When can you combine formulas?
Jul. 8th, 2005 02:28 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
guslIn physics, formulas are meant to be interpreted in terms of models: while "F = m a" can be interpreted into any body, "a = v^2 / R" only holds of objects in Uniform Circular Motion (UCM).
You can easily "make the laws of physics lie" by combining formulas to make absurd derivations.
My theory is that truth of formulas is defined on models
(e.g.
and that absurdities will only happen when the models assumed by the formulas are incompatible. But how do we decide whether two models can be combined? My current research goal is to define such a compatibility criterion.
I may need to create a meta-language (syntax for talking about models) in order to do this, but this sounds like it could be too much work.
You can easily "make the laws of physics lie" by combining formulas to make absurd derivations.
My theory is that truth of formulas is defined on models
(e.g.
UCM(body1, body2) |= a_body1 = v_body1^2 / distance(body1,body2) says that this formula is true in the set of models where body1 is in UCM around body2),and that absurdities will only happen when the models assumed by the formulas are incompatible. But how do we decide whether two models can be combined? My current research goal is to define such a compatibility criterion.
I may need to create a meta-language (syntax for talking about models) in order to do this, but this sounds like it could be too much work.
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(no subject)
Date: 2005-07-08 01:32 pm (UTC)It seems to me that physics models are compatible when the boundary-conditions within which each model is valid are the same. Boundary and/or starting conditions are an integral part of a model.
Classical Mechanics
Date: 2005-07-08 03:14 pm (UTC)Re: Classical Mechanics
Date: 2005-07-08 03:52 pm (UTC)how?
and what do you need to add to Newton's laws in order for them to make sense?
Re: Classical Mechanics
Date: 2005-07-08 04:34 pm (UTC)Knill has an introduction to the topic. (http://abel.math.harvard.edu/archive/118r_spring_05/handouts/escape.pdf)
If you want electromagnetism, you may have to go to quantum mechanics or quantum field theory. Although those theories may not make sense either. If you want gravity, I’m not sure what you need. I don’t know much general relativity.
Re: Classical Mechanics
Date: 2005-07-08 04:52 pm (UTC)This seems to assume some objects have either 0 or infinite mass. And I would be very surprised if the tiny mass can drive the big planets away from each other at infinite speed. Where would the infinite kinetic energy come from?
Re: Classical Mechanics
Date: 2005-07-08 05:04 pm (UTC)You can try reading Infinite Distance in Finite Time (http://www.meta-religion.com/Physics/Cosmological_physics/infinite_distance_in_finite_time.htm).
A point particle with a finite mass has an infinite gravitational potential well. Arbitrary amounts of kinetic energy can be extracted by moving two point particles arbitrarily close together.
Re: Classical Mechanics
Date: 2005-07-08 05:31 pm (UTC)It's interesting to see that this particular idealization of point masses leads to absurdities. But I think there must be a nice theory of classical mechanics, with no absurdities.
Of course, it won't be true of the real world, especially at really small and really large scales.
(no subject)
Date: 2005-07-09 04:09 am (UTC)Almost all equations in physics are derivable from a very small set of axioms already. a = v^2/R for cicular motion follows as soon as you define a to be the derivative of v (where by bolding them I mean the vector quantities. It's not really a physics formula at all, just a special case of the standard mathematical formula for taking a derivative.
F=ma is only valid in certain circumstances. And in the cases where it is valid, you can derive it from much more basic formulas.
So why would you want to put in a collection of special case formulas into a formal language and then try to sort out where and when to use them? Why not start with the basic axioms and then derive the rest. Then you won't have to worry about "which models" to use, because you can just do it for the most general case.
What might be worth doing, though, is formalizing the way we specialize to certain cases... in other words, encoding formally how you take a general formula and find all the various specializations of it in cases where you're assuming a particular set of restrictions. I don't know, maybe this is what you were trying to do after all but I got confused from the way you describe your plan.
(no subject)
Date: 2005-07-09 10:36 am (UTC)This is part of what I'm trying to do.
Almost all equations in physics are derivable from a very small set of axioms already. a = v^2/R for cicular motion follows as soon as you define a to be the derivative of v (where by bolding them I mean the vector quantities. It's not really a physics formula at all, just a special case of the standard mathematical formula for taking a derivative.
How does it follow? In any case, this is only true in the context of circular motion, so circular motion must be assumed in/as one of the axioms used to derive this formula. In fact, the proper formalization of this formula makes it explicit that it only holds in UCM.
F=ma is only valid in certain circumstances. And in the cases where it is valid, you can derive it from much more basic formulas.
Ok. But ∑F = m a is always valid, right? "F₀ = m a" is valid exactly when F₀ is the only force on the body (i.e. in models where the body in question has exactly one force on it).
So why would you want to put in a collection of special case formulas into a formal language and then try to sort out where and when to use them?
Because this gives us a corpus. After seeing all these examples, our system can become "experienced" in the kinds of reasoning that physicists do, and can reproduce it when working on novel problems.
My goal is coming up with a formal criterion for when you can combine assumptions/models.
For example, "body1 is in uniform circular motion around body2" is compatible with "there is a unique force between body1 and body, namely gravitation", because there are models where both assumptions are true.
However, "body1 is in uniform circular motion around body2" is incompatible with "body1 is repelled by body2", or with "body1 is going down a straight slope"
Why not start with the basic axioms and then derive the rest. Then you won't have to worry about "which models" to use, because you can just do it for the most general case.
To "derive the rest", you need assumptions about the model. Formulas are only true with respect to a certain model.
