Acceleration (differential geometry)
In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend".Consider a differentiable manifold with a given connection. Let be a curve in with tangent vector, i.e. velocity,, with parameter.
The acceleration vector of is defined by, where denotes the covariant derivative associated to.
It is a covariant derivative along, and it is often denoted by
With respect to an arbitrary coordinate system, and with being the components of the connection relative to this coordinate system, defined by
for the acceleration vector field one gets:
where is the local expression for the path, and.
The concept of acceleration is a covariant derivative concept. In other words, in order to define acceleration an additional structure on must be given.
Using abstract index notation, the acceleration of a given curve with unit tangent vector is given by.
