[in_a_hole]

Does anyone know a text which justifies why the Lagrangian approach works? This text and many others I have encountered just start with the Principle of Least Action taken as given and go from there but I'm left wondering why we define the Action as this object and why we should expect it to be minimised for the physical trajectory in the first place.

Failing a full derivation from the ground up, a proof of the equivalence to Newtonian mechanics would be interesting.

[nchagnet]

Regarding the "why the action is this object" part of the question, I find that the easiest way to think about it is from the Hamiltonian perspective. There you can think of it as minimising energy along a trajectory. From that point, a Lagrangian is just a mathematical trick to express the symplectic structure differently.

But if your question was more about "why minimizing something yields trajectories", I personally would argue this is beyond physics. As an empirical science, physicists have seen this kind of behaviour broadly (optics, classical mechanics, quantum mechanics) and just unified it as an overarching principle.

Finally regarding the proof to newtonian mechanics, I don't have anything handy from the pure Newtonian perspective beyond the usual "minimises the lagrangian and your equations of motions look the same". However, you might be interested in proofs which show newtonian gravity as low energy approximation of general relativity. And since general relativity has a nice action formulation, it all gets nicely tied in.

Hope this helps!

[in_a_hole]

But simply getting to the Lagrangian picture from the Hamiltonian picture would just leave me wondering why the Hamiltonian picture works!

My motivation for getting to the bottom of all this is to fill the gaps in my physics understanding at least up to quantum mechanics. I have a grasp of QM but I would like to have some insight into the conceptual leaps that brought us there from classical mechanics. QM works in the Hamiltonian picture and I recall from my undergrad days that you get there from a Legendre transformation on the Lagrangian (or something to that effect) so I'm trying to understand the justification of that approach before moving up the conceptual ladder.

Ideally I would like to be able to trace my way from simple postulates based on observation of the physical world all the way to QM, then maybe to QFT after that.

[omnicognate]

It's an old book and I can't vouch for it as I only just discovered it myself, but it appears to be very highly regarded, it focuses on precisely the questions you (and I) have, and just from the preface I like the author already [1]: The Variational Principles of Mechanics, by Cornelius Lanczos.

There's a PDF here: https://pages.jh.edu/rrynasi1/PhysicalPrinciples/literature/...

[1] An appetising quote:

> The author is well aware that he could have shortened his exposition considerably, had he started directly with the Lagrangian equations of motion and then proceeded to Hamilton’s theory. This procedure would have been justified had the purpose of this book been primarily to familiarize the student with a certain formalism and technique in writing down the differential equations which govern a given dynamical problem, together with certain “recipes” which he might apply in order to solve them. But this is exactly what the author did not want to do. There is a tremendous treasure of philosophical meaning behind the great theories of Euler and Lagrange, and of Hamilton and Jacobi, which is completely smothered in a purely formalistic treatment, although it cannot fail to be a source of the greatest intellectual enjoyment to every mathematically-minded person. To give the student a chance to discover for himself the hidden beauty of these theories was one of the foremost intentions of the author.

[DarmokJalad1701]

The author of this book is the same Lanczos that the "Lanczos Resampling" algorithm (used in image resizing for example), is named after:

https://en.wikipedia.org/wiki/Lanczos_resampling

[sn9]

Here's a fun book review: https://www.thepsmiths.com/p/review-the-variational-principl...

[in_a_hole]

I had heard about this book and that quote makes me want to read it. Thank you.

[ano-ther]

It’s been a while but I seem to remember that the first book of Landau-Lifschitz‘ Theroretical Mechanics starts with a 20 page discussion that does this and culminates in the Lagrangian.

[in_a_hole]

I recently got hold of a copy of that. I started Hand & Finch - Analytical Mechanics but their woolly discussion of virtual work and virtual displacement was very frustrating and unenlightening. Perhaps I'll have a better time with L&L.

[computerfriend]

It's a great introduction to Lagrangian mechanics, but as I recall (it's also been a while for me), the motivation for extremising the action is also somewhat vaguely presented.

[lucozade]

> why we define the Action as this object and why we should expect it to be minimised for the physical trajectory in the first place.

The most coherent explanation I've heard was from Feynnman [0]. As far as I understand it (and I may well not have understood it at all well), at the quantum level, all paths are taken by a particle but the contributions of the paths away from the stationary point tend to cancel each other. So, at a macroscopic level, the net effect appears to be be that the particle is following the path of least action.

> a proof of the equivalence to Newtonian mechanics

The Lagrangian method isn't really equivalent to Newton's method. Again, Feynman talks about this in [0]. It's that for a certain class of action, the Euler-Lagrange equations are equivalent to Newton's laws.

It's perfectly plausible to come up with actions that recover systems that represent Einsteinian relativity or quantum mechanics. This is the main reason (as I understand it) why it's considered a more powerful formalism.

