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The list of formulations makes a reference to an article with clear neutrality issues. That list should only point to notable formulations that are on par in their fundamental impact with the Lagrangian or Hamilton-Jacobi formulations (as some examples). I suggest the removal of the reference to the Udwadia-Kalaba equation for lack of notability. Reading the linked article, it looks like an attempted promotional push. As is well known, there is no unique formulation of analytical mechanics and the list should only point of especially notable formulations such as the Hamilton-Jacobi equation. The reference to the little-known Udwadia-Kalaba equation in the same vein as the other substantially more notable ones is inappropriate.

- V madhu (talk) 11:38, 2 December 2019 (UTC)[reply]

uoioioioioioioioioioioioioioioio.com.net.@gmail 47.17.255.96 (talk) 23:23, 17 May 2023 (UTC)[reply]

the unreferenced paragraph starting with "A key concept of inertial frames is the method for identifying" makes at least two dubious claims. first that acceleration is involved in Einstein relativity. second that stellar fixed coordinates have some meaning in mechanics. absent citations I believe both claims are incorrect. I propose to delete or significantly modify the paragraph. Johnjbarton (talk) 21:53, 14 June 2023 (UTC)[reply]

I removed the incorrect relativity comment and added citations for Goldstein on inertial frames.

Resolved

Johnjbarton (talk) 18:08, 15 June 2023 (UTC)[reply]

The current "Description of the theory" is really a description of Newtonian mechanics. As presented it distorts the over all topic.

I think the overall topic would be better presented by lifting "Branches" to the spot where Description lives and expanding the subheadings to summary sections linking Main. The current "Description" material would shrink to a paragraph, with possible movement to subarticles. Johnjbarton (talk) 19:02, 4 January 2024 (UTC)[reply]

This article does rather seem to suffer from not having been written with an overall plan in mind. Like so many broad-scope articles, it has become a repository for every little thing that somebody wanted to say. I like the idea of elevating the "Branches" section and expanding with {{main}} and/or {{further}} links. XOR'easter (talk) 02:02, 5 January 2024 (UTC)[reply]

OK, I made a stab at doing something like that. Maybe "Description of the theory" can be tightened now. XOR'easter (talk) 02:36, 5 January 2024 (UTC)[reply]

What do you think about moving the Lagrangian, Hamiltonian and and Hamilton-Jacobi sections of Newton's laws of motion#Relation to other formulations of classical physics in to this article? The Newton's law section would be just the current top paragraph and a main link to classical mechanics. This article would get a summary of Newtonian mechanics and the three sections on analytical mechanics. Johnjbarton (talk) 03:13, 5 January 2024 (UTC)[reply]

I prefer those where they are, since they're specifically about the guises in which Newton's laws present themselves in the alternate formalisms, rather than how to do anything in those formalisms other than re-derive the Newtonian approach. XOR'easter (talk) 03:18, 5 January 2024 (UTC)[reply]

I added sections on Hamiltonian and Lagrangian mechanics, because I think it makes sense to have them; this article calls for high-level overviews, while the other zooms in to the granular details of a specific aspect. What the page needs next is, I believe, a careful condensation of the long kinematics passages and the Newtonian section, with an eye to removing tangents and overly textbook-like material. XOR'easter (talk) 04:34, 5 January 2024 (UTC)[reply]

yes, I think we are at fork in the road: the Branches content is short and the rest is too long while also repeating somewhat. Moving more into branches will make it harder to navigate. So my suggestion is to take both forks

• Leave Branches.
• Change the name of "Description of the theory" to "Comparison among branches of classical mechanics"
• Convert the remaining content up to Limitations into summary paragraphs matching the Branches organization. These summaries should focus on helping reader choose, not on explaining.

Johnjbarton (talk) 19:19, 5 January 2024 (UTC)[reply]

I changed the heading of "Description of the theory" and split it in two, since the first part was about kinematics and could apply more broadly than just the Newtonian formalism (e.g., we still use reference frames and coordinates in the Lagrangian method). XOR'easter (talk) 20:50, 5 January 2024 (UTC)[reply]

I am opening this discussion as it is being a source of conflict.

F=d(mv)/dt is not the most general form of the second law, as it is sometimes believed, of which f=ma would be a special case for a constant mass. One can convince oneself of this very easily by expanding the derivative, finding F=m*a+dm/dt*v, where the last term is not invariant under Galilean transformation (i.e. one can change the acceleration by changing inertial frame).

The form F=d(mv)/dt most likely originates from special relativity, and has wrongly made its way to classical mechanics textbooks. This is (almost) never an issue as variable mass systems are rarely considered.

The most general law for a variable mass system is F=m*a-dm/dt*ve (Sommerfeld, "mechanics"), where ve is the ejection (or accretion) velocity, relative to the center of mass. This is the law used in the derivation of the rocket equation, for example, not F=d(mv)/dt.

The intention in F=d(mv)/dt is to express the fact that the force is defined as the rate of change of momentum. This definition is correct, however, to be applied correctly it must include all the parts of a system that carry momentum. This is not what is done when writing directly F=d(mv)/dt, as it only considers the "main" body of the system. In the case of a rocket for example, F=d(mv)/dt=m*a+dm/dt*v involves the velocity v of the rocket itself, and not that of the ejected mass. Only by considering the total momentum P of the rocket AND the ejected mass, and applying F=dP/dt, does one end up with Sommerfeld's law given above.

