DOCSLIB.ORG

Variational Theory in the Context of General Relativity Abstract

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Variational Theory in the Context of General Relativity Abstract Variational theory in the context of general relativity Hiroki Takeda1, ∗ 1Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan (Dated: October 25, 2018) Abstract This is a brief note to summarize basic concepts of variational theory in general relativity for myself. This note is mainly based on Hawking(2014)"Singularities and the geometry of spacetime" and Wald(1984)"General Relativity". ∗ [email protected] 1 I. BRIEF REVIEW OF GENERAL RELATIVITY AND VARIATIONAL THEORY A. Variation defnition I.1 (Action) Let M be a differentiable manifold and Ψ be some tensor fields on M . Ψ are simply called the fields. Consider the function of Ψ, S[Ψ]. This S is a map from the set of the fields on M to the set of real numbers and called the action. a···b Ψ actually represents some fields Ψ(i) c···d(i = 1; 2; ··· ; n). I left out the indices i that indicate the number of fields and the abstract indices for the tensors. The tensor of the a···b a···b tensor fields Ψ(i) c···d(i = 1; 2; ··· ; n) at x 2 M is written by Ψ(i)jx c···d(i = 1; 2; ··· ; n). Similarly I denote the tensor of the tensor fields Ψ at x 2 M by Ψjx. defnition I.2 (Variation) Let D ⊂ M be a submanifold of M and Ψjx(u), u 2 (−ϵ, ϵ), x 2 M be the tensors of one-parameter family of the tensor fields Ψ(u), u 2 (−ϵ, ϵ) at x 2 M such that (1) Ψjx(0) = Ψ in M ; (1) (2) Ψjx(u) = Ψ in M − D: (2) Then I define the variation of the fields as follows, dΨ(u) δΨ := : (3) du u=0 In this note, I suppose the existence of the derivatives dS=duju=0 for all one-parameter family Ψ(u). defnition I.3 (Functional derivative) I assume the existence of the smooth tensor fields χ which is dual to such that Z dS = χδΨ; (4) du u=0 D for any one-parameter family. 1 Then we say that S is functionally differentiable at Ψ(0). We call χ the functional derivative of S and denote it as δS := χ.2 (5) δΨ Ψ0 1 If the type of tensor field of Ψ is (k; l), that of χ is (l; k). Here contraction of all indices in the integral is understood. 2 More generally, if there exists a tensor distribution χ such that dS=duju=0 = χ[δΨ], we say that S is 2 defnition I.4 (Lagrangian) Consider a functional form of S Z S[Ψ] = L[Ψ] (6) D where L is a local function of and a finite number of its derivatives, k Ljx = L(Ψ(x); rΨ(x); ··· ; r Ψ(x)): (7) L is called by Lagrangian density. defnition I.5 (Variational principle) Let S be functionally differentiable. We require that the fields Ψ maximize S, dS = 0: (8) du u=0 This principle is called by variational principle. Hence, the following equation is hold by the definition of the functional derivative and the following fundamental lemma of variational calculus. δS = 0: (9) δΨ Ψ0 This gives us the fundamental equations for the fields. lemma I.1 (Fundamental lemma of variational calculus) Let D ⊂ M be a subman- ifold of M and Ψjx(u), u 2 (−ϵ, ϵ), x 2 M be any one-parameter family satisfying a proper boundary conditions. The following is held. Z dS = χδΨ = 0 ) χ = 0: (10) du u=0 D This statement is called by fundamental lemma of variational calculus. B. General Relativity defnition I.6 (Space-time) Space-time is the set of events. In general relativity, the mathematical model of space-time is (M , g) where M is a connected four-dimensional Hausdorff C1 manifold and g is a Lorentz metric 3on M . functionally differentiable and call χ the functional derivative of S at Ψ(0). Here the tensor distribution is a distribution whose type is tensor, so called generalized function in Japanese. The bracket on the R right hand side denotes the scalar product (distribution) which is defined by χ[δΨ] =< χ, δΨ >= D χδΨ: (Schwarz, 2008) 3 Its signature is +2 3 First, I derive the fundamental equations for fields by variational method. Hereafter I a···b refer to the fields Ψ as Ψ(i) c···d; (i = 1; 2; ··· ; n) without