[0] https://www.feynmanlectures.caltech.edu/II_19.html

[aap_]

Unfortunately I can't help with the classical picture, but in quantum physics it all comes out very nicely: You can interpret the Lagrangian as giving all possibilities to build a trajectory through spacetime. In the path integral formulation we then follow one such trajectory from one configuration to another configuration and find its amplitude. And then we integrate over all possible trajectories that we could have picked. For incoherent trajectories there will always be another one that cancels out the amplitude. Where the amplitudes add up constructively you will find stationary action and the classical behavior in the limit. So this is a depth-first approach: first follow one trajectory completely, then add up all possible trajectories.

The Hamiltonian approach in contrast is breadth-first: you single out a time axis, start with some initial state, and consider all possibilities that a particle (or field in QFT) could evolve forwards in time just a tiny bit (this is what the Hamiltonian operator does). Then you add up all these possibilities to find the next state, and so you move forwards through time by keeping track of all possible evolutions all at once. This massive superposition of everything that is possible (with corresponding amplitudes) is what you call a state (or wavefunction) and the space that it lives in is the Hilbert (or Fock) space.

So Lagrangian/path-integral: follow full trajectories, then add up all possible choices. depth-first

Hamiltonian/time-evolution: add up all choices for a tiny step in time, then simply do more steps: breadth-first

I imagine it a bit like a scanline algorithm calculating an image as it moves down the screen (Hamiltonian) vs something like a stochastic raytracer that can start with an empty image and refine it pixel by pixel by shooting more rays (Lagrangian)

This is my layman explanation anyways...hopefully it helps, even though i can't say much about their relationship in classical physics.

[whatshisface]

The least action principle conceptually emerged from the least time principle for light. Light refracts along the path that gets it from the starting to the ending point the quickest, and the index of refraction is what regulates its speed. The question went like this: we know that potential and kinetic energy work together to regulate the speed of moving objects. Is there a way to combine the two quantities into something like an index of refraction? The analogy between potential fields and optics isn't just conceptual - beams of charged particles are focused using electromagnetic "lenses," made out of fields.

[in_a_hole]

Do you know any references that discuss this in detail? I'm interested in the history of these developments. Who noticed this? Who asked this question?

[wbpaelias]

I believe Veritasium had a series where they derive the equivalence to Newton's laws

[abdullahkhalids]

There is no explanation for this, same as there is no real explanation for why energy is conserved or why closed systems have non-decreasing entropy. As others have pointed out, you can show correspondence to Newtonian mechanics under some assumptions, but the Lagrangian approach is applicable to a wide variety of areas in physics - classical mechanics, optics, quantum mechanics, quantum field theory, etc.

The universe has these weird laws, and for now, all we can do is accept them as is. But hopefully, in the future, someone will figure out deeper and simpler principles.

[nh23423fefe]

Seems false. Energy conservation is explained via time translation symmetry and noether's theorem. 2nd law is explained by boltzmann's H-theorem.

[Cleonis]

While most authors posit the stationary action concept as a given, it is in fact possible to go from the newtonian formulation to the Lagrangian formulation, and from there to Hamilton's stationary action.

That is, the relations between the various formulations of classical mechanics are all bi-directional.

At the hub of it al is the work-energy theorem.

I created a resource with interactive diagrams. Move a slider to sweep out variation. The diagram shows how the kinetic energy and the potential energy respond to the variation that is applied.

Starter page: http://cleonis.nl/physics/phys256/stationary_action.php The above page features a case that allows particularly vivid demonstration. An object is launched upwards, subject to a potential that increases with the cube of the height. The initial velocity was tweaked to achieve that after two seconds the object is back to height zero. (The two seconds implementation is for alignment with two other diagrams, in which other potentials have been implemented; linear and quadratic.)

Article with mathematical treatment: http://cleonis.nl/physics/phys256/energy_position_equation.p...

To go from F=ma to Hamilton's stationary action is a two stage proces:

- Derivation of the work-energy theorem from F=ma

- Demonstration that in cases such that the work-energy theorem holds good Hamilton's stationary action holds good also.

General remarks: In the case of Hamilton's stationary action the criterion is: The true trajectory corresponds to a point in variation space such that the derivative of Hamilton's action is zero. The criterion derivative-is-zero is sufficient. Whether the derivative-is-zero point is at a mininum or a maximum of Hamilton's action is of no relevance; it plays no part in the reason why Hamilton's stationary action holds good.

The true trajectory has the property that the rate of change of kinetic energy matches the rate of change of potential energy. Hamilton's stationary action relates to that.

The power of an interactive diagram is that it can present information simultaneously. Move a slider and you see both the kinetic energy and the potential energy change in response. It's like looking at the same thing from multiple angles all at once.

[in_a_hole]

This seems promising, I'll dig into this later.