A Prescribed Energy Problem for a Singular Hamiltonian System with a Weak Force

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Abstract

We consider the existence of periodic solutions of a Hamiltonian system q ̈ + ∇V(q) = 0 (HS) such that 1 2| q ̇(t)|2+ V(q(t)) = H for all t, where q ∈ RN (N ≥ 3), H <0 is a given number, V(q) ∈ C2(RN\{0}, R) is a potential with a singularity, and ∇V/(q) denotes its gradient. We consider a potential V(q) which behaves like -l/|q|α (α ∈ (0, 2)). In particular, in case V(q) satisfies ∇V(q) q ≤ -α1V(q) for all q ∈ RN\{0}, and ∇V(q) q ≤ -α2V(q) for all 0 < |q| ≤ R0 for α1, α2 ∈ (0, 2), R0 > 0, we prove the existence of a generalized solution that may enter the singularity 0. Moreover, under the assumption V(q) = - 1 |q|α + W(q), where 0 < α < 2 and |q|αW(q), |q|α + 1 ∇W(q), |q|α + 22W(q) → 0 as |q| → 0, we estimate the number of collisions of generalized solutions. In particular, we get the existence of a classical (non-collision) solution of (HS) for α ∈ (l, 2) when N ≥ 4 and for α ∈ (4/3, 2) when N = 3.

Original languageEnglish
Pages (from-to)351-390
Number of pages40
JournalJournal of Functional Analysis
Volume113
Issue number2
DOIs
Publication statusPublished - 1993 May 1
Externally publishedYes

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Singular Systems
Hamiltonian Systems
Generalized Solution
Energy
Periodic Solution
Collision
Singularity
Estimate

ASJC Scopus subject areas

  • Analysis

Cite this

A Prescribed Energy Problem for a Singular Hamiltonian System with a Weak Force. / Tanaka, Kazunaga.

In: Journal of Functional Analysis, Vol. 113, No. 2, 01.05.1993, p. 351-390.

Research output: Contribution to journalArticle

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