On a variational problem from one-dimensional nonlinear elasticity

We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty )$, initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms.

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The Poincaré-Cartan forms of one-dimensional variational integrals

Mathematica Slovaca ◽

10.1515/ms-2017-0439 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1381-1412

Author(s):

Veronika Chrastinová ◽

Václav Tryhuk

Keyword(s):

Differential Equations ◽

Variational Problem ◽

Ordinary Differential Equations ◽

Differential Forms ◽

Point Of View ◽

Full Generality ◽

One Dimensional ◽

Variational Integrals

AbstractFundamental concepts for variational integrals evaluated on the solutions of a system of ordinary differential equations are revised. The variations, stationarity, extremals and especially the Poincaré-Cartan differential forms are relieved of all additional structures and subject to the equivalences and symmetries in the widest possible sense. Theory of the classical Lagrange variational problem eventually appears in full generality. It is presented from the differential forms point of view and does not require any intricate geometry.

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A quasi-variational problem in nonlinear elasticity

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/bf01759310 ◽

1991 ◽

Vol 158 (1) ◽

pp. 331-389 ◽

Cited By ~ 1

Author(s):

Franco Tomarelli

Keyword(s):

Variational Problem ◽

Nonlinear Elasticity

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Asymptotic models for curved rods derived from nonlinear elasticity by Γ-convergence

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210507000194 ◽

2009 ◽

Vol 139 (5) ◽

pp. 1037-1070 ◽

Cited By ~ 17

Author(s):

Lucia Scardia

Keyword(s):

Nonlinear Elasticity ◽

Three Dimensional ◽

Curved Beam ◽

Rigorous Derivation ◽

One Dimensional ◽

Asymptotic Models ◽

Linearized Elasticity ◽

Curved Rods ◽

Γ Convergence ◽

The One

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dimension reduction starting from linearized elasticity.

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