The Rabinowitz minimal periodic solution conjecture on partially convex reversible Hamiltonian systems and brake subharmonics
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914066890489856 |
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| author | Zhou, Yuting |
| author_facet | Zhou, Yuting |
| contents | Under weaker regularity and compactness assumptions, we find the mountain-pass essential point, which is a novel extension of the classical Ambrosetti-Rabinowitz mountain pass theorem. We study the reversible superquadratic autonomous Hamiltonian systems whose Hamiltonian $H(p,q)$ is strictly convex in the position $q\in\mathbf{R}^n$ and prove that for every $T>0$, the system has a $T$-periodic brake solution $\bar x$ with minimal period $T$, provided the Hessian $H_{pp}(\bar x(t))\in\mathbf{R}^{n\times n}$ is semi-positive definite for $t\in\mathbf{R}$ or $n=1$. For brake subharmonics of general reversible nonautonomous Hamiltonian systems, we also get some new results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25567 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Rabinowitz minimal periodic solution conjecture on partially convex reversible Hamiltonian systems and brake subharmonics Zhou, Yuting Symplectic Geometry Dynamical Systems 34C25, 58E05, 53D12, 37J99 Under weaker regularity and compactness assumptions, we find the mountain-pass essential point, which is a novel extension of the classical Ambrosetti-Rabinowitz mountain pass theorem. We study the reversible superquadratic autonomous Hamiltonian systems whose Hamiltonian $H(p,q)$ is strictly convex in the position $q\in\mathbf{R}^n$ and prove that for every $T>0$, the system has a $T$-periodic brake solution $\bar x$ with minimal period $T$, provided the Hessian $H_{pp}(\bar x(t))\in\mathbf{R}^{n\times n}$ is semi-positive definite for $t\in\mathbf{R}$ or $n=1$. For brake subharmonics of general reversible nonautonomous Hamiltonian systems, we also get some new results. |
| title | The Rabinowitz minimal periodic solution conjecture on partially convex reversible Hamiltonian systems and brake subharmonics |
| topic | Symplectic Geometry Dynamical Systems 34C25, 58E05, 53D12, 37J99 |
| url | https://arxiv.org/abs/2509.25567 |