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"In Euclidean geometry, the geodesic are the straight line, but in more general spaces they need not be" -- not sure about this. A geodesic is what we mean by a "straight line" -- what else can a straight line be if not a geodesic? The fact that great circles don't appear straight is just a nasty side-effect of the Mercator projection mindset. -- Tarquin 14:54 Oct 28, 2002 (UTC)
Yes, I agree; I reformulated a bit. Also, geodesics by definition really only give locally the shortest paths, not necessarily globally. AxelBoldt 19:26 Oct 28, 2002 (UTC)
It's been a (somewhat) long time since I studied that, but aren't geodesics straight with respect to the local curvature in some sense ? That should be added to the article. --FvdP
Article said:
Moved from article:
-- Jerzy 17:34, 2004 Jan 26 (UTC)
This curvature is in turn determined by the Stress–energy–momentum tensor; this is the content of the Einstein equation.Shmuel (Seymour J.) Metz Username:Chatul ( talk) 19:53, 6 September 2020 (UTC)
I removed "In mathematics, a geodesic is a curve which is everywhere locally straight." I belive it can not be usefull, it pretends to be little mathematical, but does not have more sense than the old formulation, contrary "Geodesic stay for the curves which are "straight" in a sense." at least explaign the meaning of the word and no mathwords used.
Tosha 22:39, 22 Feb 2004 (UTC)
I reverted that, because it makes absolutely no sense as a statement in the English language. Anthony DiPierro 22:54, 22 Feb 2004 (UTC)
I do not know English, its true. Ok, is it better now?
Tosha 23:18, 22 Feb 2004 (UTC)
Not much. What does "stay for the curves" mean? I'll let someone else, who knows more about geodesics than I do, fix it. Anthony DiPierro 23:32, 22 Feb 2004 (UTC)
Ok, I hope now everybody happy(?) I wanted to get rid from word locally here (see above)
Tosha 01:24, 23 Feb 2004 (UTC)
What you do is much worse, if you want to grab an idea of geodesic then it is "straight in some sense" to do math you should define what is the "sense" and infect there are many different senses for this on the same space depending on structure you have/choose, now you have this strange curvature ... what does it mean for metric space for instance...
I will not change it back, I'm tired (hope someone will do it)
Tosha 03:13, 23 Feb 2004 (UTC)
There is no meaning for geodesic curvature in metric space, but even if you make one the curves with zero geod.curvature might not be geodesic.
Tosha 03:22, 23 Feb 2004 (UTC)
BTW geodesic curvature is nearly defined in the article, so one could just add one line in subsection "Riemannian and pseudo-Riemannian manifolds" instead of giving ref.
Tosha 04:22, 23 Feb 2004 (UTC)
Tosha, i can't tell whether you or Decumanus is the more authoritative editor for this article, but i'd urge you to work out the wording you have in mind on this page.
For instance, after staring for a while at the first paragraph of your first edit of Geodesic, which reads
, i begin to suspect that you intended the meaning of that sentence to be close to
In proposing that reading, i note that the concepts of "stay" and "stand" are related (in fact, the probably come from the same ancient word) even tho a native speaker would never consider using "stay" to cover the metaphorical sense of "stand" that occurs in my suggested interpretation.
(And to me, that sounds like a good introductory "motivating the idea" approach for an opening paragraph, and one that is consistent with shifting to a much more rigorous approach, such as Decumanus seems to be pushing for, in discussing the specific meaning of geodesic in the various contexts.
(By the way, i also note that your use of "in a sense", which has caused some objections here, is not something you suddenly added to the article, but rather a variation or elaboration of
which was introduced by User:AxelBoldt in an edit of 18:55, 2002 Oct 28, as far as i so far notice, remaaining in the article without objection for over a year. And part of my point to you is that fact does not seem to have come out here. IMO you're going to have to work hard not just at the technical content, but also both in communicating it clearly in this weird language of English (that you so bravely have undertaken to learn to an impressive degree), and finding out why other editors are being so seemingly stupid in not following your reasoning.)
You may have things to add to the article that no one else on WP is prepared to contribute, but right now, we can't tell whether or not that's the case.
I don't want to try and address the question of whether Decumanus has tried as hard as they should to understand what you're saying, but IMO there will be no hope of finding that out without your taking time to make sure that the ideas you are bringing forward are clearly understood. IMO, that will take a lot of patience on your part to help Decumanus and others understand your meanings. -- Jerzy 05:24, 2004 Feb 23 (UTC)
I've done a copy edit here. I'm with Tosha on this - he's a valuable contributor in this field, and I believe he knows exactly what he is talking about. I have similarly copy edited other pages of his.
On the geodesic curvature matter; the definition now standing on that page needs work; for one thing it isn't obvious to me that it is compatible with the link from the Gauss-Bonnet theorem page (though it may be in fact). I think that point could be addressed there. For the time being, I felt linking geodesic to geodesic curvature wasn't clarifying, and I took out the link.
Charles Matthews 08:53, 23 Feb 2004 (UTC)
Many things come to my mind when I think of geodesics, and "straight" is definitely not one of them. As discussed above, "straight" could be taken to depend on context, but the most intuitive notion of "straight" to me comes from lines in Euclidean space. These are in direct contrast with the mental image of a 2-sphere embedded in Euclidean 3-space in the natural way with spherical geodesics drawn upon it. Granted, these lines on the sphere only appear to be curved as an artifact of the Euclidean viewpoint, but I would argue that this is the most intuitive viewpoint for the reader new to the material. What do you all think about this? -
Gauge 03:18, 15 Oct 2004 (UTC)
I removed 4th Hilbert problem (link to Hilbert Problems: http://aleph0.clarku.edu/~djoyce/hilbert/problems.html) from this page, it is clearly relevant, but not at all the first problem which should be mentioned here, also the note after that is not quite correct, it was solved at least in dimension 2.
