Tangent Space

Tangent Vector  Motion along a trajectory is described by position and velocity. Position uses an originPosition uses an origin References the trajectoryReferences the trajectory  Displacement points along the trajectory. Tangent to the trajectoryTangent to the trajectory Velocity is also tangentVelocity is also tangent x1x1 x2x2 x3x3

Tangent Plane  Motion may be constrained Configuration manifold QConfiguration manifold Q Velocities are not on the manifold.Velocities are not on the manifold.  Set of all possible velocities Associate with a point x  QAssociate with a point x  Q N-dimensional set V nN-dimensional set V n  Tangent plane or fiber T x Q  x  V nT x Q  x  V n  V1V1 S1S1 S2S2 x V2V2

Tangent Bundle  Fibers can be associated with all points in a chart, and all charts in a manifold. This is a tangent bundle.This is a tangent bundle. Set is T Q  Q  V nSet is T Q  Q  V n Visualize for a 1-d manifold and 1-d vector.Visualize for a 1-d manifold and 1-d vector. V1V1 S1S1

Twisted Bundles  A tangent plane is independent of the coordinates.  Coordinates are local to a neighborhood on a chart.  Charts can align in different ways. Locally the same bundleLocally the same bundle Different manifold T QDifferent manifold T Q V1V1 S1S1

Tangent Maps  Map from tangent space back to original manifold.  = T Q  Q ; (x, v) (x)  = T Q  Q ; (x, v) (x) Projection map Projection map   Map from one tangent space to another f : U  W; U, W open f is differentiable T f : TU  TW (x, v)  (f(x), Df(x)v) Tangent map T f  Df(x) is the derivative of f V1V1 S1S1

Tangent Map Composition  The tangent map of the composition of two maps is the composition of their tangent maps T f : TU  TW; T g : TW  TXT f : TU  TW; T g : TW  TX T( gf ) = T g T fT( gf ) = T g T f  Equivalent to the chain rule next