My Euler-Lagrange equation is a non-linear differential equation and I am convinced that it cannot be expressed in a closed-form solution.
My question is: Is there a theorem in the calculus of variations which is similar to the envelope theorem that we use in a static environment?
1) I want to show that my maximized value (with a solution function) increases with a parameter.
2) I want to show that the derivative of choice variable with respect to a parameter is always positive without solving the DE.
Thank you very much in advance!
FWIW, there exist dynamical extensions of the envelope theorem for static systems, say, a functional $$S[q; \alpha]~=~\int_{t_i}^{t_f}\! \mathrm{d}t~L(q(t),\dot{q}(t),t; \alpha) $$ in case of holonomic constraints$^1$
$$\chi(q(t);\alpha)~=~0.$$ Here $\alpha$ are external parameters. The extended functional reads $$\widetilde{S}[q,\lambda; \alpha]~=~\int_{t_i}^{t_f}\! \mathrm{d}t~\widetilde{L}(q(t),\lambda(t),\dot{q}(t),t; \alpha),$$ $$ \widetilde{L}(q(t),\lambda(t),\dot{q}(t),t; \alpha)~=~L(q(t),\dot{q}(t),t; \alpha) ~+~\lambda(t) \chi(q(t);\alpha), $$ where $\lambda$ are Lagrange multipliers.
In this dynamical situation the envelope theorem becomes $$ \frac{d^{\rm tot} S[q^{\ast}(\alpha); \alpha]}{d\alpha}~=~\frac{d^{\rm tot} \widetilde{S}[q^{\ast}(\alpha),\lambda^{\ast}(\alpha); \alpha]}{d\alpha} ~=~\frac{\partial^{\rm expl} \widetilde{S}[q^{\ast}(\alpha),\lambda^{\ast}(\alpha); \alpha]}{\partial\alpha}.$$ The proof is a straightforward application of Euler-Lagrange (EL) equations for constrained systems, and derivatives thereof, e.g. $$\frac{d^{\rm tot} \chi(q^{\ast}(t;\alpha);\alpha)}{d\alpha}~=~0.$$
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$^1$ Here I'm thinking of 2-sided constraints for simplicity. I haven't thought about whether it can be extended to 1-sided constraints (i.e. inequalities) as on the Wikipedia page. I also avoid $\dot{q}$-dependent constraints $\chi$ as they would make the Lagrange multipliers $\lambda$ dynamical.