How should I find this first-integral of a functional?

Question

Let $0<a<b$. Consider the functional $$S[y]=\int_{a}^{b}x^5(y'^2-\frac{2}{3}y^3)dx$$ Prove that a first-integral of $S[y]$ is $$4x^5yy'+x^6(y'^2+\frac{2}{3}y^3)=c,$$ where $c$ is constant, stating any theorems that you use.

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I know that the Euler-Lagrange equation for $S[y]$ is $$\frac{d^2y}{dx^2}+\frac{5}{x}\frac{dy}{dx}+y^2=0$$ but how should I prove this? Does it has anything to do with this equation: $$f-y'\frac{\partial f}{\partial y'}=\text{constant}~?$$

If so, then what's $f, y', \frac{\partial f}{\partial y'}$?

Answer 1

Hint: Let
$$ g(x):=4x^5yy'+x^6\left(y'^2+\frac{2}{3}y^3\right). \tag{1} $$ Compute $g'(x)$ and use the Euler-Lagrange equation $y''=-\frac{5}{x}y'-y^2$ to simplify terms containing $y''$. If you do it correctly, you will find that $g'(x)=0$, hence $g(x)=c$.

Remark: The equation $f-y'\frac{\partial f}{\partial y'}=C$, known as Beltrami identity, is true for functionals of the form $$ S[y]=\int_a^bf[x,y(x),y'(x)]\,dx \tag{2} $$ if $\frac{\partial f}{\partial x}=0$. That's not the case for the functional in your question.

Answer 2

1. Given an explicit quantity, it is easy to verify whether or not it is conserved (as a function of time $x$), cf. Gonçalo's answer.

2. If we don't already know the explicit form of the conserved quantity/first integral, we can find it using Noether's theorem. This is what we will do in this answer.

3. It is easy to see that OP's functional $$ S[y;a,b] ~=~S[\tilde{y};\tilde{a},\tilde{b}]$$ has an exact symmetry $$\tilde{y}(\tilde{x})~=~\lambda^{\color{red}{2}}y(x), \qquad \tilde{x}~=~\lambda^{\color{red}{-1}}x, $$ where $\lambda> 0$ is a positive parameter.

4. Since OP's Lagrangian is $$L~=~x^5(\dot{y}^2-\frac{2}{3}y^3),$$ the corresponding momentum and energy are $$ p~:=~\frac{\partial L}{\partial \dot{q}}~=~2x^5\dot{y}$$ and $$h~:=~p\dot{y}-L~=~x^5(\dot{y}^2+\frac{2}{3}y^3),$$ respectively. The corresponding Noether charge $$ Q~=~\color{red}{2y}p - \color{red}{(-x)}h~=~4x^5y\dot{y}+x^6(\dot{y}^2+\frac{2}{3}y^3)$$ is OP's sought-for conserved quantity/first integral.

5. For more details, see e.g. this related Phys.SE post.