Variation of a functional

Question

I have to find the variation of the following functional:

There are two conditions a > 0 and b > 0. The question is "find the differential equation with respect to x(t), so that the functional is minimized".

Answer 1

Set $y(\tau) = x(\tau) + \varepsilon \eta(\tau)$ where $\eta(\tau)$ is an arbitrary but twice differentiable function. Then $$ S[y(\tau)]=\int_0^t a \left(\dot{x}(\tau)+\varepsilon \dot{\eta}(\tau)\right)^2 + b\left(\ddot{x}(\tau)+\varepsilon \ddot{\eta}(\tau)\right)^2 + c(x(\tau)+\varepsilon \eta(\tau))^4 d\tau. $$ Take the derivative wrt to $\varepsilon$ and let $\varepsilon \rightarrow 0$, yielding $$ \nabla_\eta S[x(\tau)] = \int_0^t 2a\dot{x}(\tau)\dot{\eta}(\tau) + 2 b \ddot{x}(\tau)\ddot{\eta}(\tau) + 4 c x(\tau)^3\eta(\tau) d\tau. $$ This is the directional/Gateaux derivative of $S[]$ wrt $\eta$ evaluated at $x$.

Are you optimizing the functional? The idea is that if you are at $x$ and take a step $\varepsilon$ in the direction $\eta$ in the appropriate space of functions, you are evaluating the value of moving from $x$ towards $y$. Setting $\nabla_\eta S[x^*]=0$ means that there are no directions away from $x^*$ that locally improve $S$. Do you have initial conditions, or other constraints?

Answer 2

Calling

$$ x_1 = x\\ x_2 = \dot x\\ x_3 = \ddot x $$

$X = (x_1,x_2,x_3)$ we have

$$ S = \int _0^t (a x_2^2+b x_3^2+c x_1^4)d\tau\ \ \ \text{s. t.}\ \ \ \dot X = A X $$

with $A = \left(\begin{array}{ccc}0 & 1 & 0\\ 0 & 0 & 1\\\end{array}\right)$

Now considering the lagrangian

$$ L = \int _0^t (a x_2^2+b x_3^2+c x_1^4+\lambda_1(t)(\dot x_1-x_2)+\lambda_2(t)(\dot x_2-x_3))d\tau $$

and using the Euler-Lagrange equations we get at

$$ \cases{\dot\lambda_1 = 4cx_1^3\\ \dot\lambda_2 +\lambda_1 = 2 a x_2\\ \lambda_2=2bx_3} $$

and after substitution we obtain the following differential equation:

$$ bx''''-a x''+2c x^3=0 $$