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I am a graduate student and I want to understant the concept of acceleration of a point moving through a smooth path $\gamma : [a,b]\to M$ where $M$ is a smooth manifold without further structures. I know that in order to differentiate a vector field along a vector we need a connection on the tangent bundle $TM$ of $M$. In this way we can calculate the acceleration and put mathematical constraints to acceleration such as Newton's Second Law of motion. But acceleration is just the change of velocity so it seems natural to define acceleration as an element of the tangent space of the tangent space of $M$, $TTM$. Why do we need a connection on $TM$ instead of defining acceleration as an element of $TTM$? Thank you for your answers.

Arctic Char
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Ali Arda Karalar
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  • Related discussion: https://math.stackexchange.com/questions/4538152/how-is-acceleration-connected-to-the-second-order-tangent-bundle https://math.stackexchange.com/questions/4735571/derivative-of-a-curve-on-tangent-bundle-tq-and-second-order-equations-where-i https://math.stackexchange.com/questions/4309198/on-which-tangent-bundles-of-mathbb-r2-does-position-velocity-acceleration. Generally, I think that "acceleration must be the tangent of velocity" is just a holdover from Euclidean space. "Acceleration is the limit of changes in velocity" requires a connection to define. – whpowell96 Apr 19 '25 at 02:21
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    You can of course define the second derivative to a curve, that, as a section, lands on $TTM$, but this is just extra work. $TTM$ has a so called vertical sub bundle given by the kernel of the pushforward of the projection on the tangent bundle. Anything that happens in this vertical subbundle is irrelevant for any curve in $M$, since the pushforward kills all that information down to the base point. This leads us back to connections (choices of a complementary subspace to this vertical space) that induce isomorphisms between the fibers (that are isomorphic to TM). – Lourenco Entrudo Apr 19 '25 at 03:04
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    So at any rate you're led back to thinking of accelerations as sections of the tangent bundle, and needing a connection to set it up – Lourenco Entrudo Apr 19 '25 at 03:06

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My answer would be that one can define acceleration exactly as explained by @LourencoEntrudo. The question for me is what can you do with it?

On a manifold without any other geometric structure, all parameterized curves with nonzero velocity are locally equivalent with respect to diffeomorphisms. So there are no interesting local geometric invariants if the velocity is locally nonzero. This means the local study of curves is interesting and in fact quite nontrivial only at a point where the velocity of the curve is zero. I don’t know offhand but I wouldn’t be surprised if acceleration is an interesting and useful concept here. The other nontrivial setting is the global study of curves, I.e., dynamical systems. It is also possible that acceleration is indirectly useful here.

But many of us don’t work in these areas so we don’t find the concept useful.

Deane
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