Calculus of variations on time scales: weak local piecewise C1 rd solutions with variable endpoints

A nonlinear calculus of variations problem on time scales with variable endpoints is considered. The space of functions employed is that of piecewise rd-continuously Δ-differentiable functions (C1 prd). For this problem, the Euler–Lagrange equation, the transversality condition, and the accessory problem are derived as necessary conditions for weak local optimality. Assuming the coercivity of the second variation, a corresponding second order sufficiency criterion is established.  2003 Elsevier Inc. All rights reserved.