Tosha 17:04, 8 Jun 2004 (UTC)
The article applies "geodesic" only to lines. It can be also be applied to higher-dimensional submanifolds. I could write something on this, but I'm not sure whether there's a difference between geodesic surfaces and totally geodesic surfaces (see e.g. [1], p.6). Does anyone know? Fpahl 15:48, 14 Sep 2004 (UTC)
The second sentence under " For metric spaces" begins with "More precisely". It seems to me that the precise definition is actually something new, and that it is a non-trivial statement that a curve satisfying that definition "is everywhere locally a distance minimizer". Also, the definition seems to be too narrow in that it requires the curve parameter to be proportional to the arc length; wouldn't any reparametrization of such a geodesic still count as a geodesic? Fpahl 15:56, 14 Sep 2004 (UTC)
Linas, you make the statement: "masses follow timelike geodesics, period. no ifs-ands or buts". It is my understanding that only the world lines of free particles are geodesics of spacetime. Thus, here on the surface of the Earth, I feel the 'weight' of acceleration and my world line(s) is certainly not a geodesic. Alfred Centauri 14:27, 6 Jun 2005 (UTC)
Suggestions/comments ? Mpatel 15:58, 6 Jun 2005 (UTC)
Domain is not introdcued before it is used in the first definition. It should be introduced to make the article self-contained and accessible to people who know mathematics but do not use geometry daily. Wazow 10:54, 11 August 2005 (UTC)
This statement is incorrect in the preamble. Though this is often used to characterize them, they are not inherently defined this way. Perhaps we need to write an explanation of variation of arclength and the Jacobi equation. Cypa 04:27, 28 September 2005 (UTC)
I just re-wrote the section that held the geodesic eqn. I'm hoping this makes it easier to understand. The fear-inducing mention of parallel transport has been banished to a later section. linas 03:51, 30 September 2005 (UTC)
It would be nice to have, in this article, or in a related one,
Further Edification:
Exotic geometry:
Mention of whiz-bang uses and applications:
linas 04:45, 30 September 2005 (UTC)
I did not watch this article for quite a while, now it has many problems: bit wrong intro is just a little thing, main thing is completely wrong section Riemannian geometry and the geodesic equation.
On intro: the curent intro is has partly wrong def, it is not sutabale for all cases, in particular it is very wrong for pseudo-riemanniann manifold (and for general gravity) as well as for spaces with connection. It was a long discussion before we converge to the one I like and I think we have to keep it.
Did you read what I wrote? Tosha 03:22, 21 October 2005 (UTC)
Main problem: it does not work for pseudo-riemanniann manifold (and for general gravity) as well as for spaces with connection. Note that I did not change it so far, but I will do it later Tosha 15:30, 21 October 2005 (UTC)
On Riemannian geometry and the geodesic equation One can not define it trough lenght, at least yoyu should ask for constant speed and it does not work for pseudorieamannian case.
I do not blame you, I'm just correcting mistake Tosha 03:22, 21 October 2005 (UTC)
It is explained above, if you do not see the mistake, you do not know what is geodesic, why then you edit this article? Tosha 15:30, 21 October 2005 (UTC)
In addition, the geodesic flow section can be removed almost completely, and refer to Hamiltonian mechanics
So why not leave all technical stuff there there is no need to repeat it here Tosha
the Hamiltonian can be mentioned in def of geodesic (I'm not against anything which is correct) note that there is no technical detales about lagrangian as well. There is reason to do this def for Riemannian geometry and most books do it this way (and it is not becouse they are stupid, main thing: it is more usefull). Also, as it is stated in the end metric def coinsided with Riemaniann
This section had many things not really related to the topic now at least it is clear what geodesic flow is Tosha 15:30, 21 October 2005 (UTC)
So I revert this secton, and thin a bit about intro Tosha 01:16, 20 October 2005 (UTC)
Not this way, the article had serious problems, see above, so before reverting you should describe the reason for making it wrong! Tosha 03:22, 21 October 2005 (UTC)
Just read carefully what I wrote, I will not repeat it again! Tosha 15:30, 21 October 2005 (UTC)
More over in my edit it is not even in def there is only comment about lagrange and we can add one about Hamilton. Tosha 15:44, 21 October 2005 (UTC)
There is metric def, so why should we repeat it for Riem case? the given def is what people use in Riem. Geom, it is used in most of books, clearly metric def gives a motivation. And sinse it works just as well for pseudo-Riemannian case why not top leave it this way?
Ok, I will not revert it my-self, I think my edit is much better, and curent is wrong. Hope someone will do it for me Tosha 15:41, 21 October 2005 (UTC)
My feeling is there are some problems with arranging material in this article, some of which I corrected plus the section on (pseudo-)Riemannian geometry is almost completely wrong. Sinse I promised to not revert I will not, but everybody else is wellcome to compare and revert Tosha 22:06, 21 October 2005 (UTC)
Tosha, when you say the section on (pseudo-)Riemannian geometry is almost completely wrong, do you mean the section called Riemannian geometry and the geodesic equation? If we remove the word "pseudo" from that section, does it become almost completely right?
Overall, I'd like the article to start with something as simple as possible, and move to more correct, more general definitions. This would include a discussion of why the simple definition fails, e.g. null geodesics, lack of existance proofs.
Here's a proposal I'd like to make:
Would that work? linas 00:42, 22 October 2005 (UTC)
I think at the moment, the best we can do is revert my edit. Tosha 01:26, 22 October 2005 (UTC)
Luna, with all respect, I do not want to keep a wrong article, even if it is really easy to read. BTW, if you noticed I did make some changes, which you were mentioned in the above discussion. Tosha 02:26, 25 October 2005 (UTC)
a curve c(t)=(t^3,0) in the plane minimize length but is no a geodesic, isn't it? you can correct def, by making it const.spped but then it will repeat def in metric geometry part. Also the statements about ways to solve it were completely wrong... Tosha 22:58, 25 October 2005 (UTC)
My two cents as an undergraduate maths student: please forgive me if I say anything stupid (because I admit I know little about geodesics compared to all the experts around). With geodesics I learnt them the "parallel transport" way first and the "variational" way afterwards, and yet I know plenty of classmates who learnt them the other way round, or one and not the other, even though we are doing the same degree at the same time. So I think having separate sections that describe them one way independently of the other way would be great, and then a section can follow on how they are related to each other. I was looking up "geodesic" here myself hoping to understand more the connection between the two approaches that I've learnt. -- KittySaturn 00:03, 23 October 2005 (UTC)
Given a ( pseudo-) Riemannian manifold M, a geodesic may be defined as a solution to a differential equation, the geodesic equation. The geodesic equation may be derived, using variational principles, as the Euler-Lagrange equations of motion applied to the length of a curve.
wrong you can not use length
that maps an interval I of the real number line to the manifold M, one writes the length of the curve as the integral
where is the tangent vector to the curve at point . Here, is the metric tensor on the manifold M.
Using the length given above as if it were an action, and solving for the Euler-Lagrange equations, one gets the geodesic equation.
wrong again it is not what you will get.
In terms of the local coordinates on M, this is
Here, the xa(t) are the coordinates of the curve γ(t) and are the Christoffel symbols. Repeated indices imply the use of the summation convention.
Rather than working with the length, one may obtain exactly the same equations by minimizing the energy functional
In physics, this energy functional is called the action, although this term is not commonly used in mathematics. Working with the action instead of the length has the minor advantage of avoiding a pesky square-root and the major advantage of using the same conceptual language used in other important areas of geometry and topology.
foggy, but wrong again
The geodesic is a second order ordinary differential equation. Equivalently, geodesics may be defined to be the solutions to a pair of first order differential equations; these equations, the Hamilton-Jacobi equations, are obtained by working with the Hamiltonian formulation instead of the Lagrangian formulation, and are given in the next section.
