Given an affine manifold $(M,\nabla)$, the geodesic equation $\ddot{x}^j+\dot{x}^k \dot{x}^l\Gamma_{kl}^j=0$ completely characterizes the geodesics on the manifold. This is often called the Euler-Lagrange equation. I was wondering what the connection between geodesics and Lagrangian mechanics are. Given a mechanical system, its solutions can be found by solving the resulting Euler-Lagrange equation. Is there some affine connection $\nabla$ we can equip the configuration space $C$ so that all solutions are hence geodesics on the "manifold" $(C,\nabla)$? Reading this paper, it seems one can construct a Riemannian metric from the kinetic energy. However, it seems this really only works when there's no potential in the system, which I find kinda odd since what about the case where there is a potential.
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$\begingroup$ As for references, you could read the first chapter or two of Lanczos' "The Variational Principles of Mechanics". I remember that he does touch on this topic, briefly, in the introductory chapter(s). And I think he does mention how the potential changes/breaks the interpretation of a system as following geodesics in a metric space. Another book is Classical Mechanics by Goldstein, which might contain similar information. $\endgroup$– MyridiumCommented Feb 7, 2023 at 1:47
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$\begingroup$ A geometric perspective that does work out for generic mechanical systems (including ones with a potential) is the symplectic geometry of Hamiltonian mechanics, which you might find interesting: physics.stackexchange.com/q/564834 $\endgroup$– AndrewCommented Feb 7, 2023 at 2:43
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$\begingroup$ Comments to the post (v4): 1. Non-Levi-Civita connections and potentials seem to be 2 separate issues. Consider to only ask 1 question per post. 2. Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$– Qmechanic ♦Commented Feb 7, 2023 at 3:03
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$\begingroup$ Possible duplicates: Can Lagrangian be thought of as a metric? , Lagrangian mechanics and geodesics in configuration space? , Auto-parallel Transport or Principle of Extremum Action? and links therein. $\endgroup$– Qmechanic ♦Commented Feb 7, 2023 at 3:34
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$\begingroup$ The equations of motion are $\nabla_{\dot{\gamma}}\dot{\gamma}=-X\circ \gamma$, where $X=\text{grad}_g(V)$ is the metric-gradient of the potential $V$ defined on $M$. By writing this out in coordinates, you can see that for non-trivial $V$, it cannot be interpreted as the geodesic equation of a modified connection on $M$ (the LHS depends on $\dot{x}$ and $\ddot{x}$, while the RHS does not). However, what you can do is consider the manifold $N=\Bbb{R}\times M$, and you can “lift” the connection $\nabla^M$ on $M$ to a connection $\nabla^N$ by defining… $\endgroup$– peek-a-booCommented Feb 7, 2023 at 4:09
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