Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  Is Lagrangian mechanics **circular** at its heart (in that it *assumes* Newton's laws)?

+ 1 like - 0 dislike
1917 views

Sorry for just copy-pasting this question from SE here. But I don't know why the user QMechanic there closed my question since "This question does not appear to be about physics within the scope defined in the help center."

The following is what I posted.

This is a question on Lagrangian formulation of mechanics and not Newton's formulation. So, we don't a priori take Newton's laws to be true.

This SE post has answers which brilliantly define mass explicitly using Newton's laws.

Now all the resources which I've come across, which "claim" to formulate Lagrangian mechanics (including Susskind himself) begin by postulating Lagrangian as $L(q, \dot q):={1\over 2}m\dot q^2-V(q)$, and just state $m$ to be the mass of the object.

That's either just brushing details under the rug or total ignorance! We can't just take $m$ to be granted from Newton's laws to formulate the supposedly "independent" (but equivalent to Newtonian mechanics) Lagrangian mechanics! Otherwise it'll be circular -- we'd have used Newton's laws (in the form that there exists a quantity called mass for each object; see the second top-voted answer in the linked post, which takes it to be the second law) in formulating Lagrangian mechanics! We need to *define* mass independently here (and later need to show that this definition is equivalent to that in Newtonian formalism).

Question: So what is a mathematically precise (at least at precise as the answers in the linked post are!) definition of mass in Lagrangian mechanics?

I'll also greatly appreciate if you can show that Lagrangian and Newtonian definitions of mass are equivalent.

asked Jun 25, 2020 in Theoretical Physics by Atom (5 points) [ no revision ]

Mass is always defined as that number that makes momentum conservation come out correctly (in collisions, for example). This is the only definition that is possible. There are not two different definitions of mass that need to be shown to be equivalent.

1 Answer

+ 0 like - 0 dislike

Tha "Lagrangian Mechanics" (LM) is not "circular", but "secondary" with respect to the Newton Mechanics (NM). Maximum that LM can do is to reproduce the NM equations of motion, and it uses the Newtonian initial conditions. In Physics no information about the system at $t=t_2$ is known (but searched), and in LM the initial velocities are hence arbirtary; thus the trajectories are arbitrary too. There is no unique trajectory in such a problem setup. Although mathematically the final contitions at $t=t_2$ make sense, it is unprobable situation in Physics. So the Newtonian aproach prevails.

As to the masses, in the non relativistic case they are additive - when you calculate the center of inertia of a compound system. In the relativistic formulation they are not completely additive, but they are still constants by definition (conserved quantities), as any other constant coefficients in the equations ;-).

answered Jun 26, 2020 by Vladimir Kalitvianski (102 points) [ revision history ]
edited Jun 26, 2020 by Vladimir Kalitvianski

Concerning the first paragraph in you answer, things are the other way round:

The Lagrangian framework is very general, Newton s law can be derived as the corresponding EOMs when applying it to classical mechanics. But the Lagrangian framework can be applied beyond classical mechanics ...
 

@Dilaton: You have not got my point at all.

@Dilaton: Also, in order to say the LM is more general, i.e., it describes "something else", you have first to have this "something else" to compare with LM. It means that "something else" constructed by physicists, had been constructed without LM. And normally, "something else" constructions contain inequalities outlining the "something else" notions applicability region. This is absent in LM.

As my Physics professor said (while I was a student), the main physical equations are guessed (or "obtained"), rather than "derived".

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysic$\varnothing$Overflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...