The dynamics of a large class of mechanical systems can be described as a geodesic motion in some ambient space. This is the essence of the Kaluza-Klein theory.
The basic and most elementary example is the case of
a charged particle in $3D$ coupled to a magnetic field which can be described as a neutral particle geodesically moving in a background metric in $4D$, such that the fourth dimension has the shape of a circle. Please see Marsden and Ratiu (Introduction to mechanics and symmetry - page 200). The $4$-th component momentum of the particle along the circle becomes the electric charge.
Thus this theory can account for origin of charge. The circle $S^1$ spanning the $4$-th dimension that we started with ends up to be the internal space $U(1)$ of the electric charge. (Also, since, in quantum mechanics, momenta along compact spaces are quantized, this theory also explains the quantization of the electric charge).
Generalization of the Kaluza-Klein theory describing non-Abelian charges and intercations such as spin and color also exist. Please, see for example the following
article by Harnad and Pare.(The generalization of the Kaluza Klein theory to include non-Abelian charges and interactions with Yang-Mills fields was initiated by Kerner).
The generalization to multiple particles interacting through
gravitational, electromagnetic and Yang-Mills fields is straightforward. One only has to couple them to the same metric in the ambient space. Also, there is no difficulty to formulate the theory relativistically, since the relativistic rules for coupling to a metric are known.
In mathematics, this theory is called "SubRiemannian geometry", Please
see the following review by I. Markina. The most general construction is to allow the
particle to start from any point in the ambient space, and move along
geodesic lines, but to constrain its velocity to lie on certain
subspaces of its tangent bundle. Thus this theory in its full generality
describes nonholonomic constraints and has many applications in geometric mechanics.
Due to all the possible interactions that can be explained by the
Kaluza-Klein approach, it seemed very attractive for the unification of all the fundamental forces. But, there is a strong argument by Witten that we cannot obtain massless fermions chirally coupled to gauge fields in this approach. (We can, of course, introduce this interaction by hand, but then we would loose the unification principle). Since, this type of interaction Lies in the basis of the standard model, this approach was virtually abandoned.
This post imported from StackExchange Physics at 2014-04-08 05:09 (UCT), posted by SE-user David Bar Moshe