A generalization of the classical Cesàro–Volterra path integral formula (original) (raw)
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CESÀRO–VOLTERRA PATH INTEGRAL FORMULA ON A SURFACE
Mathematical Models and Methods in Applied Sciences, 2009
If a symmetric matrix field e = (e ij ) of order three satisfies the Saint-Venant compatibility relations in a simply-connected open subset Ω of R 3 , then e is the linearized strain tensor field of a displacement field v of Ω, i.e. e = 1 2 (∇v T + ∇v) in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is smooth, the unknown displacement field v(x) at any point x ∈ Ω can be explicitly written as a path integral inside Ω with endpoint x, and whose integrand is an explicit function of the functions e ij and their derivatives. Now let ω be a simply-connected open subset in R 2 and let θ : ω → R 3 be a smooth immersion. If two symmetric matrix fields (γ αβ ) and (ρ αβ ) of order two satisfy appropriate compatibility relations in ω, then (γ αβ ) and (ρ αβ ) are the linearized change of metric and change of curvature tensor field corresponding to a displacement vector field η of the surface θ(ω).
A Cesàro–Volterra formula with little regularity
Journal de Mathématiques Pures et Appliquées, 2010
If a symmetric matrix field e of order three satisfies the Saint-Venant compatibility conditions in a simplyconnected domain Ω in R 3 , there then exists a displacement field u of Ω with e as its associated linearized strain tensor, i.e., e = 1 2 (∇u T + ∇u) in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is sufficiently smooth, the displacement u(x) at any point x ∈ Ω can be explicitly computed as a function of the matrix fields e and CURL e, by means of a path integral inside Ω with endpoint x. We assume here that the components of the field e are only in L 2 (Ω) (as in the classical variational formulation of three-dimensional linearized elasticity), in which case the classical path integral formula of Cesàro and Volterra becomes meaningless. We then establish the existence of a "Cesàro-Volterra formula with little regularity", which again provides an explicit solution u to the equation e = 1 2 (∇u T + ∇u) in this case. We also show how the classical Cesàro-Volterra formula can be recovered from the formula with little regularity when the field e is smooth. Interestingly, our analysis also provides as a by-product a variational problem that satisfies all the assumptions of the Lax-Milgram lemma, and whose solution is precisely the unknown displacement field u. It is also shown how such results may be used in the mathematical analysis of "intrinsic" linearized elasticity, where the linearized strain tensor e (instead of the displacement vector u as is customary) is regarded as the primary unknown. Résumé Une formule de Cesàro-Volterra avec peu de régularité. Si un champ e de matrices symétriques d'ordre trois vérifie les conditions de compatibilité de Saint-Venant dans un ouvert Ω simplement connexe de R 3 , alors il existe un champ de déplacements u de Ω ayant e comme tenseur linéarisé des déformations associé, i.e., e = 1 2 (∇u T + ∇u) dans Ω. Un résultat classique de Cesàro et Volterra affirme que, si le champ e est suffisamment régulier, le déplacement u(x) en chaque point x ∈ Ω peutêtre calculé explicitement en fonction des champs de matrices e et CURL e, au moyen d'une intégrale curviligne dans Ω ayant x comme extrémité. On suppose ici que les composantes du champ e sont seulement dans L 2 (Ω) (comme dans la formulation variationnelle classique de l'élasticité linéarisée tri-dimensionnelle), auquel cas la formule classique de Cesàro-Volterra n'a plus de sens. Onétablit alors une "formule de Cesàro-Volterra avec peu de régularité", qui donnè a nouveau une solution explicite u de l'équation e = 1 2 (∇u T + ∇u) dans ce cas. On montre aussi comment la
06 09 02 3 v 1 4 S ep 2 00 6 Path integrals and boundary conditions
2006
The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman’s approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.
We discuss the path integral formulation of quantum mechanics and use it to derive the S matrix in terms of Feynman diagrams. We generalize to quantum field theory, and derive the generating functional Z[J] and n-point correlation functions for free scalar field theory. We develop the generating functional for self-interacting fields and discuss φ 4 and φ 3 theory.
On the Laplacian vector fields theory in domains with rectifiable boundary
Mathematical Methods in The Applied Sciences, 2006
Given a domain Ω in ℝ3 with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ⊂ ℝ3 into a sum u = u++u− were u± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd.
Some Applications of Integral Formulas in Riemannian Geometry and PDE?s
Milan Journal of Mathematics, 2003
We show some applications of integral formulas, most notably the Hsiung-Minkowski formulas, the Rellich-Pohozaev identities, and variations thereof, to the study of geometric problems and PDE's. Our presentation aims at underlining the common geometrical and analytical features of such formulas. 220 S. Pigola, M. Rigoli, and A.G. Setti Vol. 71 (2003) This can be simply understood with the following two examples. Let (M, g) be compact and m-dimensional and let λ 1 denote the first (nonzero) eigenvalue of the problem
Path integrals and boundary conditions
Eprint Arxiv Quant Ph 0609023, 2006
The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.