In this paper we study the relation between the coercivity and positivity of a time scale quadratic functional $J$, which could be a second variation for a nonlinear time scale calculus of variations problem (P). We prove for the case of general jointly varying endpoints that $J$ is coercive if and only if it is positive definite and the time scale version of the strengthened Legendre condition holds. In order to prove this, we establish a time scale embedding theorem and apply it to the Riccati matrix equation associated with the quadratic functional $J$. Consequently, we obtain sufficiency criteria for the nonlinear problem (P) in terms of the positivity of $J$ or in terms of the time scale Riccati equation. This result is new even for the continuous time case when the endpoints are jointly varying .

Last change: May 22, 2007. (c) Roman Hilscher