I changed
to
which I believe to be a correct. But as I am no expert of on Riemannian manifolds, I figured I would alert others to the change.
Jrohrs 18:04, 1 August 2006 (UTC)
Tosha, you also removed the following section:
This is a strage statement, it does not define exp.map, or at least give very non-complete def.
The Hopf-Rinow theorem states, among other things, that any two points on a Riemannian manifold are joined by a geodesic.
I guess one should add "complete"
Well if not Riemannian then geodesics might be not defined, which spaces are ment here? pseudo-Riemanian? Finsler? metric spaces? what is the meaning of non-uniqueness and so on, it is just too foggy...
I am guessing that you removed this because the article failed to define the meaning of the exponential map in the context of a Riemann surface? linas 13:13, 26 October 2005 (UTC)
(I thought a section on completeness would be interesting, because some systems aren't, e.g. constrained mechanical systems, such as robot arms, aren't geodesically complete. Ditto for GR.) linas 15:11, 26 October 2005 (UTC)
I think it enough to mention Hopf-Rinow theorem in see also, or at most give a def og geodesically complete space here and refer to Hopf-Rinow. Tosha 18:57, 26 October 2005 (UTC)
There should be a section (or at least a subsection) on the geodesic equation, as this is certainly something that a general reader should see stand out. Looking at the table of contents, the concepts mentioned are important, but geodesic equation should be there too. MP (talk) 09:39, 19 February 2006 (UTC)
There should be a derivation of the geodesic equation from the action since it is pretty important...if no one objects ill add a section on it. — Preceding unsigned comment added by 207.161.187.38 ( talk) 00:41, 15 August 2014 (UTC)
While at first glance it sounds hokey, it is a legitimate word: See Google search. While it may not be widely used, I think it is worth mentioning——I'm not attempting to hijack this article with the term (though, generally, in articles talking about "geodesy" and "geodetic" concepts not specifically referring to Earth, I think "planeto-" should be used instead of "ge-"), I'm just pointing out the term's origin and more generic form. ~Kaimbridge~ 19:22, 30 July 2006 (UTC)
"Planetodesic" yields zero books.google.com hits, and the only google hits are to this article and this talk page. So I went ahead and removed it. If someone wants to wedge the term "planetodesy" back in I suppose that's okay, although (a) it probably doesn't belong in the lead and (b) it's so obscure it's probably not necessary at all. — Steve Summit ( talk) 13:10, 7 February 2007 (UTC)
In a general space, there is a difference between an autoparallel curve (which parallel transports its tangent vector) and a geodesic (which is defined as a curve which extremizes length, regardless of which type of space you're working in). In the specific case of a (pseudo-)Riemannian manifold with torsion-free and metric-compatible connection, the difference vanishes. I've tried to establish this distinction more clearly. For a nice treatment, I recommend the first dozen pages of Ortin's "Gravity and Strings". Note that Wald assumes torsion-free and metric-compatible very early on, so he never separates the two cases. -- MOBle 21:26, 1 August 2006 (UTC)
Hello,
I posted this in another page, so this is just a repost.
There a few things in the definition of geodesic provied I do not quite agree with. Firstly, when one talks about points, I think points are "in" a space, not on it, as this assumes some kind of ambient space. We live in R^3, not on it.
I also think that one cannot support the statement that geodesic are defined as the shortest paths. Locally or not, because of reparametrisation, for example. I like the definition given in "John M. Lee. Riemannian Manifolds: An Introduction to Curvature. Number 176 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. " where geodesic are simply the curves whose acceleration is zero (well, the covariant derivative of the velocity). Geodesics are, locally, the shortest paths, but the opposite is not true. From the point of view of accelaration, it is natural then to see that the lenght of the velocity vector of a geodesic is constant.
Regards
Krzysztof Krakowski
129.180.1.224
02:40, 19 April 2007 (UTC)
In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space.
Uh, or the longest? "Extremal" does not mean "shortest". -- 76.224.88.42 ( talk) 21:03, 20 December 2007 (UTC)
"Geodesic" (adj) in "geodesic grid" refers to geodesy (n), the shape of the Earth. The points in such a grid are geodetic (adj): they lie on the surface of the Earth; unfortunately -- for it makes for confusion -- it is also common to say they are geodesic (adj) points. The "geodesic" in "geodesic grid" does not refer to geodesics (n), the subject of this article. (And when I said "inflection", I wasn't referring to the mathematical concept, but the grammatical one.) It would've helped to follow the link to the geodesic grid article to see what it was about.
Silly rabbit, indeed. :P 128.83.68.169 02:21, 12 July 2007 (UTC)
I find this article to be too focussed on mathematics. If you are mentioning "metrics" and the "affine connection" in the first paragraph you are already in trouble. While I can appreciate Riemannian geometry and so forth, I remind you all that "geodesic" has "geo" for "earth" in front of it, hence remind you of its origins. That is, in geodesy. I suggest two changes to the article: (1) that the introduction be reordered so that the simpler "path over the earth" definition/discusson is given as the first paragraph, followed by the ramp up to the mathematics (you are scaring people away now), and (2) the inclusion of a new section, one coming right after the introduction, on earth geodesics. The new section would talk about such things as geodesics on a sphere, and spheroids such as WGS84, with some simple examples, perhaps even a figure... In short...you math people have to back off! :) The article might even benefit from a split into two articles, one being the simple, earth-centric discussion, the other being the more complex math-centric discussion.
(Consider the poor high-schooler writing his report the night before it is due who just wants to know what geodesic means...and he goes to his favorite reference wikipedia, only to find this article...Yikes!)
