Talk:Geodesic/Archive 2

Archive 1

Archive 2

Minimizing energy versus length

The section on Riemannian geometry was recently changed rather dramatically, and I'm not certain that I agree with the edit summary there:

• There was a critical error in the "Riem. geom." section. Minimizing the integral of a squared function only provides an upper bound on the sqrd integral, it does not minimize the sqrd integral

For a piecewise $C^{1}$ curve (more generally, a $W^{1,2}$ curve), the Cauchy-Schwarz inequality gives

$L(\gamma )^{2}\leq 2(b-a)E(\gamma )$

with equality if and only if $g(\gamma ',\gamma ')$ is equal to a constant a.e. It happens that minimizers of $E(\gamma )$ also minimize $L(\gamma )$, 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 $\gamma$, 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 $L(\gamma )$ are generally not very regular, because arbitrary reparameterizations are allowed. (One could probably also argue that the variational problem for $L(\gamma )$ is, by itself, rather ill-posed.) Sławomir
Biały 15:28, 23 January 2016 (UTC)

I find the new section much harder to read than the old one. I've reverted to the previous version, but I've also added your explanation above to the article. Hopefully this answers any concerns about correctness. Ozob (talk) 17:53, 23 January 2016 (UTC)

There is a derivation of the geodesic equation from the Riemannian (with positive definite metric) arc length integral in the "Geodesics in general relativity" article. The geodesic equation is a differential geometry topic, so its derivation belongs here, not in the "Geodesics in general relativity" page. Plus the derivation I posted here is much clearer than the one in the general relativity page. I think it's very important to clarify that the only reason the integrand can be squared is because the curve parameter is arc length. With any of the infinite number of possible parameterizations by something other than a constant multiple of arc length, the geodesic equation is not valid. Jrheller1 (talk) 19:56, 23 January 2016 (UTC)

The analysis of Lorentzian manifolds is completely different than in Riemannian signature. There the problem of finding "length minimimizing" paths really is ill-posed as a variational problem. Sławomir
Biały 20:41, 23 January 2016 (UTC)

When using (+,-,-,-) sign convention, the distance between two points is still $\textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {g_{\alpha \beta }(u^{\alpha })'(u^{\beta })'}}}\operatorname {d} \!\tau$ (the number under the square root is always non-negative). When using (-,+,+,+) sign convention, the distance is $\textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {-g_{\alpha \beta }(u^{\alpha })'(u^{\beta })'}}}\operatorname {d} \!\tau$ (the number under the square root is again always non-negative). So the derivation of the geodesic equation I posted is valid for Lorentzian manifolds. Jrheller1 (talk) 23:18, 23 January 2016 (UTC)

The length functional is very badly non-convex in Lorentzian manifolds. Generally, there are no shortest paths at all in a homotopy class. For example, with the standard Minkowski metric on $\mathbb {R} ^{4}$, the standard helix defined by $(t,\cos t,\sin t,0)$ is a curve joining the two timelike separated points $(0,1,0)$ and $(2\pi ,1,0)$, but whose tangent vector is everywhere null (so it has "length" zero). Helices like this are $C^{0}$ dense in the space of all timelike curves. Sławomir
Biały 00:18, 24 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 $\gamma (t_{1})$ and $\gamma (t_{2})$ on a curve $\gamma$ is $\textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}}}\operatorname {d} \!\tau ?$ Jrheller1 (talk) 01:00, 24 January 2016 (UTC)

That's the length of the curve. But it's not the distance between the points. The distance between the points is the length of the shortest geodesic between them (in Riemannian signature). However, on Lorentzian manifolds, unlike Riemannian manifolds, the geodesic equations are not determined variationally. In physics, one still pretends that they are, and formally computes Euler-Lagrange equations for a functional with no minima. But ultimately, it's the Euler-Lagrange equations themselves that define the geodesics. One can still get a variational characterization of geodesics on Lorentzian manifolds, but it is more involved. The basic fact in this area is that timelike geodesics (in signature (+---)) maximize the proper time among causal curves. Sławomir
Biały 12:52, 24 January 2016 (UTC)

