Necessary optimality conditions for the calculus of variations on time scales

We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems of the calculus of variations with delta-differential side conditions (Lagrange problem of the calculus of variations on time scales).

💡 Research Summary

The paper extends the calculus of variations on arbitrary time scales by establishing necessary optimality conditions for two broader classes of problems. First, it treats variational functionals whose Lagrangian depends on higher‑order delta derivatives, i.e., L(t, x, x^{Δ}, x^{ΔΔ}, …, x^{Δ^n}). By introducing a variation η(t) together with its delta derivatives up to order n and repeatedly applying the integration‑by‑parts formula on time scales, the authors derive a generalized Euler‑Lagrange equation:

∑_{k=0}^{n} (−1)^k Δ^k