Comment by t_mann
2 months ago
The thing is, Lagrangian mechanics makes exactly the same predictions as Newtownian, and it starts from a foundation of just one principle (least action) instead of three laws, so it's arguably a sparser theory. It just makes calculations easier, especially for more complex systems, that's its raison d'être. So in a world where we don't know about relativity yet, both make the best predictions we know (and they always agree), but Newton's laws were discovered earlier. Do they suddenly stop being natural laws once Lagrangian mechanics is discovered? Standard physics curricula would not agree with you btw, they practically always teach Newtownian mechanics first and Lagrangian later, also because the latter is mathematically more involved.
I will argue that 'has least action as foundation' does not in itself imply that Lagrangian mechanics is a sparser theory:
Here is something that Newtonian mechanics and Lagrangian mechanics have in common: it is necessary to specify whether the context is Minkowski spacetime, or Galilean spacetime.
Before the introduction of relativistic physics the assumption that space is euclidean was granted by everybody. The transition from Newtonian mechanics to relativistic mechanics was a shift from one metric of spacetime to another.
In retrospect we can recognize Newton's first law as asserting a metric: an object in inertial motion will in equal intervals of time traverse equal distances of space.
We can choose to make the assertion of a metric of spacetime a very wide assertion: such as: position vectors, velocity vectors and acceleration vectors add according to the metric of the spacetime.
Then to formulate Newtonian mechanics these two principles are sufficient: The metric of the spacetime, and Newton's second law.
Hamilton's stationary action is the counterpart of Newton's second law. Just as in the case of Newtonian mechanics: in order to express a theory of motion you have to specify a metric; Galilean metric or Minkowski metric.
To formulate Lagrangian mechanics: choosing stationary action as foundation is in itself not sufficent; you have to specify a metric.
So: Lagrangian mechanics is not sparser; it is on par with Newtonian mechanics.
More generally: transformation between Newtonian mechanics and Lagrangian mechanics is bi-directional.
Shifting between Newtonian formulation and Lagrangian formulation is similar to shifting from cartesian coordinates to polar coordinates. Depending on the nature of the problem one formulation or the other may be more efficient, but it's the same physics.
You seem to know more about this than me, but it seems to me that the first law does more than just induce a metric, I've always thought of it as positing inertia as an axiom.
There's also more than one way to think about complexity. Newtownian mechanics in practice requires introducing forces everywhere, especially for more complex systems, to the point that it can feel a bit ad hoc. Lagrangian mechanics very often requires fewer such introductions and often results in descriptions with fewer equations and fewer terms. If you can explain the same phenomenon with fewer 'entities', then it feels very much like Occam's razor would favor that explanation to me.
Indeed inertia. Theory of motion consists of describing the properties of Inertia.
In terms of Newtonian mechanics the members of the equivalence class of inertial coordinate systems are related by Galilean transformation.
In terms of relativistic mechanics the members of the equivalence class of inertial coordinate systems are related by Lorentz transformation.
Newton's first law and Newton's third law can be grouped together in a single principle: the Principle of uniformity of Inertia. Inertia is uniform everywhere, in every direction.
That is why I argue that for Newtonian mechanics two principles are sufficient.
The Newtonian formulation is in terms of F=ma, the Lagrangian formulation is in terms of interconversion between potential energy and kinetic energy
The work-energy theorem expresses the transformation between F=ma and potential/kinetic energy The work-energy theorem: I give a link to an answer by me on physics.stackexchange where I derive the work-energy theorem https://physics.stackexchange.com/a/788108/17198
The work-energy theorem is the most important theorem of classical mechanics.
About the type of situation where the Energy formulation of mechanics is more suitable: When there are multiple degrees of freedom then the force and the acceleration of F=ma are vectorial. So F=ma has the property that the there are vector quantities on both sides of the equation.
When expressing in terms of energy: As we know: the value of kinetic energy is a single value; there is no directional information. In the process of squaring the velocity vector directional information is discarded, it is lost.
The reason we can afford to lose the directional information of the velocity vector: the description of the potential energy still carries the necessary directional information.
When there are, say, two degrees of freedom the function that describes the potential must be given as a function of two (generalized) coordinates.
This comprehensive function for the potential energy allows us to recover the force vector. To recover the force vector we evaluate the gradient of the potential energy function.
The function that describes the potential is not itself a vector quantity, but it does carry all of the directional information that allows us to recover the force vector.
I will argue the power of the Lagrangian formulation of mechanics is as follows: when the motion is expressed in terms of interconversion of potential energy and kinetic energy there is directional information only on one side of the equation; the side with the potential energy function.
When using F=ma with multiple degrees of freedom there is a redundancy: directional information is expressed on both sides of the equation.
Anyway, expressing mechanics taking place in terms of force/acceleration or in terms of potential/kinetic energy is closely related. The work-energy theorem expresses the transformation between the two. While the mathematical form is different the physics content is the same.
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Laws (in science, not government) are just a relationship that is consistently observed, so Newton's laws remain laws until contradictions were observed, regardless of the existence of or more alternative models which would predict them to hold.
The kind of Occam’s Razor-ish rule you seem to be trying to query about is basically a rule of thumb for selecting among formulations of equal observed predictive power that are not strictly equivalent (that is, if they predict exactly the same actually observed phenomenon instead of different subsets of subjectively equal importance, they still differ in predictions which have not been testable), whereas Newtonian and Lagrangian mechanics are different formulations that are strictly equivalent, which means you may choose between them for pedagogy or practical computation, but you can't choose between them for truth because the truth of one implies the truth of the other, in either direction; they are the exactly the same in sibstance, differing only in presentation.
(And even where it applies, its just a rule of thumb to reject complications until they are observed to be necessary.)
Newtownian and Lagrangian mechanics are equivalent only in their predictions, not in their complexity - one requires three assumptions, the other just one. Now you say the fact that they have the same predictions makes them equivalent, and I agree. But it's clearly not compatible with what the other poster said about looking for the simplest possible way to explain a phenomenon. If you believe that that's how science should work, you'd need to discard theories as soon as simpler ones that make the same predictions are found (as in the case of Newtownian mechanics). It's a valid philosophical standpoint imho, but it's in opposition to how scientists generally approach Occam's razor, as evidenced eg by common physics curricula. That's what I was pointing out. Having to exclude Newtownian mechanics from what can be considered science is just one prominent consequence of the other poster's philosophical stance, one that could warrant reconsidering whether that's how you want to define it.
> Do they suddenly stop being natural laws once Lagrangian mechanics is discovered?
Not my question to answer, I think that lies in philosophical questions about what is a "law".
I see useful abstractions all the way down. The linked Asimov essay covers this nicely.