As an example of a figure, see: http://909ers.apl.washington.edu/~dushaw/perth_bermuda2.jpg The upper path is the geodesic on a sphere, the lower path is a geodesic on the wgs84 ellipsoid. I am the author of the figure and could release it to wikipedia if you thought it useful. The path goes from Perth, Australia to Bermuda. Bdushaw 21:35, 6 October 2007 (UTC)
"in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle." Strictly speaking, this isn't quite right. The Earth is not a perfect sphere (it's an oblate spheroid or ellipsoid - i.e slightly squashed at the poles), which means that the shortest distance between two points (a geodesic) is not quite a great circle. But I'm not sure how best to explain this in the intro without over-complicating it. Wardog ( talk) 11:15, 21 February 2008 (UTC)
I don't really see why there should be a separate geodesic (general relativity) article. The existence of two articles suggest that they are different concepts, while they are exactly the same concept. ( TimothyRias ( talk) 15:25, 17 July 2008 (UTC))
Well, OK, there clearly is no consensus for such a merger so I'll remove the merge template. I do think this is a shame though. Much of the opposition to the merger seems to be based on the misconception that there are distinct concepts of geodesic in mathematics and physics, while in fact there is only one such concept, the one in geometry (strictly speaking you could say there are two such concepts which coincide for the Levi-Civita connection, but both are relevant to GR). Pretty much everything that is currently said in the geodesic article is relevant to GR in someway, and similarly the geodesic article needs a good applications section discussing the role of geodesics in applications such as GR. But, instead with have some territorial pissings from the mathematics and physics camp, that lead to the propagation of the illusion that there are two distinct concepts here. Such a shame. TimothyRias ( talk) 15:26, 5 December 2010 (UTC)
should be as the Gamma symbols are not tensors. Therefore there is no need to stagger the indices: they can't be lowered anyway. If everyone agrees I will delete the confusing space before mu. Tkuvho ( talk) 13:03, 24 February 2010 (UTC)
The section on Riemannian geometry was recently changed rather dramatically, and I'm not certain that I agree with the edit summary there:
For a piecewise curve (more generally, a curve), the Cauchy-Schwarz inequality gives
with equality if and only if is equal to a constant a.e. It happens that minimizers of also minimize , because they turn out to be affinely parameterized, and the inequality is an equality.
The usefulness of this approach is that the problem of seeking minimizers of E is a more robust variational problem. Indeed, E is a "convex function" of , so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers within isotopy classes. In contrast, "minimizers" of the functional are generally not very regular, because arbitrary reparameterizations are allowed. (One could probably also argue that the variational problem for is, by itself, rather ill-posed.)
Sławomir
Biały
15:28, 23 January 2016 (UTC)
Yes, I've heard that the vast majority of possible spacetimes are pathological. But isn't it true that for any spacetime actually used in general relativity the distance between two points and on a curve is Jrheller1 ( talk) 01:00, 24 January 2016 (UTC)
This article should be primarily oriented towards explaining geodesics on surfaces in 3-dimensional space, because that is what 99.99% of useful geodesics are. The other 0.01% of useful geodesics are geodesics in general relativity, and 99.9% of useful geodesics in GR are geodesics of the Schwarzschild metric. Don't you agree that the derivation of the geodesic equation I posted (which is very similar to Kreyszig's derivation) is valid for the Schwarzschild metric? If there are other useful spacetimes for which this derivation is not valid, they should be dealt with in the "Geodesics in general relativity" page. The derivation of the geodesic equation on the "Geodesics in general relativity" page is just a less clear and concise version of Kreyszig's derivation. It certainly does not address any of the issues you (Slawekb) raised in your previous post. Jrheller1 ( talk) 21:39, 24 January 2016 (UTC)
A parameterization of a curve in the (x,y) plane of the form (t,y(t)) can represent any possible curve for which there is only one value for a given value. What this means is you can draw any piecewise smooth curve in the xy-plane that has only one value for a given value and find a parameterization for it of the form (t,y(t)) or equivalently (x,y(x)).
There is no need to use the more general form (x(t),y(t)). This will only produce a more complicated ODE (a system of two second order ODEs) with the same result: minimizing the arc length integral will produce the right solution (straight lines through the origin) and minimizing the "energy" integral will produce the wrong solution. You can use an ODE solver with initial conditions y(0)=0 and y'(0)=a for some constant for the "energy" ODE above and see for yourself that it produces a curve that deviates more and more from the right answer as the curve approaches the hemisphere boundary.
A computationally simpler example of the result of applying the Euler-Lagrange equation to both and is for the minimal surface problem. The minimal surface problem is to find the surface with minimum area for given boundary conditions. To do this it is necessary to minimize the surface area integral The E-L equation (function of multiple variables version) applied to the surface area integral results in the PDE (in other words, mean curvature is zero everywhere). The E-L equation applied to produces the Laplace equation This is the wrong answer. The Laplace equation is only an approximation to the minimal surface equation for height field boundaries with relatively slow variations in z. This is just like the solution to the "energy" ODE from my last post. It is a fairly good approximation to the geodesic close to the origin (where z is varying slowly) but gets worse farther away. Jrheller1 ( talk) 04:56, 26 January 2016 (UTC)
Obviously a lot of technically skilled people have edited this article. But that seems to have blinded them to the obvious. For the majority of readers coming to this article all they are interested is great the circle idea of a geodesic. They just need a simple sentence to inform or confirm their notion of what a geodesic is.
The don't need or want a topic sentence like this : "...In the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are (locally) the shortest path between points in the space....". This sentence contains a raft of references to concepts and words that a general reader will be unfamiliar with. A general reader is likely to either skip the article or start on a trip through the internet trying to figure out what the heck this sentence means.
What the general reader is probably interested in is contained nicely in this paragraph: "The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph."
It seems to this reader that the problems of the lead section could be fixed easily by just reversing the order of the two paragraphs. The technical reader can easily skip past the opening section that he probably is well aware of and begin to try to understand the more technical uses of the word in mathematics and physics with the second paragraph of the lead section serving as a summary of the technical information to follow. — Preceding unsigned comment added by Davefoc ( talk • contribs) 05:47, 15 April 2016 (UTC)
JRSpriggs made the edit suggested above. Thank you. Davefoc ( talk) 05:01, 20 September 2016 (UTC)
Across, much of the article "t" is used as the affine parameter along a geodesic curve. This is a rather unfortunate choice as to many readers it will suggest a relation with time, where no such connection need to exist (or even make sense in the case or Riemannian geometry). I would suggest changing it to something more "neutral" such as λ. T R 11:26, 29 September 2016 (UTC)
The current short description is misleading in several regards. First, in a lorentzian manifold a geodesic maximizes the interval rather than minimizing it. Second, in a manifold with a connection but no metric, geodesics are defined but path length is not. Can anybody come up with wording that is more accurate but still concise? Shmuel (Seymour J.) Metz Username:Chatul ( talk) 23:01, 4 September 2020 (UTC)
The article has the short title Shortest path on a curved surface or a Riemannian manifold
. That is correct for a manifold with a
positive definite metric, but it is incorrect for a manifold with a
Lorentzian metric. I've considered changing it to "Extremal path", but that's not quite right either. Your thoughts?