omitting the indices. I assume the form of the action is Lagrangian form and the Lagrangian depends on the fields and their first covariant derivatives. a···b Consider the variation of the fields Ψ(i) c···d; (i = 1; 2; ··· ; n). Then Z ! dS @L @L a···b − a···b = Σ ··· δΨ(i) c···d ··· δ(Ψ(i) c···d;e) : (11) i a b a b du u=0 D @Ψ(i) c···d @Ψ(i) c···d;e a···b a···b Sinceδ(Ψ(i) c···d;e) = (δΨ(i) c···d);e, the second term can be written as 2 3 Z ! ! @L @L 4 a···b − a···b 5 Σ a···b δΨ(i) c···d a···b δΨ(i) c···d : (12) i D @Ψ(i) c···d;e @Ψ(i) c···d;e ;e ;e According to Stokes' theorem, the first term can be written as Z Z a a Q ;a = Q dσa; (13) D @D where e @L a···b Q := Σ a···b δΨ(i) c···d: (14) i @Ψ(i) c···d;e a···b This integral have to vanish because δΨ(i) c···d = 0 is held at the boundary @D from the second condition of variation Eq. (2). Finally we get the following expression, 0 1 Z ! dS @L @L @ − A a···b = Σ a···b a···b δΨ(i) c···d = 0: (15) du i D @Ψ(i) c···d @Ψ(i) c···d;e u=0 ;e Thus, we can get the following Euler-Lagrange equations from fundamental lemma of vari- ational calculus Eq. (10), ! @L − @L χ = a···b a···b = 0: (16) @Ψ(i) c···d @Ψ(i) c···d;e ;e Then we get the energy-momentum tensor from the Lagrangian by considering a variation with respect to the metric. Suppose a variation of the metric gab(u; x) does not change the a···b fields Ψ(i) c···d. Then, Z ! Z dS @L a···b @L @ϵ = Σ ··· δ(Ψ(i) c···d;e) + δgab + L δgab: (17) i a b du u=0 D @Ψ(i) c···d;e @gab D @gab 4 R R The last term comes from the fact that the integral is defined by f = fϵ. Since ϵ = −1 1=2 1 2 3 4 (4!) η, ηabcd = (−g) 4!δ[a δb δc δd] and g := det(gab), @ηabcd 1 −1=2 @g 1 2 3 4 = − (−g) 4!δ[a δb δc δd] : (18) @gef 2 @gef Thus, we obtain @ϵ 1 = gabϵ: (19) @gab 2 Here I use the the fact that the cofactor of the matrix can be expressed as a product of the component of the inverse matrix and the determinant od the matrix, that is the relation ef a···b @g=@gef = g g to evaluate @g=@gef . The first term arises because δ(Ψ(i) c···d;e) does a···b not necessarily become zero unlike δΨ(i) c···d due to a variation of the components of the a a a connection Γ bc induced by a variation of the metric. @Γ bc=@u = δΓ bc should be a tensor because the difference between two connections should be a tensor. This can be calculated as follows, 1 δΓa = gadf(δg ) + (δg ) − (δg ) g: (20) bc 2 db ;c dc ;b bc ;d One can verify this equation by checking that this equation is held in normal coordinates at a p where the components Γ bc and the coordinate derivatives of the components gab vanish because this equation is a tensor equation. For Riemann normal coordinate, ( ) 1 @δg @δg @δg (l:h:s) = gad cd + bd − bc ; (21) 2 @xb @xc @xd a and, since the components Γ bc at p induced by the covariant derivatives vanish, 1 (r:h:s) = gadf(δg ) + (δg ) − (δg ) g = (l:h:s:): (22) 2 db ;c dc ;b bc ;d If the tensor equation is held in a specified coordinate, it is held in any coordinates. This method is often used to derive equations in relativity. However, One should be careful to calculations including the variation and the covariant derivative. Note that δ(A + B) = δA + δB, δ(AB) = (δA)B + A(δB) and δ(@A) = @(δA). Especially, the variation operator does not commute the covariant derivative for any tensors g (here its type is (0; 2) as an example), r r d d a (δgbc) = δ ( agbc) + gdc δΓba + gbd δΓac; (23) when one consider the variation of the metric because a change of the connection is induced. This can be easily derived by the above three equations. On the other hand, the variation 5 operator does commute the covariant derivative when you consider the variation of the fields except for the metric as I mentioned above. Eq. (19) can be also derived by this equation straightforward. One also should be careful to that the since the metricity condition holds only for the unperturbed metric gab, r(g(0)) = 0 does not mean that r(g(∆u)) = 0. Using to leave only terms multiplied by δgab, The first term in Eq.
