Let $\varphi : \Bbb{R}^2 \rightarrow \Bbb{R}$ defining an implicite curve $C = \{ (x,y), \varphi(x,y) = 0 \}$, and $u : \Bbb{R}^2 : \rightarrow \Bbb{R}$
Does the line integral $\int_C u(x,y)\ dC$ exists ?
And if it exist, is there a way to write it :
$ \int_C u(x,y) \ dC = \int_{\Bbb{R}^2}f(x,y,\varphi,\varphi')\ dxdy \ $ ? (to apply Euler-Lagrange equation)
I thought of something like $ \int_{\Bbb{R}^2}u(x,y).\delta_0(\varphi(x,y))\ dxdy \ $ ($\delta_0$ the dirac distribution), as $\delta_0 \circ \varphi $ is like the indicator function of C. But i guess this formula isn't true ?
If $C$ is a rectifiable curve, then yes. Note that for a general smooth function $\varphi$ the zero set $\varphi=0$ can be quite horrible: it can be an arbitrary closed subset of $\mathbb R^2$. But if you assume that $\nabla \varphi\ne 0$ on $C$, then the implicit function theorem implies that $C$ is a smooth curve.
Not really, but you can approximate it by integrals of that kind. The formal tool is the coarea formula but the basic idea is to integrate $u$ over $\epsilon$-neighborhood of the curve (using the Lebesgue measure), divide by $2\epsilon$, and take the limit $\epsilon\to 0$. The problem is, it's difficult to express the $\epsilon$-neighborhood of $C$ in terms of $\varphi$. It is easier to take the region between two level sets $\varphi = \pm \epsilon$ as an approximation to $C$, but this region has variable width. The width is roughly $\epsilon |\nabla \varphi|^{-1}$. To offset this, we integrate not $u$ itself, but the product $u |\nabla \varphi|$. This works:
$$\int_C u \,dC = \lim_{\epsilon\to 0} \frac{1}{2\epsilon}\iint_{\mathbb R^2}u |\nabla \varphi| \chi_{ \{|\varphi|<\epsilon \} }$$
If my guess is right, this will not do you any good. Please don't think that by formal manipulations you can mask the fact that you have a variational problem with free boundary $C$. These are delicate problems that require tools of both analysis and geometric measure theory. Use variations, of course, but make them geometric variations: move $C$ along its normal vector field, etc.