Shmuel (Seymour J.) Metz Username:Chatul (
talk)
01:10, 28 January 2021 (UTC)
{{
about|geodesics in general|geodesics in general relativity|Geodesic (general relativity)|Geodesey|Geodesy|other uses}}
, remove "and pseudo-Riemannian manifolds". Shmuel (Seymour J.) Metz Username:Chatul ( talk) 21:33, 28 January 2021 (UTC)
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"In Euclidean geometry, the geodesic are the straight line, but in more general spaces they need not be" -- not sure about this. A geodesic is what we mean by a "straight line" -- what else can a straight line be if not a geodesic? The fact that great circles don't appear straight is just a nasty side-effect of the Mercator projection mindset. -- Tarquin 14:54 Oct 28, 2002 (UTC)
Yes, I agree; I reformulated a bit. Also, geodesics by definition really only give locally the shortest paths, not necessarily globally. AxelBoldt 19:26 Oct 28, 2002 (UTC)
It's been a (somewhat) long time since I studied that, but aren't geodesics straight with respect to the local curvature in some sense ? That should be added to the article. --FvdP
Article said:
Moved from article:
-- Jerzy 17:34, 2004 Jan 26 (UTC)
This curvature is in turn determined by the Stress–energy–momentum tensor; this is the content of the Einstein equation.Shmuel (Seymour J.) Metz Username:Chatul ( talk) 19:53, 6 September 2020 (UTC)
I removed "In mathematics, a geodesic is a curve which is everywhere locally straight." I belive it can not be usefull, it pretends to be little mathematical, but does not have more sense than the old formulation, contrary "Geodesic stay for the curves which are "straight" in a sense." at least explaign the meaning of the word and no mathwords used.
Tosha 22:39, 22 Feb 2004 (UTC)
I reverted that, because it makes absolutely no sense as a statement in the English language. Anthony DiPierro 22:54, 22 Feb 2004 (UTC)
I do not know English, its true. Ok, is it better now?
Tosha 23:18, 22 Feb 2004 (UTC)
Not much. What does "stay for the curves" mean? I'll let someone else, who knows more about geodesics than I do, fix it. Anthony DiPierro 23:32, 22 Feb 2004 (UTC)
Ok, I hope now everybody happy(?) I wanted to get rid from word locally here (see above)
Tosha 01:24, 23 Feb 2004 (UTC)
What you do is much worse, if you want to grab an idea of geodesic then it is "straight in some sense" to do math you should define what is the "sense" and infect there are many different senses for this on the same space depending on structure you have/choose, now you have this strange curvature ... what does it mean for metric space for instance...
I will not change it back, I'm tired (hope someone will do it)
Tosha 03:13, 23 Feb 2004 (UTC)
There is no meaning for geodesic curvature in metric space, but even if you make one the curves with zero geod.curvature might not be geodesic.
Tosha 03:22, 23 Feb 2004 (UTC)
BTW geodesic curvature is nearly defined in the article, so one could just add one line in subsection "Riemannian and pseudo-Riemannian manifolds" instead of giving ref.
Tosha 04:22, 23 Feb 2004 (UTC)
Tosha, i can't tell whether you or Decumanus is the more authoritative editor for this article, but i'd urge you to work out the wording you have in mind on this page.
For instance, after staring for a while at the first paragraph of your first edit of Geodesic, which reads
, i begin to suspect that you intended the meaning of that sentence to be close to
In proposing that reading, i note that the concepts of "stay" and "stand" are related (in fact, the probably come from the same ancient word) even tho a native speaker would never consider using "stay" to cover the metaphorical sense of "stand" that occurs in my suggested interpretation.
(And to me, that sounds like a good introductory "motivating the idea" approach for an opening paragraph, and one that is consistent with shifting to a much more rigorous approach, such as Decumanus seems to be pushing for, in discussing the specific meaning of geodesic in the various contexts.
(By the way, i also note that your use of "in a sense", which has caused some objections here, is not something you suddenly added to the article, but rather a variation or elaboration of
which was introduced by User:AxelBoldt in an edit of 18:55, 2002 Oct 28, as far as i so far notice, remaaining in the article without objection for over a year. And part of my point to you is that fact does not seem to have come out here. IMO you're going to have to work hard not just at the technical content, but also both in communicating it clearly in this weird language of English (that you so bravely have undertaken to learn to an impressive degree), and finding out why other editors are being so seemingly stupid in not following your reasoning.)
You may have things to add to the article that no one else on WP is prepared to contribute, but right now, we can't tell whether or not that's the case.
I don't want to try and address the question of whether Decumanus has tried as hard as they should to understand what you're saying, but IMO there will be no hope of finding that out without your taking time to make sure that the ideas you are bringing forward are clearly understood. IMO, that will take a lot of patience on your part to help Decumanus and others understand your meanings. -- Jerzy 05:24, 2004 Feb 23 (UTC)
I've done a copy edit here. I'm with Tosha on this - he's a valuable contributor in this field, and I believe he knows exactly what he is talking about. I have similarly copy edited other pages of his.
On the geodesic curvature matter; the definition now standing on that page needs work; for one thing it isn't obvious to me that it is compatible with the link from the Gauss-Bonnet theorem page (though it may be in fact). I think that point could be addressed there. For the time being, I felt linking geodesic to geodesic curvature wasn't clarifying, and I took out the link.
Charles Matthews 08:53, 23 Feb 2004 (UTC)
Many things come to my mind when I think of geodesics, and "straight" is definitely not one of them. As discussed above, "straight" could be taken to depend on context, but the most intuitive notion of "straight" to me comes from lines in Euclidean space. These are in direct contrast with the mental image of a 2-sphere embedded in Euclidean 3-space in the natural way with spherical geodesics drawn upon it. Granted, these lines on the sphere only appear to be curved as an artifact of the Euclidean viewpoint, but I would argue that this is the most intuitive viewpoint for the reader new to the material. What do you all think about this? -
Gauge 03:18, 15 Oct 2004 (UTC)
I removed 4th Hilbert problem (link to Hilbert Problems: http://aleph0.clarku.edu/~djoyce/hilbert/problems.html) from this page, it is clearly relevant, but not at all the first problem which should be mentioned here, also the note after that is not quite correct, it was solved at least in dimension 2.