Load more

Recommended publications
  • A Short Introduction to the Quantum Formalism[S]
    A short introduction to the quantum formalism[s] François David Institut de Physique Théorique CNRS, URA 2306, F-91191 Gif-sur-Yvette, France CEA, IPhT, F-91191 Gif-sur-Yvette, France [email protected] These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May- June 2012; (ii) a previous letter [Dav11] on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed. Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics). These notes do not cover the historical and philosophical aspects of quantum physics. arXiv:1211.5627v1 [math-ph] 24 Nov 2012 Preprint IPhT t12/042 ii CONTENTS Contents 1 Introduction 1-1 1.1 Motivation . 1-1 1.2 Organization . 1-2 1.3 What this course is not! . 1-3 1.4 Acknowledgements . 1-3 2 Reminders 2-1 2.1 Classical mechanics .
    [Show full text]
  • Functional Integration on Paracompact Manifods Pierre Grange, E
    Functional Integration on Paracompact Manifods Pierre Grange, E. Werner To cite this version: Pierre Grange, E. Werner. Functional Integration on Paracompact Manifods: Functional Integra- tion on Manifold. Theoretical and Mathematical Physics, Consultants bureau, 2018, pp.1-29. hal- 01942764 HAL Id: hal-01942764 https://hal.archives-ouvertes.fr/hal-01942764 Submitted on 3 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Functional Integration on Paracompact Manifolds Pierre Grangé Laboratoire Univers et Particules, Université Montpellier II, CNRS/IN2P3, Place E. Bataillon F-34095 Montpellier Cedex 05, France E-mail: [email protected] Ernst Werner Institut fu¨r Theoretische Physik, Universita¨t Regensburg, Universita¨tstrasse 31, D-93053 Regensburg, Germany E-mail: [email protected] ........................................................................ Abstract. In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More recently Cartier and DeWitt-Morette (CDWM) focused on the definition of a proper integration measure and established a rigorous mathematical formulation of functional integration. CDWM’s central observation relates to the distributional nature of fields, for it leads to the identification of distribution functionals with Schwartz space test functions as density measures.
    [Show full text]
  • Well-Defined Lagrangian Flows for Absolutely Continuous Curves of Probabilities on the Line
    Graduate Theses, Dissertations, and Problem Reports 2016 Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Amsaad, Mohamed H., "Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line" (2016). Graduate Theses, Dissertations, and Problem Reports. 5098. https://researchrepository.wvu.edu/etd/5098 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Adrian Tudorascu, Ph.D., Chair Harumi Hattori, Ph.D. Harry Gingold, Ph.D. Tudor Stanescu, Ph.D. Charis Tsikkou, Ph.D. Department Of Mathematics Morgantown, West Virginia 2016 Keywords: Continuity Equation, Lagrangian Flow, Optimal Transport, Wasserstein metric, Wasserstein space.
    [Show full text]
  • Functional Derivative
    Appendix Functional Derivative Appendix A: Calculus of Variation Consider a mapping from a vector space of functions to real number. Such mapping is called functional. One of the most representative examples of such a mapping is the action functional of analytical mechanics, A ,which is defined by Zt2 A½xtðÞ Lxt½ðÞ; vtðÞdt ðA:1Þ t1 where t is the independent variable, xtðÞis the trajectory of a particle, vtðÞ¼dx=dt is the velocity of the particle, and L is the Lagrangian which is the difference of kinetic energy and the potential: m L ¼ v2 À uxðÞ ðA:2Þ 2 Calculus of variation is the mathematical theory to find the extremal function xtðÞwhich maximizes or minimizes the functional such as Eq. (A.1). We are interested in the set of functions which satisfy the fixed boundary con- ditions such as xtðÞ¼1 x1 and xtðÞ¼2 x2. An arbitrary element of the function space may be defined from xtðÞby xtðÞ¼xtðÞþegðÞt ðA:3Þ where ε is a real number and gðÞt is any function satisfying gðÞ¼t1 gðÞ¼t2 0. Of course, it is obvious that xtðÞ¼1 x1 and xtðÞ¼2 x2. Varying ε and gðÞt generates any function of the vector space. Substitution of Eq. (A.3) to Eq. (A.1) gives Zt2 dx dg aðÞe A½¼xtðÞþegðÞt L xtðÞþegðÞt ; þ e dt ðA:4Þ dt dt t1 © Springer Science+Business Media Dordrecht 2016 601 K.S. Cho, Viscoelasticity of Polymers, Springer Series in Materials Science 241, DOI 10.1007/978-94-017-7564-9 602 Appendix: Functional Derivative From Eq.
