Newtonian mechanics on curved spacetime

[mma]

The spacetime model of Newtonian mechanics is a fiber bundle with a one-dimensional base space (=time) and a 3-dimensional fiber (=space). On the tangent bundle of the spacetime a curvature-free connection is given. Newton's 1st law states that free pointlike particles move on geodesics determinded by this connection. Newton's second law states that the covariant derivative of the tangent vector of the world-line by intself equals the force acting on the particle. For the connection is curvature-free, an absolute parallelism can be defined on the tangent bundle of the spacetime, so it becomes an affine space, hence Newton's 1st law states that te world-lines of pointlike free particles are straight lines in this 4-dimensional affine space, and one can give a trivialization of the spacetime on which these world-lines are also straight lines and the covariant derivative is equal with the coordinate-derivative. These trivializations belong to the inertial observers.

But, what is, if we restrict our mechanics e.g. to the surface of the Earth? Then the fiber will be S² and the connection will be curved. However, Newton's first law can be kept in the form that the pointlike particles move on geodesics, and Newton's second law also using covariant derivative, no absolute parallelism can be defined on the tangent bundle of the spacetime, so inertial frames also don't exist. Have someone ever met with this kind of mechanics?

 

[pat_connell]

Yes, this is known as the theory of spherical mechanics. In this case, Newton's 1st law states that free pointlike particles move on geodesics of the sphere and Newton's second law states that the covariant derivative of the tangent vector of the world-line by itself equals the force acting on the particle. Since the connection is curved, there is no absolute parallelism on the tangent bundle and therefore no inertial frames. Spherical mechanics can be used to model the motion of a particle on the surface of the Earth. The most common example is the motion of a satellite around the Earth, where the gravitational force can be modeled using Newton's second law. Other applications include the motion of gyroscopes and the motion of non-inertial frames in general relativity.