Axisymmetry of critical points for the Onsager functional

A simple proof is given of the classical result (Fatkullin I, Slastikov V. 2005 Critical points of the Onsager functional on a sphere. Nonlinearity 18 , 2565–2580 ( doi:10.1088/0951-7715/18/6/008 ); Liu H et al. 2005 Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential. Commun. Math. Sci. 3 , 201–218 ( doi:10.4310/CMS.2005.v3.n2.a7 )) that critical points for the Onsager functional with the Maier-Saupe molecular interaction are axisymmetric, including the case of stable critical points with an additional dipole-dipole interaction (Zhou H et al. 2007 Characterization of stable kinetic equilibria of rigid, dipolar rod ensembles for coupled dipole-dipole and Maier-Saupe potentials. Nonlinearity 20 , 277–297 ( doi:10.1088/0951-7715/20/2/003 )). The proof avoids spherical polar coordinates, instead using an integral identity on the sphere S 2 . For general interactions with absolutely continuous kernels the smoothness of all critical points is established, generalizing a result in (Vollmer MAC. 2017 Critical points and bifurcations of the three-dimensional Onsager model for liquid crystals. Archive for Rational Mechanics and Analysis 226 , 851–922 ( doi:10.1007/s00205-017-1146-8 )) for the Onsager interaction. It is also shown that non-axisymmetric critical points exist for a wide variety of interactions including that of Onsager. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

ECONOMIC DYNAMICS AND THE CALCULUS OF VARIATIONS IN THE INTERWAR PERIOD

Analogies with rational mechanics played a pivotal role in the search for formal models in economics. In the period between the two world wars, a small group of mathematical economists tried to extend this view from statics to dynamics. The main result was the extensive application of calculus of variations to obtain a dynamic representation of economic variables. This approach began with the contributions put forward by Griffith C. Evans, a mathematician who, in the first phase of his scientific career, published widely in economics. Evans’s research was further developed by his student Charles Roos. At the international level, this dynamic approach found its main followers in Italy, within the Paretian tradition. During the 1930s, Luigi Amoroso, the leading exponent of the Paretian School, made major contributions, along with his student Giulio La Volpe, that anticipated the concept of temporary equilibrium. The analysis of the application of the calculus of variations to economic dynamics in the interwar period raises a set of questions on the application of mathematics designed to study mechanics and physics to economics.

The Economic Dynamics and the Calculus of Variations in the Interwar Period

Analogies with rational mechanics played a pivotal role in the search for formal models in economics. In the period between the two world wars, a small group of mathematical economists tried to extend this view from statics to dynamics. The main result was the extensive application of calculus of variations to obtain a dynamic representation of e-conomic variables. This approach began with the contributions put forward by Griffith C.Evans, a mathematician who in the first phase of his scientific career published wi-dely in economics. Evans' research was further developed by his student, Charles Roos. At the international level, this dynamic approach found its main followers in Italy, within the Paretian tradition. During the 1930s, Luigi Amoroso, the leading exponent of the Paretian School, made major contributions along with his student, Giulio La Volpe that anticipated the concept of temporary equilibrium. The analysis of the application of the calculus of variations to economic dynamics in the interwar period raises a set of questions on the application of mathematics designed to study mechanics and physics to economics