    [Show full text]
  • 35 Nopw It Is Time to Face the Menace
    35 XIV. CORRELATIONS AND SUSCEPTIBILITY A. Correlations - saddle point approximation: the easy way out Nopw it is time to face the menace - path integrals. We were tithering on the top of this ravine for long enough. We need to take the plunge. The meaning of the path integral is very simple - tofind the partition function of a system locallyfluctuating, we need to consider the all the possible patterns offlucutations, and the only way to do this is through a path integral. But is the path integral so bad? Well, if you believe that anyfluctuation is still quite costly, then the path integral, like the integral over the exponent of a quadratic function, is pretty much determined by where the argument of the exponent is a minimum, plus quadraticfluctuations. These belief is translated into equations via the saddle point approximation. Wefind the minimum of the free energy functional, and then expand upto second order in the exponent. 1 1 F [m(x),h] = ddx γ( m (x))2 +a(T T )m (x)2 + um (x)4 hm + ( m (x))2 ∂2F (219) L ∇ min − c min 2 min − min 2 − min � � � I write here deliberately the vague partial symbol, in fact, this should be the functional derivative. Does this look familiar? This should bring you back to the good old classical mechanics days. Where the Lagrangian is the only thing you wanted to know, and the Euler Lagrange equations were the only guidance necessary. Non of this non-commuting business, or partition function annoyance. Well, these times are back for a short time! In order tofind this, we do a variation: m(x) =m min +δm (220) Upon assignment wefind: u F [m(x),h] = ddx γ( (m +δm)) 2 +a(T T )(m +δm) 2 + (m +δm) 4 h(m +δm) (221) L ∇ min − c min 2 min − min � � � Let’s expand this to second order: d 2 2 d x γ (mmin) + δm 2γ m+γ( δm) → ∇ 2 ∇ · ∇ ∇ 2 +a (mmin) +δm 2am+aδm � � ∇ u 4 ·3 2 2 (222) + 2 mmin +2umminδm +3umminδm hm hδm] − min − Collecting terms in power ofδm, thefirst term is justF L[mmin].
    [Show full text]
  • On the Solution and Application of Rational Expectations Models With
    On the Solution and Application of Rational Expectations Models with Function-Valued States⇤ David Childers† November 16, 2015 Abstract Many variables of interest to economists take the form of time varying distribu- tions or functions. This high-dimensional ‘functional’ data can be interpreted in the context of economic models with function valued endogenous variables, but deriving the implications of these models requires solving a nonlinear system for a potentially infinite-dimensional function of infinite-dimensional objects. To overcome this difficulty, I provide methods for characterizing and numerically approximating the equilibria of dynamic, stochastic, general equilibrium models with function-valued state variables by linearization in function space and representation using basis functions. These meth- ods permit arbitrary infinite-dimensional variation in the state variables, do not impose exclusion restrictions on the relationship between variables or limit their impact to a finite-dimensional sufficient statistic, and, most importantly, come with demonstrable guarantees of consistency and polynomial time computational complexity. I demon- strate the applicability of the theory by providing an analytical characterization and computing the solution to a dynamic model of trade, migration, and economic geogra- phy. ⇤Preliminary, subject to revisions. For the latest version, please obtain from the author’s website at https://sites.google.com/site/davidbchilders/DavidChildersFunctionValuedStates.pdf †Yale University, Department of Economics. The author gratefully acknowledges financial support from the Cowles Foundation. This research has benefited from discussion and feedback from Peter Phillips, Tony Smith, Costas Arkolakis, John Geanakoplos, Xiaohong Chen, Tim Christensen, Kieran Walsh, and seminar participants at the Yale University Econometrics Seminar, Macroeconomics Lunch, and Financial Markets Reading Group.
    [Show full text]
  • Characteristic Hypersurfaces and Constraint Theory
    Characteristic Hypersurfaces and Constraint Theory Thesis submitted for the Degree of MSc by Patrik Omland Under the supervision of Prof. Stefan Hofmann Arnold Sommerfeld Center for Theoretical Physics Chair on Cosmology Abstract In this thesis I investigate the occurrence of additional constraints in a field theory, when formulated in characteristic coordinates. More specifically, the following setup is considered: Given the Lagrangian of a field theory, I formulate the associated (instantaneous) Hamiltonian problem on a characteristic hypersurface (w.r.t. the Euler-Lagrange equations) and find that there exist new constraints. I then present conditions under which these constraints lead to symplectic submanifolds of phase space. Symplecticity is desirable, because it renders Hamiltonian vector fields well-defined. The upshot is that symplecticity comes down to analytic rather than algebraic conditions. Acknowledgements After five years of study, there are many people I feel very much indebted to. Foremost, without the continuous support of my parents and grandparents, sister and aunt, for what is by now a quarter of a century I would not be writing these lines at all. Had Anja not been there to show me how to get to my first lecture (and in fact all subsequent ones), lord knows where I would have ended up. It was through Prof. Cieliebaks inspiring lectures and help that I did end up in the TMP program. Thank you, Robert, for providing peculiar students with an environment, where they could forget about life for a while and collectively find their limits, respectively. In particular, I would like to thank the TMP lonely island faction.