I meant the distance the object has traveled moving between the two points on the curve. The minimum over all possible curves $\gamma$ of $\textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}}}\operatorname {d} \!\tau$ definitely exists (at least in a reasonable spacetime such as the Schwarzschild metric) and it is exactly the same as the minimum of $\textstyle {\int _{\tau _{1}}^{\tau _{2}}\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}\operatorname {d} \!\tau .$ The only reason squaring the integrand is valid is because of parameterization by arc length. If you apply the Euler-Lagrange equations to $\textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}}}\operatorname {d} \!\tau ,$ you get exactly the same result as when you apply it to $\textstyle {\int _{\tau _{1}}^{\tau _{2}}\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}\operatorname {d} \!\tau .$ The math is just messier. In Kreyszig's derivation of the geodesic equation, he did actually apply the E-L equations to the functional with the square root integrand and then later switched to arc length parameterization (which eliminates a few terms in the computation). Jrheller1 (talk) 18:55, 24 January 2016 (UTC)

This is more than a case of the math being messier. The least proper time of a causal curve between two timelike separated points is actually equal to zero. (Imagine connecting them by a timelike curve and taking a tight coil of null rays around that curve - see optical black hole.) What you're after is curves that maximize the length, among timelike curves. These may not even exist. For example, consider a space-time that just has one point deleted. Such a manifold is "geodesically incomplete". Even assuming that some version of "completeness" holds (or that we could formulate completeness without knowing in advance what the geodesics are, as in the Hopf-Rinow theorem), existence of timelike geodesics between two timelike separated points seems hard to me. Roughly, one should use a heat flow argument, with the quadratic energy $E(\gamma )$, starting with some initial timelike curve and evolving in an isotopy class. It seems like this sort of argument would at least work with compact manifolds with boundary, and then extended to complete manifolds by a suitable truncation argument. But there's a lot in the details, and it's definitely going to be extremely non-trivial. You can find somewhat of a discussion of these issues in Roger Penrose's "Techniques of differential topology in general relativity". What he does there is to take the timelike geodesics as given (via a Hamiltonian approach), and then considers rectifiability using the resulting ambient paths. There is nothing wrong with that approach, but it's somewhat unsatisfactory as a variational characterization of timelike geodesics because one is already assuming what they are, locally. Sławomir
Biały 20:55, 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)

You're the one that brought up Lorentzian space-times. The analysis of geodesics is completely different in Lorentzian space-times. For geodesics on surfaces, it is a theorem that the length functional is dominated by the energy functional, which is more well-behaved analytically. The "space of paths" between two points is a manifold in an appropriate sense, and the energy is a Morse function on the manifold that can be minimized by taking an appropriate flow (see, e.g., [1]). In contrast, the length functional is not a well-behaved Morse function. So, yes, a variational characterization of geodesics even on "simple" things like compact surfaces uses the energy rather than arc length. Sławomir
Biały 22:24, 24 January 2016 (UTC)

When you parameterize by arc length, minimizing arc length of a curve on a surface or manifold via the Euler-Lagrange equation produces exactly the same result as minimizing the "energy" of the curve. So when you parameterize by arc length, both ways are exactly equivalent. When you parameterize a curve by something other than arc length, minimizing the "energy" will give you the wrong answer, minimizing the arc length will give you the right answer. So in the context of geodesics, this concept of "energy" is purely a device for simplifying mathematical calculations when parameterizing by arc length. Jrheller1 (talk) 23:14, 24 January 2016 (UTC)

'When you parameterize a curve by something other than arc length, minimizing the "energy" will give you the wrong answer, minimizing the arc length will give you the right answer.' This is not true. The critical points of the energy functional on the space of continuous, piecewise $C^{1}$ paths $\gamma :[0,1]\to M$ with fixed endpoints $\gamma (0)=p$ and $\gamma (1)=q$ are exactly the affinely parameterized geodesics. (These also turn out to be critical points of the functional L, but that fact is not actually useful, because L is very degenerate and it's pretty easy to cook up ridiculous "critical points" of L (e.g., a cuspidal curve, where the variational derivative of L doesn't even exist!)) One still has to say in what sense the resulting geodesic is "minimizing". At worst, that analysis is the same whether length or energy is used. But because the energy is much "smoother" than the length functional, it's a more robust tool. (There are, e.g., homotopy arguments, Morse theory, heat flow arguments, as well as the Hopf-Rinow type arguments.) Indeed, existence of minimizing geodesics was an early triumph of energy methods in the calculus of variations (see, for example, Lyusternik and Fet (1951) "Variational problems on closed manifolds" Dolklady Akad. Nauk SSSR 81:17-18.). Sławomir
Biały 00:39, 25 January 2016 (UTC)

