What is Geodesic equation: Definition and 67 Discussions

In geometry, a geodesic () is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line" to a more general setting.
The noun "geodesic" and the adjective "geodetic" come from geodesy, the science of measuring the size and shape of Earth, while many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). 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.
In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature. More generally, 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. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.
Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.

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  1. tom.stoer

    Geodesic Equation: Generalizing for Functions F

    The geodesic equation follows from vanishing variation ##\delta S = 0## with ##S[C] = \int_C ds = \int_a^b dt \sqrt{g_{ab}\,\dot{x}^a\,\dot{x}^b}## In many cases one uses the energy functional with ##\delta E = 0## instead: ##E[C] = \int_a^b dt \, {g_{ab}\,\dot{x}^a\,\dot{x}^b}## Can...
  2. L

    Question about Geodesic Equation Derivation using Lagrangian

    I'm trying to derive the Geodesic equation, \ddot{x}^{α} + {Γ}^{α}_{βγ} \dot{x}^{β} \dot{x}^{γ} = 0. However, when I take the Lagrangian to be {L} = {g}_{γβ} \dot{x}^{γ} \dot{x}^{β}, and I'm taking \frac{\partial {L}}{\partial \dot{x}^{α}}, I don't understand why the partial derivative of...
  3. S

    Derivation of the Geodesic equation using the variational approach in Carroll

    Hello Everybody, Carroll introduces in page 106 of his book "Spacetime and Geometry" the variational method to derive the geodesic equation. I have a couple of questions regarding his derivation. First, he writes:" it makes things easier to specify the parameter to be the proper time τ...
  4. P

    Geodesic Equation from conservation of energy-momentum

    Hi everyone, While reading http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html reference I bumped into a result. Can anyone get from Eq.19.1 to Eq.19.3? I've also been struggling to get from that equation to the one before 19.4 (which isn't numbered)...anyone? Thank...
  5. A

    Alternate form of geodesic equation

    Homework Statement We're asked to show that the geodesic equation \frac{du^{a}}{dt} +\Gamma^{a}_{bc}u^{b}u^{c}=0 can be written in the form \frac{du_{a}}{dt}=\frac{1}{2}(\partial_{a}g_{cd})u^{c}u^{d} Homework Equations...
  6. E

    Geodesic Equation - Physics Explained

    http://mykomica.org/boards/shieiuping/physics/src/1335180831965.jpg http://mykomica.org/boards/shieiuping/physics/src/1335180965708.jpg
  7. R

    Geodesic Equation for Straight Lines in Euclidean Space

    Homework Statement If a general parameter ##t=f(s)## is used to parameterize a straight line in Euclidean space, then the geodesic equation takes the form ##\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=h(s)\frac{du^i}{dt}##, where...
  8. A

    Metric Connection from Geodesic Equation

    For the following two-dimensional metric ds^2 = a^2(d\theta^2 + \sin^2{\theta}d\phi^2) using the Euler-Lagrange equations reveal the following equations of motion \ddot{\phi} + 2\frac{\cos{\theta}}{\sin{\theta}}\dot{\theta}\dot{\phi} = 0 \ddot{\theta} -...
  9. Philosophaie

    The three vectors in the Null Geodesic equation

    From a metric maybe the Schwarzschild, you can find g in co and contra varient forms. From that you can calculate Affinity. My question is from the Null Geodesic equation (ds=0) what do the three contravarient vectors represent? Do they represent the path of a planet around the sun or the...
  10. L

    Derivation of geodesic equation from hamiltonian (lagrangian) equations

    Homework Statement Hello, I would like to derive geodesics equations from hamiltonian H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu} using hamiltonian equations. A similar case are lagrangian equations. With the definition L=g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu I tried to solve the...
  11. Y

    If already known the Action and unkown the Metric, how to get geodesic equation?

    If already known the form of Action and unkown the Metric, how to derive the geodesic equation?
  12. haushofer

    Geodesic equation via conserved stress tensor

    Hi, I have a question which was raised after reading the article "Derivation of the string equation of motion in general relativity" by Gürses and Gürsey. The geodesic equation for point particles can apparently be obtained as follows. First one takes the stress tensor of a point particle...
  13. P

    Geodesic equation in new coordinates question

    Homework Statement Suppose \bar{x}^{\mu} is another set of coordinates with connection components \bar{\Gamma}^{\mu}_{\alpha\beta}. Write down the geodesic equation in new coordinates. Homework Equations Using the geodesic equation: 0 = \frac{d^{2}x^{\mu}}{ds^{2}} +...
  14. A

    How Do Geodesics Behave in a 2D Metric with Signature (-,+)?

    Homework Statement Consider the 2-dim metric {{\it ds}}^{2}=-{\frac {{a}^{2}{{\it dr}}^{2}}{ \left( {r}^{2}-{a}^{2}\right) ^{2}}}+{\frac {{r}^{2}{d\theta }^{2}}{{r}^{2}-{a}^{2}}}, where r > a. What is its signature? Show that its geodesics satisfy {\frac {{a}^{2}{{\it dr}}^{2}}{{d\theta...
  15. J

    Solution help for the Geodesic Equation

    Let’s say an asteroid is about to enter earth’s atmosphere(it will burn up of course). The initially sitting is at: point = [x_{0},y_{0},z_{0},t_{0}] = [r_{0},\\theta_{0},\\phi_{0},t_{0}] With a 4-velocity: V = [v_{1},v_{2},v_{3},v_{4}] The momentum at 0+ p_{0+} = mass*V =...
  16. P

    Solving the Geodesic Equation: Raising Contravariant Indices

    Homework Statement I would like to manipulate the geodesic equation. Homework Equations The geodesic equation is usually written as k^{a}{}_{;b} k^{b}=\kappa k^{a} (it is important for my purpose to keep it in non-affine form). It is clear that by contracting with the metric we may...
  17. K

    What does the geodesic equation for a surface involve?

    I don't understand the equation of the geodesic y=y(x) for the surface given by z=f(x,y) : a(x)y''(x)=b(x)y'(x)^3+c(x)y'(x)^2+d(x)dxdy-e(x) the functions a,b,c,d,e are here not very important, what I don't understand, is that there is terms in \frac{dy}{dx} and dxdy...What does this mean ?
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