Tosha 17:04, 8 Jun 2004 (UTC)
The article applies "geodesic" only to lines. It can be also be applied to higher-dimensional submanifolds. I could write something on this, but I'm not sure whether there's a difference between geodesic surfaces and totally geodesic surfaces (see e.g. [1], p.6). Does anyone know? Fpahl 15:48, 14 Sep 2004 (UTC)
The second sentence under " For metric spaces" begins with "More precisely". It seems to me that the precise definition is actually something new, and that it is a non-trivial statement that a curve satisfying that definition "is everywhere locally a distance minimizer". Also, the definition seems to be too narrow in that it requires the curve parameter to be proportional to the arc length; wouldn't any reparametrization of such a geodesic still count as a geodesic? Fpahl 15:56, 14 Sep 2004 (UTC)
Linas, you make the statement: "masses follow timelike geodesics, period. no ifs-ands or buts". It is my understanding that only the world lines of free particles are geodesics of spacetime. Thus, here on the surface of the Earth, I feel the 'weight' of acceleration and my world line(s) is certainly not a geodesic. Alfred Centauri 14:27, 6 Jun 2005 (UTC)
Suggestions/comments ? Mpatel 15:58, 6 Jun 2005 (UTC)
Domain is not introdcued before it is used in the first definition. It should be introduced to make the article self-contained and accessible to people who know mathematics but do not use geometry daily. Wazow 10:54, 11 August 2005 (UTC)
This statement is incorrect in the preamble. Though this is often used to characterize them, they are not inherently defined this way. Perhaps we need to write an explanation of variation of arclength and the Jacobi equation. Cypa 04:27, 28 September 2005 (UTC)
I just re-wrote the section that held the geodesic eqn. I'm hoping this makes it easier to understand. The fear-inducing mention of parallel transport has been banished to a later section. linas 03:51, 30 September 2005 (UTC)
It would be nice to have, in this article, or in a related one,
Further Edification:
Exotic geometry:
Mention of whiz-bang uses and applications:
linas 04:45, 30 September 2005 (UTC)
I did not watch this article for quite a while, now it has many problems: bit wrong intro is just a little thing, main thing is completely wrong section Riemannian geometry and the geodesic equation.
On intro: the curent intro is has partly wrong def, it is not sutabale for all cases, in particular it is very wrong for pseudo-riemanniann manifold (and for general gravity) as well as for spaces with connection. It was a long discussion before we converge to the one I like and I think we have to keep it.
Did you read what I wrote? Tosha 03:22, 21 October 2005 (UTC)
Main problem: it does not work for pseudo-riemanniann manifold (and for general gravity) as well as for spaces with connection. Note that I did not change it so far, but I will do it later Tosha 15:30, 21 October 2005 (UTC)
On Riemannian geometry and the geodesic equation One can not define it trough lenght, at least yoyu should ask for constant speed and it does not work for pseudorieamannian case.
I do not blame you, I'm just correcting mistake Tosha 03:22, 21 October 2005 (UTC)
It is explained above, if you do not see the mistake, you do not know what is geodesic, why then you edit this article? Tosha 15:30, 21 October 2005 (UTC)
In addition, the geodesic flow section can be removed almost completely, and refer to Hamiltonian mechanics
So why not leave all technical stuff there there is no need to repeat it here Tosha
the Hamiltonian can be mentioned in def of geodesic (I'm not against anything which is correct) note that there is no technical detales about lagrangian as well. There is reason to do this def for Riemannian geometry and most books do it this way (and it is not becouse they are stupid, main thing: it is more usefull). Also, as it is stated in the end metric def coinsided with Riemaniann
This section had many things not really related to the topic now at least it is clear what geodesic flow is Tosha 15:30, 21 October 2005 (UTC)
So I revert this secton, and thin a bit about intro Tosha 01:16, 20 October 2005 (UTC)
Not this way, the article had serious problems, see above, so before reverting you should describe the reason for making it wrong! Tosha 03:22, 21 October 2005 (UTC)
Just read carefully what I wrote, I will not repeat it again! Tosha 15:30, 21 October 2005 (UTC)
More over in my edit it is not even in def there is only comment about lagrange and we can add one about Hamilton. Tosha 15:44, 21 October 2005 (UTC)
There is metric def, so why should we repeat it for Riem case? the given def is what people use in Riem. Geom, it is used in most of books, clearly metric def gives a motivation. And sinse it works just as well for pseudo-Riemannian case why not top leave it this way?
Ok, I will not revert it my-self, I think my edit is much better, and curent is wrong. Hope someone will do it for me Tosha 15:41, 21 October 2005 (UTC)
My feeling is there are some problems with arranging material in this article, some of which I corrected plus the section on (pseudo-)Riemannian geometry is almost completely wrong. Sinse I promised to not revert I will not, but everybody else is wellcome to compare and revert Tosha 22:06, 21 October 2005 (UTC)
Tosha, when you say the section on (pseudo-)Riemannian geometry is almost completely wrong, do you mean the section called Riemannian geometry and the geodesic equation? If we remove the word "pseudo" from that section, does it become almost completely right?
Overall, I'd like the article to start with something as simple as possible, and move to more correct, more general definitions. This would include a discussion of why the simple definition fails, e.g. null geodesics, lack of existance proofs.
Here's a proposal I'd like to make:
Would that work? linas 00:42, 22 October 2005 (UTC)
I think at the moment, the best we can do is revert my edit. Tosha 01:26, 22 October 2005 (UTC)
Luna, with all respect, I do not want to keep a wrong article, even if it is really easy to read. BTW, if you noticed I did make some changes, which you were mentioned in the above discussion. Tosha 02:26, 25 October 2005 (UTC)
a curve c(t)=(t^3,0) in the plane minimize length but is no a geodesic, isn't it? you can correct def, by making it const.spped but then it will repeat def in metric geometry part. Also the statements about ways to solve it were completely wrong... Tosha 22:58, 25 October 2005 (UTC)
My two cents as an undergraduate maths student: please forgive me if I say anything stupid (because I admit I know little about geodesics compared to all the experts around). With geodesics I learnt them the "parallel transport" way first and the "variational" way afterwards, and yet I know plenty of classmates who learnt them the other way round, or one and not the other, even though we are doing the same degree at the same time. So I think having separate sections that describe them one way independently of the other way would be great, and then a section can follow on how they are related to each other. I was looking up "geodesic" here myself hoping to understand more the connection between the two approaches that I've learnt. -- KittySaturn 00:03, 23 October 2005 (UTC)
Given a ( pseudo-) Riemannian manifold M, a geodesic may be defined as a solution to a differential equation, the geodesic equation. The geodesic equation may be derived, using variational principles, as the Euler-Lagrange equations of motion applied to the length of a curve.
wrong you can not use length
that maps an interval I of the real number line to the manifold M, one writes the length of the curve as the integral
where is the tangent vector to the curve at point . Here, is the metric tensor on the manifold M.
Using the length given above as if it were an action, and solving for the Euler-Lagrange equations, one gets the geodesic equation.
wrong again it is not what you will get.
In terms of the local coordinates on M, this is
Here, the xa(t) are the coordinates of the curve γ(t) and are the Christoffel symbols. Repeated indices imply the use of the summation convention.
Rather than working with the length, one may obtain exactly the same equations by minimizing the energy functional
In physics, this energy functional is called the action, although this term is not commonly used in mathematics. Working with the action instead of the length has the minor advantage of avoiding a pesky square-root and the major advantage of using the same conceptual language used in other important areas of geometry and topology.
foggy, but wrong again
The geodesic is a second order ordinary differential equation. Equivalently, geodesics may be defined to be the solutions to a pair of first order differential equations; these equations, the Hamilton-Jacobi equations, are obtained by working with the Hamiltonian formulation instead of the Lagrangian formulation, and are given in the next section.
