From the initial condition problem of the Euler-Lagrange equation to the principle of least action

I browsed through many similar questions about the Initial Condition Problem (ICP) and Boundary Value Problem (BVP) for Euler-Lagrange equations, here some interesting but (in my opinion) incomplete examples: E.g.1 or E.g.2 or E.g.3. This question is within the subject but slightly different.

We know that studying the first variation of the action functional between two positions $$q(t_{1})=q_{1}$$ and $$q(t_{2})=q_{2}$$, where a particle is found at the instants $$t_{1}$$ and $$t_{2}$$, with $$t_{1}, is equivalent to study the BVP with the Euler-Lagrange equation $$$$\frac{\partial L}{\partial q}-\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)=0$$$$ and the boundary conditions $$q(t_{1})=q_{1}$$ and $$q(t_{2})=q_{2}$$.

My question is, if we look at the matter "backwards": from an ICP with the Euler-Lagrange equation above and an initial condition $$q(t_{3})=q_{3}$$ and $$\dot{q}(t_{3})=\dot{v}_{3}$$, with $$t_{1}, assuming all the regularity conditions for the Picard–Lindelöf theorem are satisfied and the solution exists and it's unique, what can we say in terms of the action functional? Will the first variation of the action functional be zero if we plug the solution that we found from the ICP?

• Did you mean "implies the BVP" instead of "is equivalent to study the BVP"...? Otherwise I think you have answered your own question, but I may have missed a subtlety. Commented Jun 15, 2021 at 13:37
• @MariusLadegårdMeyer I mean it's equivalent. There is a theorem which states that the necessary and sufficient condition for the action to be minimized is the E.L. equation to satisfy the BVP with withe same extremes of the integral of the action.
– Simo
Commented Jun 15, 2021 at 13:50

• This implies big philosophical consequences. If we force the presence of the FBC, the E.L. equations are no longer deterministic since a BVP is not deterministic, at least not after $t_{2}$. Btw, are you aware of any quantitative proof of what you said?