Talk:Lagrangian mechanics

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Possible error in the introduction using the terms ″variation″

In the introduction of this page the term $\delta {\mathcal {S}}:[t_{0},t_{1}]\rightarrow \mathbb {R}$ is defined and called the ″variation″, which is possibly erroneous (although $\delta {\mathcal {S}}$ is indeed a Gateaux derivative). Comparing $\delta {\mathcal {S}}$ with the definition of ″variation″ (of $x_{0}$ ) in the Wikipedia page virtual displacement, one finds instead that a variation is a function as well of the parameter ε. In other words, the "variation" is not equal to the "Gateaux derivative". However, it might be that the term "variation" is defined in some other way (which is then inconsistent with the definition in virtual displacement). Doubledipp (talk) 16:45, 16 April 2022 (UTC)Doubledipp[reply]

There is some discrepancy between math and physics in how the term "variation" is used. In math,

\delta {\cal {S}}

could have been (and is in some sources) called infinitesimal variation. StrokeOfMidnight (talk) 06:46, 20 April 2022 (UTC)[reply]

Lead section rewritten

I've rewritten the lead section per MOS:INTRO to get rid of the equations and the more complicated mathematical language. All of that stuff appears later in the article, and the lead needs to provide a less technical overview of the topic.PianoDan (talk) 20:05, 25 April 2022 (UTC)[reply]

Suggestion about

"Any variation of the functional gives an increase in the functional integral of the action."

Delete this sentence and discuss saddle point in the sentence before it. 210.61.187.232 (talk) 11:07, 17 June 2024 (UTC)[reply]

I deleted the sentence. Johnjbarton (talk) 15:01, 17 June 2024 (UTC)[reply]

Source for nightmares

@WikieMouse added this paragraph

Analyzing the behaviour of complex dynamical systems was nightmarishly complicated until the emergence of Lagrangian Dynamics - for example, calculation of the motion of a swinging pendulum since the time-varying constraint forces like the friction of the wire, the tension of the pendulum rod, the drag in the air, etc. - Lagrangian dynamics uses the principle of least action proposed by Pierre de Fermat to explain the the properties of light waves.

Adding this source:

Parsons, Paul; Dixon, Gail (2016). 50 ideas you really need to know : science. London: Quercus. pp. 4–7. ISBN 9781784296148.

What does the source talk about, the last sentence, or every aspect of the paragraph? I know that Lagrangian dynamics profoundly altered theoretical analysis but this is first I've heard that it also had similarly fundamental effects on practical applications. Johnjbarton (talk) 18:11, 23 September 2024 (UTC)[reply]

This is what the source said, although I would say I rephrased it a little and added an example. And as far as I know, Lagrangian mechanics does make calculating practical things much more easier than Newtonian mechanics since it only makes use of a set of generalized equations. I stand mine although I could be a little wrong, you see I am not an experienced editor yet. WikieMouse (talk) 11:22, 26 September 2024 (UTC)[reply]

@WikieMouse I think different kinds of physics problems are more easily tackled in one framework or another. While not commonly covered in introductory physics, each approach adopts certain criteria for solutions. Newtonian mechanics focuses on forces and a single point in time, trying to predict motion in the next instant. Thus when the forces are known and the goal is predicting a small increment in time, the technique is superior to all others. Conversely integral methods like Lagrangian mechanics are based on energy and are much better at answering global questions like the time course of a pendulum. Johnjbarton (talk) 15:23, 26 September 2024 (UTC)[reply]

That's what I am saying! There is nothing wrong in my paragraph. I got the necessary stuff covered. WikieMouse (talk) 09:30, 27 September 2024 (UTC)[reply]

I don't know if this is because the additions could be better written or because the source (I have no access to it) got things wrong, but it doesn't make much sense to me. A swinging pendulum is not a very complex system. What is the friction of the wire? Though there are tricks to write lagrangians for some non-conservative forces, their main use is for the conservative ones (with well-defined potentials). Where are we "completely summing the overall paths of possible motion of the particles"? Langrangian mechanics (people) are least concerned with path integrals when solving their problems. I don't think any engineers use langrangians. The Fermat principle note disrupts the flow of the text, it's also a little ambiguous. Revert, find better sources? Ponor (talk) 11:23, 27 September 2024 (UTC)[reply]

I am very sorry, let me change my example so that you're more happy. WikieMouse (talk) 12:18, 27 September 2024 (UTC)[reply]

Alright I changed the paragraph, have a look and tell me if it needs anything else. WikieMouse (talk) 14:15, 27 September 2024 (UTC)[reply]

I added a ref and made some changes. BTW the variational principle that leads to the principle of least action is a separate concept. It is an extension of the Lagrangian approach. See eg Action principles. Johnjbarton (talk) 15:55, 27 September 2024 (UTC)[reply]

Thanks for the correction. WikieMouse (talk) 17:01, 27 September 2024 (UTC)[reply]