Not all hypersurfaces in $\mathbb{R}^n$ are of contact type.

Weinstein, in the paper: "On the hypothesis of Rabinowitz' periodic orbit theorems", where he defined the concept of contact type, gives a criterion. If $H^1(\Sigma)=0$, and $\Sigma$ is of contact type, then the characteristic line bundle comes with a distinguished orientation, determined by those vectors $\xi$ such that $\alpha(\xi)>0$ for all contact forms $\alpha$. This is independent of the contact form. For periodic orbits this induces a positivity criterion. In the same paper he also constructs an hypersurface in $\mathbb{R}^4$ which is not of contact type, by showing that the criterion is violated.

Many hypersurfaces are of contact type, as you remarked. Another nice example are mechanical hypersurfaces. These are hypersurfaces arising as level sets from hamiltonians $H=T+V$, where $T$ is the kinetic energy term $T=1/2|p|^2$, and $V$ is a potential depending only on $q$. This works in general cotangent bundles.

Not all hypersurfaces in $\mathbb{R}^{2n}$ are of contact type.

Weinstein, in the paper: "On the hypothesis of Rabinowitz' periodic orbit theorems", where he defined the concept of contact type, gives a criterion. If $H^1(\Sigma)=0$, and $\Sigma$ is of contact type, then the characteristic line bundle comes with a distinguished orientation, determined by those vectors $\xi$ such that $\alpha(\xi)>0$ for all contact forms $\alpha$. This is independent of the contact form. For periodic orbits this induces a positivity criterion. In the same paper he also constructs an hypersurface in $\mathbb{R}^4$ which is not of contact type, by showing that the criterion is violated.

Many hypersurfaces are of contact type, as you remarked. Another nice example are mechanical hypersurfaces. These are hypersurfaces arising as level sets from hamiltonians $H=T+V$, where $T$ is the kinetic energy term $T=\frac{1}{2}\vert p\vert^2$, and $V$ is a potential depending only on $q$. This works in general cotangent bundles.

Weinstein, in the paper where he defined the concept, gives a criterion, if $H^1(\Sigma)=0$, the characteristic line bundle comes with a distinguished orientation. In the same paper he also constructs an hypersurface in $\mathbb{R}^4$ which is not of contact type, by showing that the criterion is violated.

Many hypersurfaces are of contact type, as you remarked. Another nice example are mechanical hypersurfaces. These are hypersurfaces arising from hamiltonians $H=T+V$, where $T$ is the kinetic energy term $T=1/2|p|^2$, and $V$ is a potential depending only on $q$. This works in general cotangent bundles.

Not all hypersurfaces in $\mathbb{R}^n$ are of contact type.

Weinstein, in the paper: "On the hypothesis of Rabinowitz' periodic orbit theorems", where he defined the concept of contact type, gives a criterion. If $H^1(\Sigma)=0$, and $\Sigma$ is of contact type, then the characteristic line bundle comes with a distinguished orientation, determined by those vectors $\xi$ such that $\alpha(\xi)>0$ for all contact forms $\alpha$. This is independent of the contact form. For periodic orbits this induces a positivity criterion. In the same paper he also constructs an hypersurface in $\mathbb{R}^4$ which is not of contact type, by showing that the criterion is violated.

Many hypersurfaces are of contact type, as you remarked. Another nice example are mechanical hypersurfaces. These are hypersurfaces arising as level sets from hamiltonians $H=T+V$, where $T$ is the kinetic energy term $T=1/2|p|^2$, and $V$ is a potential depending only on $q$. This works in general cotangent bundles.