Lagrangian, geodesics and relativity

Question

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?