Calculus of variations and envelope theorem application 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! 
 A: 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.
