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0 pointsCleonis1y ago0 comments

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.

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t_mann1y ago

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.

CleonisOP1y ago

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.

t_mann1y ago

Nicely said, but I think then we are in agreement that Newtownian mechanics has a bit of redundancy that can be removed by switching to a Lagrangian framework, no? I think that's a situation where Occam's razor can be applied very cleanly: if we can make the exact same predictions with a sparser model.

Now the other poster has argued that science consists of finding minumum complexity explanations of natural phenomena, and I just argued that the 'minimal complexity' part should be left out. Science is all about making good predictions (and explanations), Occam's razor is more like a guiding principle to help find them (a bit akin to shrinkage in ML) rather than a strict criterion that should be part of the definition. And my example to illustrate this was Newtonian mechanics, which in a complexity/Occam's sense should be superseded by Lagrangian, yet that's not how anyone views this in practice. People view Lagrangian mechanics as a useful calculation tool to make equivalent predictions, but nobody thinks of it as nullifying Newtownian mechanics, even though it should be preferred from Occam's perspective. Or, as you said, the physics content is the same, but the complexity of the description is not, so complexity does not factor into whether it's physics.

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