For completeness: F=d(mv)/dt is (trivially) correct when the mass is constant, or when the sum of the ejection/accretion forces is zero (for example for an isotropic ejection, or with two equal and opposite ejections). F=ma is correct (also trivially) when the mass is constant, or when the velocity of the ejected/accreted mass is zero (for example when the system releases mass without ejecting it), or when the sum of the ejection/accretion forces is zero.

This discussion came up regarding the thumbnail of the "classical mechanics" template. I understand that the intention in choosing F=d(mv)/dt was to display the most general form. As this form is not true in the general case, it should not be used. The question becomes which should be used instead. The most general form for a variable mass system is that of Sommerfeld, but I don't think it would be representative of classical mechanics as it concerns very niche cases. Therefore, I think F=ma is most appropriate.

This is a widespread misconception in mechanics textbooks and it should not be perpetuated on a high-visibility platform such as Wikipedia. The standard reference on the matter is the paper from Plastino and Muzzio "On the use and abuse of Newton's second law for variable mass problems" (1992), Celestial Mechanics and Dynamical Astronomy, 53, 227-232. M Facchin (talk) 15:15, 22 August 2024 (UTC)[reply]

First, please use the math templates to make equations easier to read. I assume you know how to typeset in LaTeX. Given that an object subject to a force is not moving at constant velocity, I do not see why Galilean relativity is relevant here? The moving object that ejects mass is not in an inertial frame of reference and therefore will appear to be subject to a force. In the case of the rocket, the second term should have a negative sign because mass is being ejected in the opposite direction. I remember seeing the derivation in introductory physics with no issue. Nerd271 (talk) 21:12, 22 August 2024 (UTC)[reply]

Considerations of symmetry are about the laws not about the objects themselves. The point about the Galilean transformation is that the law ${\displaystyle F={\frac {d(mv)}{dt}}=ma+{\frac {dm}{dt}}v}$ has this extra term proportional to ${\displaystyle v}$, which is not invariant under Galilean transformation. This means that if the same dynamics is described in a frame moving at constant velocity with respect to another frame, then the acceleration should be different in those two frames, which is impossible (acceleration doesn't change from one inertial frame to another). To make the point simpler one can consider the case where ${\displaystyle F=0}$, then we have ${\displaystyle ma=-{\frac {dm}{dt}}v}$, where the acceleration is directly proportional to velocity, which trivially breaks Galilean relativity as the velocity can be arbitrarily modified by a change of reference frame.

Most derivations of the rocket equation (including that on Wikipedia) start with a balance of momentum before and after ejection, including that of the rocket itself and the ejected mass. The reason why you might think that it makes use of ${\displaystyle F={\frac {d(mv)}{dt}}}$ is because this balance of momentum takes the form F=d(total momentum)/dt. This is, however, not what is done when writing directly ${\displaystyle F={\frac {d(mv)}{dt}}}$, which takes only into account the momentum of the rocket. This would give, as above, ${\displaystyle F=ma+{\frac {dm}{dt}}v}$, where ${\displaystyle v}$ is the velocity of the rocket itself, not the relative velocity of the ejected mass with respect to the rocket, as it should be. M Facchin (talk) 21:42, 22 August 2024 (UTC)[reply]

This section cites a book which I have access to. It basically says the same thing I did. Mass is ejected in the opposite direction, so there is a negative sign. The rocket is not in an inertial frame of reference. You cannot apply Galilean relativity. Nerd271 (talk) 21:22, 22 August 2024 (UTC)[reply]

Precisely, there is a negative sign that does not appear when using F=d(mv)/dt (I'll stick to plaintext as Tex doesn't seem to render). And the velocity attached to the last term is the relative velocity of the ejected mass, not the velocity of the object. This is explicit in the section you mention. M Facchin (talk) 21:46, 22 August 2024 (UTC)[reply]

I do not understand the objection about the applicability of Galilean relativity. Again, considerations of symmetry are about the laws, not the objects subject to the laws. Thank goodness Galilean relativity applies in other cases than objects moving at constant velocities. You may refer to the case of collisions where it is typically introduced, this is however a much more general notion. M Facchin (talk) 21:52, 22 August 2024 (UTC)[reply]

Consider an active rocket in otherwise empty space. The net force on the entire system is zero. But neither the rocket nor the exhaust constitutes an inertial frame of reference. The equation ${\displaystyle ma={\frac {dm}{dt}}u}$ is only for either the rocket or the exhaust, separately. Here, ${\displaystyle u}$ is the velocity of the exhaust. If you take them as a single system and use the center-of-mass frame, then yes, this is an inertial frame of reference, and Galilean relativity applies, because the net force is zero, meaning the velocity is constant. Nerd271 (talk) 22:28, 22 August 2024 (UTC)[reply]

${\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}}$ is a good compromise. M Facchin (talk) 20:59, 25 August 2024 (UTC)[reply]