Non-Euclidean mechanics; is it useful?

Question

Special relativity has the following single-particle Lagrangian: $$S = \int_{t_0}^{t_f}\sqrt {\langle \mathrm d\vec{s},\mathrm d\vec{s}\rangle}.$$

Clearly it is based on Euclidean norms; it is in Minkowski or Riemannian-geometry norm, but both norms are only a generalization of the Euclidean norm.

Now I can formulate another Lagrangian that Looks like this:

$$S = \int_{t_0}^{t_f} ({\langle \mathrm d\vec{s},\mathrm d\vec{s},\mathrm d\vec{s}\rangle })^{\frac{1}{3}}\;.$$

I have generalized the Standard Lagrangian of a relativistic particle to the 3-norm and tried to concept a generalized scalar products for 3-norms.

Are such field theories developed now and can such field theories be constructed? Is there any evidence to construct a physical theory based on 3-norms?

Answer 1

OP's proposal (v2) is a special case of Finsler geometry with $n=3$. The main idea is to replace the quadratic metric tensor $g^{(2)}_{\mu_1\mu_2}$ for pseudo-Riemannian manifolds, which defines (infinitesimal, possibly imaginary) distance on the manifold via

$$ds ~=~ \sqrt[2]{g^{(2)}_{\mu_1\mu_2}dx^{\mu_1}dx^{\mu_2}},$$

with (possibly a sequence of) higher metric tensors $g^{(n)}_{\mu_1\ldots\mu_n}$ with a Finsler distance formula $$ds ~=~ \sum_{n\in\mathbb{N}} \sqrt[n]{ g^{(n)}_{\mu_1\ldots\mu_n} dx^{\mu_1}\ldots dx^{\mu_n}}.$$

There exists already a huge literature on Finsler geometry and its applications to physics.