(no subject)
Date: 2005-07-09 05:56 pm (UTC)How does it follow? In any case, this is only true in the context of circular motion, so circular motion must be assumed in/as one of the axioms used to derive this formula. In fact, the proper formalization of this formula makes it explicit that it only holds in UCM.
I would not say it's an axiom, I would say it's a particular value you're plugging in. The general formula is a(t) = v'(t) , v(t) = x'(t).
If you plug in any uniform circular function for x, for instance:
x(t) = ( r cos(wt) , r sin(wt) )
and then take two derivatives, then you immediately get that the magnitude of the a vector is equal to v2 over the radius of the circle. It just comes from taking a second derivative of sin and cos.
It's sometimes handy to memorize the relation for this special case, but it only takes a moment to derive if you don't remember it, so it may or may not be worth it. For other functions x, you'll get different relations.
Ok. But ∑F = m a is always valid, right? "F₀ = m a" is valid exactly when F₀ is the only force on the body (i.e. in models where the body in question has exactly one force on it).
Nope. ΣF = ma is only valid for large objects travelling at slow velocities in inertial reference frames with rectangular coordinates. If any of those 4 conditions are different, then you have to use different formulas. In rotating reference frames you get extra terms. At large velocities, the acceleration in the direction of motion is different from the acceleration perpendicular for the same force. And for tiny things you have to add quantum corrections which make it look very different. Even in classical mechanics, if you're using other coordinates then you get different equations relating the forces and accelerations. So it is only valid under a very limited set of conditions... namely, those which Isaac Newton was aware of in the 1700s.
To "derive the rest", you need assumptions about the model. Formulas are only true with respect to a certain model.
Any equation is of course meaningless unless you know what physical quantities the variables stand for. If this is what you mean by "with respect to a certain model" then yes. But you're using model a bit differently than I would use it.
(no subject)
Date: 2005-07-10 07:56 am (UTC)x(t) = ( r cos(wt) , r sin(wt) )... and then take two derivatives,
x'(t) = (- r w sin(wt), r w cos(wt) )
v = |x'(t)| = r w
x''(t) = ( - r w^2 cos(wt) , - r w^2 sin(wt) )
a = |x''(t)| = r w^2
v^2 = r^2 w^2
thus a = v^2 /r.
Ok.
Any equation is of course meaningless unless you know what physical quantities the variables stand for. If this is what you mean by "with respect to a certain model" then yes. But you're using model a bit differently than I would use it.
I am using the logician's meaning of "model".
The interesting thing is seeing how we can combine models (in either the physics or the logic sense), or how we can readapt old models to new situations.
Tangent: why is rotation not relative (like velocity is)? When is a frame considered rotating? And do the "background stars" provide a satisfying answer to this, or are they merely a correlate of the real rotational reference frame?
I guess this is the same question as "why is acceleration not relative?"
(no subject)
Date: 2005-07-10 07:04 pm (UTC)thus a = v^2 /r.
Cool, I'm always glad when I end up giving someone just enough information so that they can figure it out on their own. Good teaching practice :)
I am using the logician's meaning of "model".
Well that explains that difference... I was assuming a much more physics based definition for model... I don't know enough about logic. I think I'm seeing more what you're getting at now. Trying to automate a lot of the things physicists do when they're confronted by a bunch of equations or need to come up with new equations for a certain situation.
Tangent: why is rotation not relative (like velocity is)? When is a frame considered rotating? And do the "background stars" provide a satisfying answer to this, or are they merely a correlate of the real rotational reference frame? I guess this is the same question as "why is acceleration not relative?"
This is an interesting question, and it has a long answer.
Before Einstein there was Galilean relativity which said the mechanical laws of physics were the same in any inertial frame. But the laws of physics were different in non-inertial frames (rotating or accelerating) because you get fictitious "forces" such as the centrifugal force and coreolis force showing up.
Einstein's first observation was that electromagnetism violated Galilean relativity... the equations for electromagnetism involve a "v" in them which everyone assumed was an absolute v defined by the aether. So he postulated special relativity which restores the property that the laws of physics are the same in any inertial frame... but with Lorentz tranformations to go between frames instead of Galilean tranformations.
He then went a big step further and postulated general relativity which says that the laws of physics are actually the same in any frame whatsoever... be it inertial, accelerating, or rotating, or anything else. But he did this only by introducing the notion of gravity as spacetime curvature. If you're in a frame that's rotating, you could right down the laws of physics as usual, but you would think there is a curvature to spacetime such that you're at the peak of a deep gravity well that gets deeper and deeper as you go further out into space. The stars far away would appear to be moving faster than light around you, but in GR this doesn't matter... as long as nothing nearby looks like it's moving at c, there are no contradictions or paradoxes.
However, even with general relativity, I'd say that it's still possible to notice whether you're rotating absolutely. All you have to do is look at the matter-energy density around to see if spacetime should be curved, and if it's curved with no stuff there causing it to curve than you must be rotating. One might be able to argue that a stubborn rotational observer still might not know whether they were rotating if they just said "well, there's a lot of dark matter out there which I can't see.. but it must be there, since I see that everything is curved." That, however, borders on delusional because I think it would be fairly obvious that the way in which the "dark matter" was distributed was very artificial. So to answer your question about rotating frames being absolute... in an idealistic sense they are relative, but in a practical sense you can always tell the difference, so they are essentially absolute.
(no subject)
Date: 2005-07-10 07:08 pm (UTC)as long as nothing nearby looks like it's moving at c, there are no contradictions or paradoxes.
I meant to write "faster than c" here.
(no subject)
Date: 2005-07-10 11:56 pm (UTC)