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A Brief Primer on the legendre transformation

Journal of Applied Mathematics and Physics 2023年第11期11卷 3505-3518页

作者： Steven J. Kilner David L. Farnsworth Department of Mathematics Monroe Community College Rochester USA School of Mathematics and Statistics Rochester Institute of Technology Rochester USA

Guidance is offered for understanding and using the legendre transformation and its associated duality among functions and curves. The genesis of this paper was encounters with colleagues and students asking about the transformation. A main feature is simplicity of exposition, while keeping in mind the purpose or application for using the transformation.

关键词： legendre transformation Dual Curve Reflectivity Line of Support Tangent Line

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The Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

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Science China Mathematics 2022年第7期65卷 1485-1516页

作者： Ming Xu School of Mathematical Sciences Capital Normal UniversityBeijing 100048China

Let M_(t) be an isoparametric foliation on the unit sphere(S^(n−1)(1),g^(st))with d principal *** the spherical coordinatesinduced by M_(t),we construct a Minkowski norm with the representation F=r√2f(t),which generalizes the notions of(α,β)-norm and(α1,α2)-*** the technique of the spherical local frame,we givean exact and explicit answer to the question when F=r√2 f(t)really defines a Minkowski *** the similar technique,we study the Hessian isometry Φ between two Minkowski norms induced by M_(t),which preservesthe orientation and fixes the spherical ξ-*** aretwo ways to describe this Φ,either by a system of ODEs,or by its restriction toany normal plane for M_(t),which is then reduced to a Hessian isometry between Minkowski norms on R^(2) satisfying certain symmetry and(d)-*** d>2,we prove that this Φ can be obtained by gluing positive scalar multiplications and compositions of the legendre transformation and positive scalar multiplications,so it must satisfy the(d)-property for any orthogonal decomposition R^(n)=V'+V'',i.e.,for any nonzero x=x'+x'' and Φ(x)=x=x'+x''with x',x'∈V'and x'',x''∈V'',we have g_(x)^(F1)(x'',x)=g_(x)^(F2)x(x'',x).As byproducts,we prove the following *** the indicatrix(S_(F,g)),where F is a Minkowski norm induced by M_(t) and g is the Hessian metric,the foliation N_(t)=S_(F)∩R>_(0)M_(0) is *** Conjecture is valid for a Minkowski norm F induced by M_(t),i.e.,if its Hessian metric g is flat on R^(n)\legendre transformationwith n>2,then F is Euclidean.

关键词： Minkowski norm Hessian isometry Hessian metric isoparametric foliation Laugwitz Conjecture legendre transformation

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Symplectic Algorithm for Computing One-Dimensional Stationary State Schr（?）dinger Equation

Symplectic Algorithm for Computing One-Dimensional Stationar...

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CCAST“保结构算法及其应用”研讨会

作者： Liu Xuesben Liu Xiaoyan Zhou Zhongyan Ding PeizhuInstitute of Atomic and Molecular Physics, Jilin University, Changchun, China, 130023

＜正＞In this paper,one-dimensional stationary stateSchr（?）dinger equation is transformed to Hamiltonian canonicalequation by legendre transformation,and two methods for computingeigenvalues and eigenvectors are introduced by symplectic *** two methods are applied for one-dimensional harmonicscillator,Hydrogen atom and the double-well anharmonic *** results are reasonable.

关键词： legendre transformation symplectic schemes shooting method.

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