1 Introduction

2 Preliminaries
2.1 The delta derivative and the delta integral
2.2 The nabla derivative and the nabla integral
2.3 Relation between delta and nabla operators

3 Direct problems of the calculus of variations on time scales
3.1 The delta approach to the calculus of variations
3.2 The nabla approach to the calculus of variations

4 Inverse problems of the calculus of variations on time scales
4.1 A general form of the Lagrangian
4.2 Necessary condition for an Euler–Lagrange equation
4.3 Discussion

5 The delta-nabla calculus of variations for composition functionals
5.1 The Euler–Lagrange equations
5.2 Natural boundary conditions
5.3 Isoperimetric problems
5.4 Illustrative examples

6 Conclusions