One of the reasons I started thinking more about philosophy in college was due to a rather simple thing: the three formulations of mechanics. Probably most of my readers haven't taken many physics classes beyond perhaps a required Freshman class. So they may not know that there are more ways to view the laws of mechanics than the traditional Newton's laws. Yet those alternative formulations open up all kinds of questions about both physics and science in general.
The basic formulation of mechanics was given to us by Newton and, with a few slight revisions due to how we do calculus, the laws of Newton are how most people do mechanics. What is interesting in Newton's formulation of mechanics are the entities with which mechanics is discussed. We have the notion of force, the notion of mass, and the notion of momentum. We derive the motion of objects by working through these concepts. Object move by balancing forces in collisions (or by gravity or electro-magnitism), by conserving changes in momentum and by conserving energy. Newton's laws aren't just some mathematical equations. They provide us a ground from which we think about motion.
What is surprising to many is that Newton's approach isn't the only one. As soon as you finish your introductory physics classes you are introduced to the Lagrangian and Euler's equation. I'll not bore you with too much physics. The equations are:
The Lagrangian:
From the Lagrangian we get the Euler-Lagrange equation:
What's significant is the form of the Lagrangian. Rather than thinking in terms of forces and balancing them one thinks in terms of an equation that minimizes the difference between kinetic energy (the first part of the Lagrangian) and potential energy (the V in the equation). In other words matter behaves in terms of minimizing the difference between kinds of energy.
Now, as any physics student knows, the Lagrangian is tremendously useful for solving many kinds of mechanics problems. What most students don't think about is that it is conceptually a radically different way of thinking about motion.
However the Lagrangian isn't the only other form. We also have the Hamiltonian. Now in practice for classical mechanics the Hamiltonian isn't used as much. Typically the Newtonian or Lagrangian forms simply result in easier to solve equations. However it is very prominent in the change to mechanics that the quantum revolution brought. (Obviously in a different form)
Momentum:
Position: 
(Note for those not familiar with the notion, the dots over the letters represent the ordinary derivative - it's a short hand used a lot in physics)
What the Hamiltonian form of mechanics does is to get an equation that sums the energies of the system. The Hamiltonian equations provide us a relationship between the change in position and momentum as being related to this total energy of the system. The benefit of this, especially in the quantum formulations, is that it allows us to see what is conserved in the system. This allows us to see symmetries and we start to think about mechanics and motion not in terms of forces and momentum, or difference between kinetic and potential energy, but in terms of symmetries and conservations.
It once again is a radically different way to think about things. Indeed, a lot of current theoretical physics thinks in terms of symmetries and patterns rather than in terms of energies, forces or the like.
Why I bring all this up though isn't simply to review lower division physics. Rather it is the implication. These three formulations of mechanics all are conceptually extremely different. So if we ask the philosophical question, "what is really going on - what are the entities in physics?" That is does the universe function in terms of mysteriously minimizing energy differences? Does it work through a wave equation (the Hamiltonian) that shows how energy evolves via time?
Of course all of this is in terms of classical physics. We know classical physics is wrong, so why is this an issue? Well, one can raise these same three formulations in quantum mechanics. So one can talk about forces and particles that convey forces between particles in a conceptual scheme not that different from the traditional billiard ball model of Newton. Richard Feynman took the Lagrangian form of mechanics and found an analogous development in quantum theory. (I'll not go into it, but many have heard of Feynman diagrams which are related) And of course the early development of quantum mechanics used a variation of the Hamiltonian as the natural way of writing down equations.
When I was an undergraduate this really bothered me because there never seemed to be a way to answer the question. It really opened my eyes to the fact that given mathematics there were several different ways to think about it. It was undetermined, to use the jargon of philosophy of science.
Now of course one can argue that these are metaphysical questions and thus not really a part of science. There's a lot of truth to that. However at the same time I think most scientists want to have some grasp on what exactly they are talking about. While some, typically called instrumentalists, say these questions don't matter (or are unanswerable) it seems hard to buy. We know what objects we're discussing when talking about regular objects from our every day experience. And Newton's laws make sense in terms of the kinds of things we encounter - force as a kind of pressure or resistance as we move (or are moved). Mass as how heavy things are. Momentum as the tendency for things to move in a straight line. And arguably Newton's appeal to those was a huge change over the earlier Aristotilean physics.
So I raise this not necessarily to say there are answers. But hopefully to pique your curiosity.
I am not denying that you have a point, perhaps I am too dense, but I don't see it. You can start with any one of the equations and derive the other two. Isn't the difference a purely mathematical one? In any given physics experiment the same quantities would be measured no matter which system was used. We measure what we can. While I want to see your point, I missed it.
