Jacobi Fields, Conjugate Points

Abstract

Let us go back to the action principle as realized by Jacobi, i.e., time is eliminated, so we are dealing with the space trajectory of a particle. In particular, we want to investigate the conditions under which a path is a minimum of the action and those under which it is merely an extremum. For illustrative purposes we consider a particle in two-dimensional real space. If we parametrize the path between points P and Q by ϑ, then Jacobi’s principle states:

$$ \delta \int_{\vartheta 1 \buildrel \wedge \over = P}^{\vartheta 2 \buildrel \wedge \over = Q} {d\vartheta \sqrt {H - V({q_1},{q_2})} } \sqrt {\sum\limits_{i,j = 1}^2 {{m_{ij}}} ({q_k})\frac{{d{q_i}}}{{d\vartheta }}} \frac{{d{q_j}}}{{d\vartheta }} = 0\;{\rm{.}} $$

(3.1)