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Do any circle centered at the origin constitute a geodesic?

Because I think it parallel transport its own tangent vector.

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- Thread starter Damidami
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In summary: The geodesics of a curved surface are, again, straight lines, but they may be curved in various directions. Thanks for the clarification!In summary, the geodesics of a surface depend only on the metric, not the coordinate system.

- #1

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Do any circle centered at the origin constitute a geodesic?

Because I think it parallel transport its own tangent vector.

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- #3

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But I still don't quite get it: the tangent vectors of the coordinate lines in polar coordinates do rotate about the origin.

Maybe the covariant derivative compensate for this rotation, but I can't figure out how to write the geodesic equation of R^2 in polar coordinates, I know it must give me a system of 2 differential equations but don't know how to get them.

Say, if I have point [tex]p=(r,\phi) = (0,0)[/tex] and vector [tex]v=(r,\phi) = (1, \pi/4)[/tex], how do I calculate it's geodesic (I know in cartesian coordinates it has to be the y=x line).

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Now how can I calculate a geodesic starting at [tex](0,0)[/tex] and with initial tangent vector [tex](1, \pi/4)[/tex] ?

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They are [tex]\Gamma^{\phi}_{r \phi} = \Gamma^{\phi}_{\phi r} = \frac{1}{r} [/tex] and [tex]\Gamma^{r}_{\phi\phi} = -r [/tex]

The others [tex]\Gamma = 0[/tex].

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What differential equation must geodesics satisfy then?

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It seems to work!

As to the geodesics differential equation for polar coordinates, I found the answer on MTW.

But I'm not very good at solving diffs eqs.

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Parallel transport in flat polar coordinates refers to the movement of a vector along a curve in a polar coordinate system without changing its direction. It is a mathematical concept used in differential geometry to understand how vectors change as they move along a curved surface.

Parallel transport is different from regular transport because it takes into account the curvature of the surface on which the vector is moving. In regular transport, the vector's direction may change as it moves along a curved surface, whereas in parallel transport, the vector's direction remains constant.

In physics, parallel transport is used to understand how quantities such as velocity, momentum, and force change as they move along a curved path. It is particularly important in understanding the behavior of objects in a gravitational field, as it helps to determine how their properties change as they move through space.

To calculate parallel transport in flat polar coordinates, one must take into account the Christoffel symbols, which represent the curvature of the surface. These symbols are used in a mathematical formula to determine the change in the vector's direction as it moves along a given curve.

Parallel transport in flat polar coordinates has applications in various fields, including physics, engineering, and computer graphics. It is used in general relativity to understand the motion of objects in a gravitational field, in computer graphics to simulate the movement of objects, and in engineering to design structures that can withstand external forces without changing their shape.

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