I changed
to
which I believe to be a correct. But as I am no expert of on Riemannian manifolds, I figured I would alert others to the change.
Jrohrs 18:04, 1 August 2006 (UTC)
Tosha, you also removed the following section:
This is a strage statement, it does not define exp.map, or at least give very non-complete def.
The Hopf-Rinow theorem states, among other things, that any two points on a Riemannian manifold are joined by a geodesic.
I guess one should add "complete"
Well if not Riemannian then geodesics might be not defined, which spaces are ment here? pseudo-Riemanian? Finsler? metric spaces? what is the meaning of non-uniqueness and so on, it is just too foggy...
I am guessing that you removed this because the article failed to define the meaning of the exponential map in the context of a Riemann surface? linas 13:13, 26 October 2005 (UTC)
(I thought a section on completeness would be interesting, because some systems aren't, e.g. constrained mechanical systems, such as robot arms, aren't geodesically complete. Ditto for GR.) linas 15:11, 26 October 2005 (UTC)
I think it enough to mention Hopf-Rinow theorem in see also, or at most give a def og geodesically complete space here and refer to Hopf-Rinow. Tosha 18:57, 26 October 2005 (UTC)
There should be a section (or at least a subsection) on the geodesic equation, as this is certainly something that a general reader should see stand out. Looking at the table of contents, the concepts mentioned are important, but geodesic equation should be there too. MP (talk) 09:39, 19 February 2006 (UTC)
There should be a derivation of the geodesic equation from the action since it is pretty important...if no one objects ill add a section on it. — Preceding unsigned comment added by 207.161.187.38 ( talk) 00:41, 15 August 2014 (UTC)
While at first glance it sounds hokey, it is a legitimate word: See Google search. While it may not be widely used, I think it is worth mentioning——I'm not attempting to hijack this article with the term (though, generally, in articles talking about "geodesy" and "geodetic" concepts not specifically referring to Earth, I think "planeto-" should be used instead of "ge-"), I'm just pointing out the term's origin and more generic form. ~Kaimbridge~ 19:22, 30 July 2006 (UTC)
"Planetodesic" yields zero books.google.com hits, and the only google hits are to this article and this talk page. So I went ahead and removed it. If someone wants to wedge the term "planetodesy" back in I suppose that's okay, although (a) it probably doesn't belong in the lead and (b) it's so obscure it's probably not necessary at all. — Steve Summit ( talk) 13:10, 7 February 2007 (UTC)
In a general space, there is a difference between an autoparallel curve (which parallel transports its tangent vector) and a geodesic (which is defined as a curve which extremizes length, regardless of which type of space you're working in). In the specific case of a (pseudo-)Riemannian manifold with torsion-free and metric-compatible connection, the difference vanishes. I've tried to establish this distinction more clearly. For a nice treatment, I recommend the first dozen pages of Ortin's "Gravity and Strings". Note that Wald assumes torsion-free and metric-compatible very early on, so he never separates the two cases. -- MOBle 21:26, 1 August 2006 (UTC)
Hello,
I posted this in another page, so this is just a repost.
There a few things in the definition of geodesic provied I do not quite agree with. Firstly, when one talks about points, I think points are "in" a space, not on it, as this assumes some kind of ambient space. We live in R^3, not on it.
I also think that one cannot support the statement that geodesic are defined as the shortest paths. Locally or not, because of reparametrisation, for example. I like the definition given in "John M. Lee. Riemannian Manifolds: An Introduction to Curvature. Number 176 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. " where geodesic are simply the curves whose acceleration is zero (well, the covariant derivative of the velocity). Geodesics are, locally, the shortest paths, but the opposite is not true. From the point of view of accelaration, it is natural then to see that the lenght of the velocity vector of a geodesic is constant.
Regards
Krzysztof Krakowski
129.180.1.224
02:40, 19 April 2007 (UTC)
In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space.
Uh, or the longest? "Extremal" does not mean "shortest". -- 76.224.88.42 ( talk) 21:03, 20 December 2007 (UTC)
"Geodesic" (adj) in "geodesic grid" refers to geodesy (n), the shape of the Earth. The points in such a grid are geodetic (adj): they lie on the surface of the Earth; unfortunately -- for it makes for confusion -- it is also common to say they are geodesic (adj) points. The "geodesic" in "geodesic grid" does not refer to geodesics (n), the subject of this article. (And when I said "inflection", I wasn't referring to the mathematical concept, but the grammatical one.) It would've helped to follow the link to the geodesic grid article to see what it was about.
Silly rabbit, indeed. :P 128.83.68.169 02:21, 12 July 2007 (UTC)
I find this article to be too focussed on mathematics. If you are mentioning "metrics" and the "affine connection" in the first paragraph you are already in trouble. While I can appreciate Riemannian geometry and so forth, I remind you all that "geodesic" has "geo" for "earth" in front of it, hence remind you of its origins. That is, in geodesy. I suggest two changes to the article: (1) that the introduction be reordered so that the simpler "path over the earth" definition/discusson is given as the first paragraph, followed by the ramp up to the mathematics (you are scaring people away now), and (2) the inclusion of a new section, one coming right after the introduction, on earth geodesics. The new section would talk about such things as geodesics on a sphere, and spheroids such as WGS84, with some simple examples, perhaps even a figure... In short...you math people have to back off! :) The article might even benefit from a split into two articles, one being the simple, earth-centric discussion, the other being the more complex math-centric discussion.
(Consider the poor high-schooler writing his report the night before it is due who just wants to know what geodesic means...and he goes to his favorite reference wikipedia, only to find this article...Yikes!)
