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Critical point (mathematics)
About: Critical point (mathematics) is a research topic. Over the lifetime, 2040 publications have been published within this topic receiving 52101 citations.
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TL;DR: In this paper, general existence theorems for critical points of a continuously differentiable functional I on a real Banach space are given for the case in which I is even.
4,081 citations
Book•
01 Jul 1986TL;DR: The mountain pass theorem and its application in Hamiltonian systems can be found in this paper, where the saddle point theorem is extended to the case of symmetric functionals with symmetries and index theorems.
Abstract: An overview The mountain pass theorem and some applications Some variants of the mountain pass theorem The saddle point theorem Some generalizations of the mountain pass theorem Applications to Hamiltonian systems Functionals with symmetries and index theorems Multiple critical points of symmetric functionals: problems with constraints Multiple critical points of symmetric functionals: the unconstrained case Pertubations from symmetry Variational methods in bifurcation theory.
3,685 citations
Book•
08 Feb 1989
TL;DR: The direct method of the Calculus of Variations, Fenchel Transform and duality, Minimax Theorems for Indefinite Functional, Borsuk-Ulam Theorem and Index Theories, Lusternik-Schnirelman Theory and Multiple Periodic Solution with Fixed Energy, Morse-Ekeland Index, Morse Theory, and Morse Theory for Second Order Systems as discussed by the authors.
Abstract: 1 The Direct Method of the Calculus of Variations- 2 The Fenchel Transform and Duality- 3 Minimization of the Dual Action- 4 Minimax Theorems for Indefinite Functional- 5 A Borsuk-Ulam Theorem and Index Theories- 6 Lusternik-Schnirelman Theory and Multiple Periodic Solutions with Fixed Energy- 7 Morse-Ekeland Index and Multiple Periodic Solutions with Fixed Period- 8 Morse Theory- 9 Applications of Morse Theory to Second Order Systems- 10 Nondegenerate Critical Manifolds
2,036 citations
TL;DR: In this paper, a natural time-dependent generalization for the well-known pair distribution function $g(mathrm{r})$ of systems of interacting particles is given, which gives rise to a very simple and entirely general expression for the angular and energy distribution of Born approximation scattering by the system.
Abstract: A natural time-dependent generalization is given for the well-known pair distribution function $g(\mathrm{r})$ of systems of interacting particles. The pair distribution in space and time thus defined, denoted by $G(\mathrm{r}, t)$, gives rise to a very simple and entirely general expression for the angular and energy distribution of Born approximation scattering by the system. This expression is the natural extension of the familiar Zernike-Prins formula to scattering in which the energy transfers are not negligible compared to the energy of the scattered particle. It is therefore of particular interest for scattering of slow neutrons by general systems of interacting particles: $G$ is then the proper function in terms of which to analyze the scattering data.After defining the $G$ function and expressing the Born approximation scattering formula in terms of it, the paper studies its general properties and indicates its role for neutron scattering. The qualitative behavior of $G$ for liquids and dense gases is then described and the long-range part exhibited by the function near the critical point is calculated. The explicit expression of $G$ for crystals and for ideal quantum gases is briefly derived and discussed.
2,015 citations
TL;DR: In this article, a two-dimensional spin model with cubic or isotropic symmetry is mapped onto a solid-on-solid model, which leads to an analytic calculation of the critical point and some critical indices.
Abstract: A two-dimensional $n$-component spin model with cubic or isotropic symmetry is mapped onto a solid-on-solid model. Subject to some plausible assumptions this leads to an analytic calculation of the critical point and some critical indices for $\ensuremath{-}2l~nl~2$.
890 citations