  Technical results are presented on motion in N(>4)D manifolds to clarify the
physics of Kaluza-Klein theory, brane theory and string theory. The so-called
canonical or warp metric in 5D effectively converts the manifold from a
coordinate space to a momentum space, resulting in a new force (per unit mass)
parallel to the 4D velocity. The form of this extra force is actually
independent of the form of the metric, but for an unbound particle is tiny
because it is set by the energy density of the vacuum or cosmological constant.
It can be related to a small change in the rest mass of a particle, and can be
evaluated in two convenient gauges relevant to gravitational and quantum
systems. In the quantum gauge, the extra force leads to Heisenberg's relation
between increments in the position and momenta. If the 4D action is quantized
then so is the higher-dimensional part, implying that particle mass is
quantized, though only at a level of 10^{-65} gm or less which is unobservably
small. It is noted that massive particles which move on timeline paths in 4D
can move on null paths in 5D. This agrees with the view from inflationary
quantum field theory, that particles acquire mass dynamically in 4D but are
intrinsically massless. A general prescription for dynamics is outlined,
wherein particles move on null paths in an N(>4)D manifold which may be flat,
but have masses set by an embedded 4D manifold which is curved.
