My background is in maths, but I have been studying some basic physics with occasional input from a friend who is studying for a physics PhD. Due to my background, I am keen to visualize things geometrically, and find the idea that the Lagrangian might behave as a (pseudo-Riemannian) metric very appealing. The question "Can Lagrangian be thought of as a metric?" discusses this, but the answers there only address the case where potential energy terms can be neglected so that the corresponding metric is positive definite. I am specifically interested in the more general case of a mixed-signature metric.

Let $M$ be a pseudo-Riemannian manifold with metric $g$. Geodesics $\gamma$ : $[\tau_0,\tau_1] \rightarrow M$ can be characterized as being stationary for the functional $E(\gamma) = \int g\left(\frac{d\gamma}{d\tau},\frac{d\gamma}{d\tau}\right)d\tau$ over curves with fixed endpoints $\gamma(\tau_0),\gamma(\tau_1)$ where $\frac{d\gamma}{d\tau}$ is never zero. The co-ordinates on $M$, and hence the components of $\frac{d\gamma}{d\tau}$, can be labelled however we like: for example, some of the latter could represent velocities (or momenta) of particles while others represent quantities like 1/distance (or wavenumber). We can think of the coefficients in $g$ as scaling factors which ensure that the corresponding components of $g\left(\frac{d\gamma}{d\tau},\frac{d\gamma}{d\tau}\right)$ have the appropriate weightings (and "units").

If the integrand in $E$ represents a Lagrangian, then the components of $\frac{d\gamma}{d\tau}$ must split into those which contribute to kinetic energy and those which contribute to potential energy. Depending on the choices of units for the scaling factors in $g$, the former look like velocity or momentum ($n$-dimensional), while the latter look like 1/distance or wavenumber (1-dimensional). The corresponding coefficients in the metric must have opposite signs; alternatively, one set of components of $\frac{d\gamma}{d\tau}$ can pick up a factor of $i$, to make all coefficients positive. Note that velocity and momentum are quantities which "involve time" while 1/distance and wavenumber are quantities which do not. Finally, momentum and wavenumber are connected via Planck's constant (the quantum of action) in the de Broglie relations, while $E$ itself corresponds to action.

To me, all of this seems very suggestive of a connection with relativity. Could any of what I've said be meaningful? Is any aspect of it discussed in the literature in any way?

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    $\begingroup$ This question appears to be asking for opinions, which makes it off-topic. $\endgroup$
    – Kyle Kanos
    Apr 26, 2015 at 16:49
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    $\begingroup$ I'm not quite sure what your question really is - the energy functional you write down is precisely what we take as the action of a test particle. What exactly do you want to know about that? $\endgroup$
    – ACuriousMind
    Apr 26, 2015 at 17:20
  • $\begingroup$ A metric is a defining mathematical property of a special linear space, which in themselves are special topological spaces. You can study the properties of these spaces all by themselves without ever encountering a physics problem. The only physical question would be why macroscopic physics seems to invariably converging on equations of motion that have these properties. Nobody can answer that for you, at the moment, because we do not know what microscopic physics (if any) causes this layer of reality. $\endgroup$
    – CuriousOne
    Apr 26, 2015 at 18:20
  • $\begingroup$ StackExchange has informed me that I can only notify one user per comment, so I'll respond to each comment separately. @KyleKanos: the question was meant as a request for known information and/or references on related material, not for opinions. $\endgroup$ Apr 26, 2015 at 18:39
  • $\begingroup$ @ACuriousMind: M is not spacetime, g is not the usual metric tensor on spacetime, γ is not the path of a single particle and τ is not proper time. I will rewrite my question to try to make that more clear. $\endgroup$ Apr 26, 2015 at 18:39