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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 .

Time scale embedding theorem and coercivity of quadratic functionals