In the simplest possible example of parameterizing a curve on a surface by something other than arc length (a curve $(x(t)=t,y(t))$ on the upper hemisphere $z={\sqrt {1-x^{2}-y^{2}}}$), minimizing the arc length integral gives the right answer: a straight line $(t,kt)$ in the xy-plane for some constant $k,$ $(t,kt,z(t,kt))$ is a great circle. Minimizing the energy integral gives the wrong answer: it starts out very close to the straight line $(t,kt)$ but as the ODE solution curve gets closer to the hemisphere boundary seriously diverges from the straight line.

The coefficients of the first fundamental form of the upper hemisphere $z={\sqrt {1-x^{2}-y^{2}}}$ are $g_{11}=1+x^{2}/z^{2},$ $g_{12}=xy/z^{2},$ and $g_{22}=1+y^{2}/z^{2}.$ The coordinates of the curve are $u^{1}(t)=t$ and $u^{2}(t)=y(t)$ so the derivatives of the coordinates are $(u^{1})'=1$ and $(u^{2})'=y'(t).$ So $g_{\alpha \beta }(u^{\alpha })'(u^{\beta })'$ is $(1+x^{2}/z^{2})+(2xy/z^{2})y'+(1+y^{2}/z^{2})(y')^{2}.$ Replacing x with t and $z^{2}$ with $1-t^{2}-y^{2}$ yields $F(t,y,y')=(1+t^{2}/(1-t^{2}-y^{2}))+2ty/(1-t^{2}-y^{2})y'+(1+y^{2}/(1-t^{2}-y^{2})(y')^{2}.$ This is is the integrand of the "energy" functional and ${\sqrt {F}}$ is the integrand of the arc length functional. Applying Euler-Lagrange equation to the "energy" functional yields $(t^{2}-1)(y^{2}+t^{2}-1)y''-(t^{2}-1)y(y')^{2}+2ty^{2}y'-y(y^{2}-1)=0.$ Applying E-L to the arc length functional yields $(y^{2}+t^{2}-1)y''-t(t^{2}-1)(y')^{3}+(3t^{2}-1)y(y')^{2}-t(3y^{2}-1)y'+y(y^{2}-1)=0.$ It is easy to show that $y=kt$ solves the second ODE (for the arc length functional) but not the first (for the "energy" functional).

You seem to think that a curve parameterized by arc length is something special. This is wrong, any piecewise smooth curve can be parameterized by arc length (or infinitely many other parameterizations). Jrheller1 (talk) 07:27, 25 January 2016 (UTC)

Yes, affine parameterization is special. Arc length parameterization is a special case. All you've shown here is that the critical points of the energy functional on the space of curves of a special form are not geodesics (with affine parameter), which isn't very surprising. You need to write down the EL equations for $(x(t),y(t))$. That will then give you the geodesic equations. And regardless, you still haven't said exactly what is being "minimized" by this procedure. You've said that this "minimizes the arclength", but haven't specified a domain. Is the domain homotopy classes of C^1 paths with fixed endpoints? Is it the free loop space (for closed geodesics)? Once you have specified the domain, you have to say in what sense a "minimum" is achieved. Is the second variation formula valid for the arclength functional, and the domain you have selected? Can you conclude that you have a global minimum in your class of functions? A local minimum? Is there a compactness result guaranteeing global minima once you have specified the parameters of the problem? None of this is addressed. Sławomir
Biały 12:48, 25 January 2016 (UTC)