As an example of a figure, see: http://909ers.apl.washington.edu/~dushaw/perth_bermuda2.jpg The upper path is the geodesic on a sphere, the lower path is a geodesic on the wgs84 ellipsoid. I am the author of the figure and could release it to wikipedia if you thought it useful. The path goes from Perth, Australia to Bermuda. Bdushaw 21:35, 6 October 2007 (UTC)
"in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle." Strictly speaking, this isn't quite right. The Earth is not a perfect sphere (it's an oblate spheroid or ellipsoid - i.e slightly squashed at the poles), which means that the shortest distance between two points (a geodesic) is not quite a great circle. But I'm not sure how best to explain this in the intro without over-complicating it. Wardog ( talk) 11:15, 21 February 2008 (UTC)
I don't really see why there should be a separate geodesic (general relativity) article. The existence of two articles suggest that they are different concepts, while they are exactly the same concept. ( TimothyRias ( talk) 15:25, 17 July 2008 (UTC))
Well, OK, there clearly is no consensus for such a merger so I'll remove the merge template. I do think this is a shame though. Much of the opposition to the merger seems to be based on the misconception that there are distinct concepts of geodesic in mathematics and physics, while in fact there is only one such concept, the one in geometry (strictly speaking you could say there are two such concepts which coincide for the Levi-Civita connection, but both are relevant to GR). Pretty much everything that is currently said in the geodesic article is relevant to GR in someway, and similarly the geodesic article needs a good applications section discussing the role of geodesics in applications such as GR. But, instead with have some territorial pissings from the mathematics and physics camp, that lead to the propagation of the illusion that there are two distinct concepts here. Such a shame. TimothyRias ( talk) 15:26, 5 December 2010 (UTC)
should be as the Gamma symbols are not tensors. Therefore there is no need to stagger the indices: they can't be lowered anyway. If everyone agrees I will delete the confusing space before mu. Tkuvho ( talk) 13:03, 24 February 2010 (UTC)
The section on Riemannian geometry was recently changed rather dramatically, and I'm not certain that I agree with the edit summary there:
For a piecewise curve (more generally, a curve), the Cauchy-Schwarz inequality gives
with equality if and only if is equal to a constant a.e. It happens that minimizers of also minimize , because they turn out to be affinely parameterized, and the inequality is an equality.
The usefulness of this approach is that the problem of seeking minimizers of E is a more robust variational problem. Indeed, E is a "convex function" of , so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers within isotopy classes. In contrast, "minimizers" of the functional are generally not very regular, because arbitrary reparameterizations are allowed. (One could probably also argue that the variational problem for is, by itself, rather ill-posed.)
Sławomir
Biały
15:28, 23 January 2016 (UTC)
Yes, I've heard that the vast majority of possible spacetimes are pathological. But isn't it true that for any spacetime actually used in general relativity the distance between two points and on a curve is Jrheller1 ( talk) 01:00, 24 January 2016 (UTC)
This article should be primarily oriented towards explaining geodesics on surfaces in 3-dimensional space, because that is what 99.99% of useful geodesics are. The other 0.01% of useful geodesics are geodesics in general relativity, and 99.9% of useful geodesics in GR are geodesics of the Schwarzschild metric. Don't you agree that the derivation of the geodesic equation I posted (which is very similar to Kreyszig's derivation) is valid for the Schwarzschild metric? If there are other useful spacetimes for which this derivation is not valid, they should be dealt with in the "Geodesics in general relativity" page. The derivation of the geodesic equation on the "Geodesics in general relativity" page is just a less clear and concise version of Kreyszig's derivation. It certainly does not address any of the issues you (Slawekb) raised in your previous post. Jrheller1 ( talk) 21:39, 24 January 2016 (UTC)
A parameterization of a curve in the (x,y) plane of the form (t,y(t)) can represent any possible curve for which there is only one value for a given value. What this means is you can draw any piecewise smooth curve in the xy-plane that has only one value for a given value and find a parameterization for it of the form (t,y(t)) or equivalently (x,y(x)).
There is no need to use the more general form (x(t),y(t)). This will only produce a more complicated ODE (a system of two second order ODEs) with the same result: minimizing the arc length integral will produce the right solution (straight lines through the origin) and minimizing the "energy" integral will produce the wrong solution. You can use an ODE solver with initial conditions y(0)=0 and y'(0)=a for some constant for the "energy" ODE above and see for yourself that it produces a curve that deviates more and more from the right answer as the curve approaches the hemisphere boundary.
A computationally simpler example of the result of applying the Euler-Lagrange equation to both and is for the minimal surface problem. The minimal surface problem is to find the surface with minimum area for given boundary conditions. To do this it is necessary to minimize the surface area integral The E-L equation (function of multiple variables version) applied to the surface area integral results in the PDE (in other words, mean curvature is zero everywhere). The E-L equation applied to produces the Laplace equation This is the wrong answer. The Laplace equation is only an approximation to the minimal surface equation for height field boundaries with relatively slow variations in z. This is just like the solution to the "energy" ODE from my last post. It is a fairly good approximation to the geodesic close to the origin (where z is varying slowly) but gets worse farther away. Jrheller1 ( talk) 04:56, 26 January 2016 (UTC)
Obviously a lot of technically skilled people have edited this article. But that seems to have blinded them to the obvious. For the majority of readers coming to this article all they are interested is great the circle idea of a geodesic. They just need a simple sentence to inform or confirm their notion of what a geodesic is.
The don't need or want a topic sentence like this : "...In the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are (locally) the shortest path between points in the space....". This sentence contains a raft of references to concepts and words that a general reader will be unfamiliar with. A general reader is likely to either skip the article or start on a trip through the internet trying to figure out what the heck this sentence means.
What the general reader is probably interested in is contained nicely in this paragraph: "The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph."
It seems to this reader that the problems of the lead section could be fixed easily by just reversing the order of the two paragraphs. The technical reader can easily skip past the opening section that he probably is well aware of and begin to try to understand the more technical uses of the word in mathematics and physics with the second paragraph of the lead section serving as a summary of the technical information to follow. — Preceding unsigned comment added by Davefoc ( talk • contribs) 05:47, 15 April 2016 (UTC)
JRSpriggs made the edit suggested above. Thank you. Davefoc ( talk) 05:01, 20 September 2016 (UTC)
Across, much of the article "t" is used as the affine parameter along a geodesic curve. This is a rather unfortunate choice as to many readers it will suggest a relation with time, where no such connection need to exist (or even make sense in the case or Riemannian geometry). I would suggest changing it to something more "neutral" such as λ. T R 11:26, 29 September 2016 (UTC)
The current short description is misleading in several regards. First, in a lorentzian manifold a geodesic maximizes the interval rather than minimizing it. Second, in a manifold with a connection but no metric, geodesics are defined but path length is not. Can anybody come up with wording that is more accurate but still concise? Shmuel (Seymour J.) Metz Username:Chatul ( talk) 23:01, 4 September 2020 (UTC)
The article has the short title Shortest path on a curved surface or a Riemannian manifold
. That is correct for a manifold with a
positive definite metric, but it is incorrect for a manifold with a
Lorentzian metric. I've considered changing it to "Extremal path", but that's not quite right either. Your thoughts?
Shmuel (Seymour J.) Metz Username:Chatul (
talk)
01:10, 28 January 2021 (UTC)
{{
about|geodesics in general|geodesics in general relativity|Geodesic (general relativity)|Geodesey|Geodesy|other uses}}
, remove "and pseudo-Riemannian manifolds". Shmuel (Seymour J.) Metz Username:Chatul ( talk) 21:33, 28 January 2021 (UTC)
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