ddt x dx dt dk 1x dtk

Consider the phase space trajectoryz(t) = x(t), dx dt (t),..., This line of argument continues up tox. If they intersect at an angle, then evolving the two states by a small timeδt << 1, and using the Lipschitz continuity off, meaning that the trajectories cannot have kinks in them (as shown in LemmaA.1), Now consider the single trajectoryz(t). Assume it intersects itself at an angle, att1 and t2. Therefore, the assumption thatz(t)can intersect itself atan angle must be wrong. Effectively an additional dimension is added to phase space, which is time.