I think it's manifestly true that minimizing the energy integral gives the right answer, but to the wrong question. The question you posed is: On the upper hemisphere, among curves of the form $(t,y(t))$, which has the least energy? But this isn't a physical setup for arc length minimization! You've constrained the first coordinate not just to move left-to-right, but to do so at a particular rate. Forcing a particle to move at a certain velocity in the x-direction acts like a driving force in the y-direction because of the curvature of the surface. That's sure to change its behavior, and your calculation confirms that. I suspect that if you built a mechanical setup to simulate this situation (say, a ball on a hemisphere with some kind of attachment that was free in one direction and rigid in the other direction, and the rigid direction was forced to obey $x(t)=t$), you would observe precisely the path you derived. Ozob (talk) 13:56, 25 January 2016 (UTC)

I would like to disagree with one of your suppositions. Perhaps it's true that 99.99% of all useful closed geodesics are on surfaces embedded in R³. Perhaps it's even true that most of the other useful geodesics are in GR, and that most of those are geodesics of the Schwarzschild metric, though I strongly doubt this. (Geodesics are useful anytime you model something by a manifold, and lots of things are modeled by manifolds.) That does not mean that this article should cater solely to geodesics on surfaces embedded in R³ and geodesics in GR. The article is not titled geodesics on surfaces or geodesics in general relativity. It is not solely about geodesics on well-behaved surfaces or spacetimes. The article is titled geodesic. It's about all geodesics on all manifolds, and its content should reflect that. Ozob (talk) 02:24, 25 January 2016 (UTC)

I don't disagree with anything you say here. I just think that the thing the majority of readers probably need most from this article is a concise but thorough explanation of the differential geometry of curves on smooth surfaces as it relates to geodesics, like that given by Kreyszig. In just a few pages, he explains the exact equivalence of a curve satisfying the geodesic equation and a curve having vanishing geodesic curvature (meaning the curve normal vector is parallel to the surface normal vector at every point where the curvature of the curve is non-zero). Jrheller1 (talk) 07:50, 25 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 $y$ value for a given $x$ value. What this means is you can draw any piecewise smooth curve in the xy-plane that has only one $y$ value for a given $x$ 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 $a$ 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 $\textstyle {\int _{t_{1}}^{t_{2}}F\operatorname {d} \!t}$ and $\textstyle {\int _{t_{1}}^{t_{2}}F^{2}\operatorname {d} \!t}$ 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 $\textstyle {\int \int _{R}{\sqrt {1+z_{x}^{2}+z_{y}^{2}}}\operatorname {d} \!x\operatorname {d} \!y}.$ The E-L equation (function of multiple variables version) applied to the surface area integral results in the PDE $k_{1}+k_{2}=0$ (in other words, mean curvature is zero everywhere). The E-L equation applied to $\textstyle {\int \int _{R}(1+z_{x}^{2}+z_{y}^{2})\operatorname {d} \!x\operatorname {d} \!y}$ produces the Laplace equation $z_{xx}+z_{yy}=0.$ 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)

The reply for minimal surfaces is the same for the reply for your example for geodesics. One seeks critical points for the Dirichlet energy functional of immersions of a domain in R^2 into R^3. Also, you keep referring to the Euler-Lagrange equations as "solving" a minimization problem, and even claim to have "derived" the Euler-Lagrange equations, but nowhere have you actually properly stated the minimization problem that you are claiming to solve, much less prove any kind of uniqueness. Indeed, the area functional and arclength functional both lead to ill-posed variational problems (there is no uniqueness), in part because the space of functions that they are naturally defined on is very wild (W^{1,1}). Sławomir
Biały 10:39, 26 January 2016 (UTC)

Again, it is the right answer to the wrong question. The energy integral you wrote down is physically useful, but not for describing minimal surfaces. Ask yourself the question: Do I expect the graph of an electrostatic potential to be a minimal surface for its boundary conditions? Some thought should convince you that the answer ought to be no. But if you allow yourself three dimensions of freedom, (x(s,t), y(s,t), z(s,t)), then you are not looking at the graph of an electrostatic potential anymore. The physics has changed, and so should your answer. Ozob (talk) 13:42, 26 January 2016 (UTC)