SPENCER OPERATOR AND APPLICATIONS: From Continuum Mechanics to Mathematical physics
نویسنده
چکیده
Let us revisit briefly the foundation of n-dimensional elasticity theory as it can be found today in any textbook, restricting our study to n = 2 for simplicity. If x = (x, x) is a point in the plane and ξ = (ξ(x), ξ(x)) is the displacement vector, lowering the indices by means of the Euclidean metric, we may introduce the ”small” deformation tensor ǫ = (ǫij = ǫij = (1/2)(∂iξj + ∂jξi)) with n(n + 1)/2 = 3 (independent) components (ǫ11, ǫ12 = ǫ21, ǫ22). If we study a part of a deformed body, for example a thin elastic plane sheet, by means of a variational principle, we may introduce the local density of free energy φ(ǫ) = φ(ǫij |i ≤ j) = φ(ǫ11, ǫ12, ǫ22) and vary the total free energy F = ∫
منابع مشابه
Notes on Noncommutative Geometry
Noncommutative geometry has roots in and is a synthesis of a number of diverse areas of mathematics, including: • Hilbert space and single operator theory; • Operator algebras (C*-algebras and von Neumann algebras); • Spin geometry – Dirac operators – index theory; • Algebraic topology – homological algebra. It has certainly also been inspired by quantum mechanics, and, besides feedback to the ...
متن کاملAlgebraic Quantum Field Theory and Operator Algebras
In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as " why mathematicians are/should be interested in algebraic quantum field theory " would be equally fitting. Besides a presentation of the framework and the main results of local quantum physics these notes may s...
متن کاملNonlinear Picone identities to Pseudo $p$-Laplace operator and applications
In this paper, we derive a nonlinear Picone identity to the pseudo p-Laplace operator, which contains some known Picone identities and removes a condition used in many previous papers. Some applications are given including a Liouville type theorem to the singular pseudo p-Laplace system, a Sturmian comparison principle to the pseudo p-Laplace equation, a new Hardy type inequality with weight an...
متن کاملar X iv : m at h - ph / 0 10 20 18 v 4 1 8 A pr 2 00 1 Lectures on Algebraic Quantum
In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as “why mathematicians are/should be interested in algebraic quantum field theory” would be equally fitting. Besides a presentation of the framework and the main results of local quantum physics these notes may ser...
متن کاملPositive Operator Measures, Generalised Imprimitivity Theorem and Their Applications
Introduction In the common textbook presentation of quantum mechanics the observables of a quantum system are represented by selfadjoint operators, or, equivalently , by spectral measures. The origin of this point of view dates back to the very beginning of quantum theory. Its rigorous mathematical formulation is mainly due to von Neumann [53], and, for a more recent and complete review, we ref...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011