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Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question. Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ( …

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different fro …

Define the map $$P:TS^{n}\to S^{n} \;\;\;\text{by}\;\; P((x,v))=\frac{x+v}{\parallel x+v \parallel}$$ where $$TS^{n}=\{(x,v)\in S^{n} \ \times \mathbb{R}^{n+1}\mid v \perp x \}$$ This map is us …

Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property: For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X_H$ on $H^{-1 …

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point. Let $p: \mathbb{R}^n \to \mathbb{R}$ be a polynomial function. We define the Ham …