I've read some papers in Geodesic active contours (Image processing), which use the minimization of an Energy, consist of Internal Energy and External energy, for example, in the paper of Kass (Snake: Active contour model - 1988), we have the following formula
$$E_{snake} = \int_0^1 (E_{int}(v(s))+E_{image}(v(s))+E_{con}(v(s)))ds $$
with $v(s) = (x(s),y(s))$ and $E_{int} = \frac{1}{2}(\alpha(s)|v_s(s)|^2+\beta(s)|v_{ss}(s)|^2)$. In the paper, they have the first term makes the snake act like a membrane and the second term makes it act like a thin-plate. I don't know how they derive the fomula $E_{int}$, and so the $E_{con}, E_{image}$ like that. What is it and where does it come from? In mechanic ? or another related field? Thanks in advance. Please forgive me if I had some grammatical mistakes.
It comes from very idealized continuum mechanics: squared first derivative is elastic energy, squared second derivative is bending energy.
To simplify things, I'll consider scalar function $x(s)$, since the contributions of both $x$ and $y$ components are simply added together in your formula. The length of the graph $x(s)$ is $$ \int_0^1 \sqrt{1+(x'(s))^2}\,ds $$ which for small values of derivative can be approximated by $$ \frac12 \int_0^1 (x'(s))^2\,ds \tag1 $$ Minimization of the functional (1) among curves constrained to pass through given points creates a piecewise linear function: this is what we get by tightening an elastic string that is nailed to a wall in a few places.
The bending energy, whatever it is, should depend on the curvature of the graph. the curvature is related to second derivative. So,
$$ \int_0^1 (x''(s))^2\,ds\to \min \tag2 $$ becomes an idealized model of a string that resists bending. Minimization of (2) with the same conditions as above leads to a pleasantly curved graph, which passes through the given points without sharp turns (sharp turns would mean rapid change of first derivative, hence large second derivative).
(Both images are from my post Connecting dots naturally.)