Manin obstruction is explain why Hasse principle dont work in non-quadratic forms. Let's write the notation first;

$V(\mathbb{Q})$ is variety for rational numbers.

$V(A_\mathbb{Q})$ is variety for adelic point (i.e. its contain real solution and p-adic solution for all p)

Its clearly $V(\mathbb{Q}) \subset V(A_\mathbb{Q})$

Manin obsruction says, we can find new variety set $V(A_\mathbb{Q})^{Br}$ such that $V(\mathbb{Q}) \subset V(A_\mathbb{Q})^{Br} \subset V(A_\mathbb{Q})$. If $V(A_\mathbb{Q})$ is not empty but $V(A_\mathbb{Q})^{Br}$ is empty, then $V(\mathbb{Q})$ must be empty. So we can equation has no rational solution.

But why its don't work for quadratic forms? I estimate in the quadratic forms, $V(A_\mathbb{Q})^{Br}$ must be equal to $V(A_\mathbb{Q})$ or $V(\mathbb{Q})$. Is this approach correct?

I could not find an explanation of this in the article and books I have examined.I think, I can use this for my thesis. therefore I have to explain to how it is not work or I must have a reference for this.

I would be very thankful if you could help.

The Manin obstruction explains why the Hasse principle doesn't work for non-quadratic forms. Let's write the notation first;

$V(\mathbb{Q})$ is variety for rational numbers.

$V(A_\mathbb{Q})$ is variety for adèlic points (i.e., it contains real solution and p-adic solution for all $p$).

It's clear that $V(\mathbb{Q}) \subset V(A_\mathbb{Q})$.

Manin obsruction says, we can find new variety set $V(A_\mathbb{Q})^{\text{Br}}$ such that $V(\mathbb{Q}) \subset V(A_\mathbb{Q})^{\text{Br}} \subset V(A_\mathbb{Q})$. If $V(A_\mathbb{Q})$ is not empty but $V(A_\mathbb{Q})^{\text{Br}}$ is empty, then $V(\mathbb{Q})$ must be empty. So we can conclude the defining equations have no rational solution.

But why doesn't it work for quadratic forms? I estimate that, for quadratic forms, $V(A_\mathbb{Q})^{\text{Br}}$ must be equal to $V(A_\mathbb{Q})$ or $V(\mathbb{Q})$. Is this approach correct?

I could not find an explanation of this in the article and books I have examined. I think, I can use this for my thesis. Therefore I have to explain why it doesn't work or I must have a reference for this.