Perhaps a comparison/contrast would be in order. Do you find something philosophically puzzling about Turing machines and Lambda calculus both being completely different ways of describing computing? Why or why not?
Again, I am NOT trying to argue, just seeking clarification.
Mathematically you are correct. The point is that in terms of the meaning of what is written, they are very different. So which of the three meanings is correct? (Well, ignoring the issue that classical physics is wrong) The point is that even back in the 19th century there was this interesting ambiguity in physics.
As I said, for instrumentalists this isn't an issue. They just say something along the lines of Feynman: shut up and calculate. The meaning is largely irrelevant and even, in the case of Kuhn, irrelevant. But for the realist this is a big issue.
The bit about computational issues and certain things in information theory strike me as interesting for exactly the same reasons. For instance I find it tremendously interesting the apparent relationship between entropy and information that Shannon brought out.
I'll be talking more about these sorts of ambiguities in science today in my next post.
I guess I just don't see the problem. From my view, there is only one meaning with three different mathematical points of view. Force, momentum, and energy all receive different emphasis in the mathematical formulations, but they are all derivable from each other. It's like seeing two different sides of the same coin, you get a different view but you still are describing a coin.
Pragmatically, and perhaps psychologically, there are differences in the three formulations. Scientists working with Lagrangian and Hamiltonian formulations did look at things differently and ended up discovering relativity and quantum mechanics.
http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics
If one of these were right and the others wrong, then there are certainly implications for many popular philosophical subjects. Even though none would differ mathematically.
David,
For me, the difference in meaning between the Lagrangian formulation and Newton's laws is that in one, you have a certain kind of entity (force) causing objects' movements. In the Lagrangian formulation, you don't speak of forces causing things to happen, not in the usual billiard-ball sense. Instead, you have a range of possible states a system can take up, and classical mechanics constrains it to move along certain paths on that manifold of possible states. So Lagrangian mechanics lacks the idea of something causing things to happen; instead, it is more a time-independent perspective that everything is already laid out, and things just happen. Another way of framing Lagrangian mechanics is that there is that everything conspires to achieve the 'goal' of minimising the Lagrangian, so much so that a professor of mine has dared to call it 'teleological'. However, I am not a fan of that take on Lagrangian mechanics, because it seems to take a myopic and anthropomorphic view of the system 'attempting' in real time to achieve a certain goal.
Newton's laws are, I feel, the more intuitive formulation, since they deal in more physically imaginable entities -- the Newtonian concept of force is very close to our intuitive idea of force, and we do tend to think of one billiard ball 'causing' the movement of the ball it crashes into, instead of imagining them to be simply a particular manifestation of a point on a path on the manifold of states of a holistic two-ball system.
The way I always thought about it in college was that the Lagrangian entailed that matter tends to minimize energy difference in a way analogous to how one might think of potential energy wanting to give itself up. However both the Lagrangian and the Hamiltonian are much, much more abstract and suggest that our common notions like force really aren't what's going on. In a way they prepared the conceptual groundwork for quantum mechanics since one of the ways to think about QM is to consider the more Hamiltonian form (the Schrodenger or Dirac equations) and talk about the evolution of the wave function. In a sense one could do the same sort of thing conceptually back in the days of classical physics.
What really struck me profoundly in that first advanced mechanics class was just how radically different these formulations are even though, as you said David, they don't make a lot of practical difference.
Whether that is significant to you really says a lot about how you view science. (Which is what I'll get at in my next post)
In a way they prepared the conceptual groundwork for quantum mechanics since one of the ways to think about QM is to consider the more Hamiltonian form (the Schrodenger or Dirac equations) and talk about the evolution of the wave function. Agree, I don't think I ever heard the word force a single time in my quantum mechanics class. Hence, starting from the Newtonian formulation using forces is probably tough going. Yet, in the standard model forces are one of the central features. So I do see how the different formulations differ pragmatically and hermeneutically in that at different times one view will have more explanatory power.
I don't see how different formulations make any ontological difference. If the entities that make up a Lagrangian explanation can be mapped to Newtonian entities via mathematical transformation then entities in one system are explicable via entities in another system. I don't see how science could possibly attach different meanings to different formulations, there is no objective way of doing this, or am I missing something?
to solve a problem in any physical system Lagrangian / Hamiltonian approach is much more convenient than Newtonian approach.
I tend to find pulleys easier in Newtonian formulations. But that's just me. The advantage to Lagrangians of course is that it's easy to use tensors and do a change of coordinate system. At least for me it's far more of a pain to do that with Newton. Of course I'll admit that there are very, very few Hamiltonian apporaches that are easier. As you say, typically Lagrangians are best. Especially if you are doing perturbations.
I've closed comments in order to avoid spam since I don't check this older blog as much anymore.
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