) You do get non-integer powers showing up even in basic physics, eg Kepler's third law (which pops out of an equation like (1)). (3) "Fundamental laws are usually linear (Maxwell equations, for instance)": GR and the standard model are the most fundamental laws we have and are non-linear. ... | |
| Feb 23, 2015 at 6:41 | comment | added | dushyanth | "this is just how things are" that is where we started right? | |
| Feb 18, 2015 at 9:40 | comment | added | orion | The energy conservation law is just an integral of the 2nd Newton's law - they are the same thing. If we defined force as $F=ma$, then we must define work as $\int F\,dx$. | |
| Feb 18, 2015 at 9:25 | comment | added | dushyanth | I understand you say that force is defined like that for convenience but then why is energy=fs.I can't apply the same to work because work (or) energy is conserved. Is that a coincidence that fs is conserved. | |
| Feb 11, 2015 at 8:29 | comment | added | orion | (1) superposition principle means linear relationship - for some phenomena it's exact, for some, it works for small perturbations (2) physics works through differential equations, which shift the powers by integers (3) if you happen to have an equation with a more complex solution, it's just as likely (or even less likely) to get a transcendental power, as it is to get any transcendental function you can imagine - why $x^a$ and not $J_{1/2}(x)\ln\Gamma(\sqrt{x})$? | |
| Feb 10, 2015 at 15:50 | comment | added | dushyanth | Can somebody please brief down answer for me | |
| Feb 8, 2015 at 8:03 | comment | added | dushyanth | can someone make the above answer brief? | |
| Feb 7, 2015 at 11:23 | comment | added | orion | Sure, the more proper definition of force would be the derivative of a particular conserved value -- the linear momentum. But that doesn't change the fact it's linear and that it's not even strictly needed as a physical quantity. Physics works quite well without ever talking about forces specifically. | |
| Feb 7, 2015 at 10:44 | comment | added | Ján Lalinský | "An example is F=ma (just defines what force is - it's convenient to define it like that)" There is no universal agreement on this point of view, there are others. For example, in one view force is not defined by $a$ but by spring deformation or weight measurements. Also, in special relativity this equation is not used any more, not because the definition of force was changed, but because Newtonian mechanics is approximate to special relativistic mechanics. | |
| Feb 5, 2015 at 13:05 | comment | added | dotancohen | This answer hits the nail on the head at the beginning: the reason is that the relationships in question are all linear. Why are they linear relationships? For the geometrical measurements, the answer is proportional scale. For the physical measurements, the answer is (usually) conservation of energy. | |
| Feb 3, 2015 at 17:36 | comment | added | orion | Good point, geometry by itself also comes all in integer powers (rational, with small-powered roots if you play with norms). And indeed the inverse square law and its relatives come directly from that. | |
| Feb 3, 2015 at 15:55 | comment | added | adam.r | Excellent answer. I'm not sure if this covers the feature that I was going to comment on -- that many of these laws describe spatial relationships, and space comes in countable dimensions. So this explains the relationship between one-dimensional and two-dimensional measures of regular shapes (circles, spheres, squares, cubes). I suspect that it also accounts for how the intensity of forces decreases with distance. | |
| Feb 3, 2015 at 13:01 | history | answered | orion | CC BY-SA 3.0 |