Lagrange multiplier method proofs that extremizing (finite dimensional) function $f(x)$ subject to: $g(x)=0$, equals extremizing: $f_a(x,\lambda)=f(x)+ \lambda \cdot g(x)$
Suppose now we want to extremise the functional $J(x)$ defined as $$ J = \int_{t_0}^{t_f} f(x(t),t) \, dt $$ subject to the constraint $$g(x(t),t) = 0$$ How to proof that this situation is equivalent to extremize $J_a$ defined as $$ J_a = \int_{t_0}^{t_f} [\,f(x(t),t)+\lambda(t)\cdot g(x(t),t)\,]\, dt $$
In particular: Why $\lambda$ is now function of variable $t$? How is that we 'enter' the integral with Lagrangian Multipliers?
Thanks for any help