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 $\vartheta$, then Jacobi’s principle states:

$$\displaystyle{ \delta \int _{\vartheta _{1}\stackrel{\hat{\ }}{=}P}^{\vartheta _{2}\stackrel{\hat{\ }}{=}Q}d\vartheta \sqrt{H - V (q_{ 1},q_{2})}\;\sqrt{\sum _{i,j=1 }^{2 }m_{ij } (q_{k } )\frac{dq_{i } } {d\vartheta } \,\frac{dq_{j}} {d\vartheta }} = 0\;. }$$

(5.1)

To save space let us simply write $g(q_{1},q_{2}, \frac{dq_{1}} {d\vartheta }, \frac{dq_{2}} {d\vartheta } )$ for the integrand. Hence the action reads

$$\displaystyle{ S\{[q_{1},q_{2}];\vartheta _{1},\vartheta _{2}\} =\int _{ \vartheta _{1}}^{\vartheta _{2} }d\vartheta \;g\left (q_{1}(\vartheta ),q_{2}(\vartheta ); \frac{dq_{1}} {d\vartheta }, \frac{dq_{2}} {d\vartheta } \right )\;. }$$

(5.2)

For our further discussion it would be very convenient to choose one coordinate, e.g., q ₁ instead of $\vartheta$, to parametrize the path: $q_{2}(q_{1})$ with $q_{1}^{(1)} \leq q_{1} \leq q_{1}^{(2)}$.