User:Mim.cis/sandbox/Geodesic Solutions of Landmarks and Surface and Volumes

=The geodesic dynamics of manifolds and Green's kernels =

Geodesic velocities are weighted Green's kernels, weights the canonical-conjugate momentum (see /Hamiltonian-Control-CA for momentum derivations).

In CA, the smoothness to insure smooth flows of diffeomorphisms, ($$), results from the fact that the different geodesic motions of the manifolds have vector fields $$ v_t \in V $$ which are rewritten in terms of the differentiable Green's kernels convolved with the conjugate momentum (Lagrange multiplier) $$ p_t \in \mathcal{Q}^* $$.


 * Landmarks: Motions of landmarks,    landmarks $$x_i, i=1,2,...$$, $$q_t(i) \doteq \phi_t(x_i)$$, $$q_0(i) = x_i$$, with Hamiltonian
 * $$H(q_t,p_t,v_t) = \sum_i p_t(i) \cdot v_t \circ q_t(i) - \frac{1}{2} (Av_t | v_t)$$ and dynamics taking the form

The momentum is a generalized function $$Av \in V^*$$ given by sum over landmarks, $$Av_t = \textstyle \sum_{u=1}^n \displaystyle \delta_{\phi_t(m_u)}p_t(u)$$ $$H(v_t,p_t,q_t) = \int_U p_t(u) \cdot v_t \circ q_t (u) du - \frac{1}{2} (Av_t | v_t)$$, and dynamics taking the form
 * Surfaces: Motions of surfaces,    chart  $$m: U \subset {\mathbb R}^2 \rightarrow {\mathbb R}^3$$, $$q_t(u) = \phi_t\circ m(u), q_0 = m$$, with Hamiltonian

The momentum is a generalized function $$Av \in V^*$$ given by a surface integral over landmarks, $$Av_t = \textstyle \int_U \displaystyle \delta_{\phi_t(m_u)}p_t(u) du$$
 * Volumes: Motions of volumes  $$x \in X$$ with $$q_t(x) \doteq \phi_t(x)$$, $$q_0 = id$$, with Hamiltonian


 * $$H(q_t,p_t,v_t) = \int_X p_t(x) \cdot v_t \circ q_t (x) dx - \frac{1}{2} (Av_t | v_t)$$, and dynamics taking the for

The momentum is a classical function $$Av_t = \mu_t dx$$.