    [Show full text]
  • Current-Density Functionals in Extended Systems
    Current-Density Functionals in Extended Systems Arjan Berger Het promotieonderzoek beschreven in dit proefschrift werd uitgevoerd in de theoretische chemie groep van het Materials Science Centre (MSC) aan de Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen. Het onderzoek werd financieel mogelijk gemaakt door de Stichting voor Fundamenteel Onderzoek der Materie (FOM). Arjan Berger, Current-Density Functionals in Extended Systems, Proefschrift Rijksuniversiteit Groningen. c J. A. Berger, 2006. RIJKSUNIVERSITEIT GRONINGEN Current-Density Functionals in Extended Systems Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op maandag 2 oktober 2006 om 16.15 uur door Jan Adriaan Berger geboren op 24 februari 1978 te Steenwijk Promotor: prof.dr.R.Broer Copromotores: dr. R. van Leeuwen dr. ir. P. L. de Boeij Beoordelingscommissie: prof. dr. A. G¨orling prof. dr. J. Knoester prof. dr. G. Vignale ISBN: 90-367-2758-8 voor mijn ouders vi Contents General Introduction xi 0.1 Introduction................................. xi 0.2 OverviewoftheThesis.. .. .. .. .. .. .. .. .. .. .. .. xiv 1 Density-Functional Theory 1 1.1 Introduction................................. 1 1.2 Preliminaries ................................ 1 1.3 Hohenberg-Kohn Theorem for Nondegenerate Ground States ..... 3 1.4 TheFunctionalDerivative . 6 1.5 Kohn-ShamTheory............................. 7 1.6 Hohenberg-Kohn Theorem for Degenerate Ground States . ..... 10 1.7 The Lieb Functional and Noninteracting v-Representability . 13 1.8 Exchange-CorrelationFunctionals. .... 17 2 Time-Dependent (Current-)Density-Functional Theory 21 2.1 Introduction................................. 21 2.2 Preliminaries ................................ 21 2.3 TheRunge-GrossTheorem . 23 2.4 Time-DependentKohn-ShamTheory. 26 2.5 The Adiabatic Local Density Approximation .
    [Show full text]
  • Strain Gradient Visco-Plasticity with Dislocation Densities Contributing to the Energy 1 Introduction
    Strain gradient visco-plasticity with dislocation densities contributing to the energy Matthias R¨ogerand Ben Schweizer1 April 18, 2017 Abstract: We consider the energetic description of a visco-plastic evolution and derive an existence result. The energies are convex, but not necessarily quadratic. Our model is a strain gradient model in which the curl of the plastic strain contributes to the energy. Our existence results are based on a time-discretization, the limit procedure relies on Helmholtz decompositions and compensated compactness. Key-words: visco-plasticity, strain gradient plasticity, energetic solution, div-curl lemma MSC: 74C10, 49J45 1 Introduction The quasi-stationary evolution of a visco-plastic body is analyzed in an energetic approach. We use the framework of infinitesimal plasticity with an additive decom- position of the strain. The equations are described with three functionals, the elastic energy, the plastic energy, and the dissipation. The three functionals are assumed to be convex, but not necessarily quadratic. In this sense, we study a three-fold non-linear system. Our interest is to include derivatives of the plastic strain p in the free energy. We are therefore dealing with a problem in the context of strain gradient plasticity. Of particular importance are contributions of curl(p) to the plastic energy, since this term measures the density of dislocations. Attributing an energy to plastic deformations means that hardening of the material is modelled. Since derivatives of p contribute to the energy, the model introduces a length scale in the plasticity problem; this is desirable for the explanation of some experimental results. We treat a model that was introduced in [16], with analysis available in [11], [21], and [22].
    [Show full text]
  • 15 Functional Derivatives
    15 Functional Derivatives 15.1 Functionals A functional G[f] is a map from a space of functions to a set of numbers. For instance, the action functional S[q] for a particle in one dimension maps the coordinate q(t), which is a function of the time t, into a number—the action of the process. If the particle has mass m and is moving slowly and freely, then for the interval (t1,t2) its action is t2 m dq(t) 2 S [q]= dt . (15.1) 0 2 dt Zt1 ✓ ◆ If the particle is moving in a potential V (q(t)), then its action is t2 m dq(t) 2 S[q]= dt V (q(t)) . (15.2) 2 dt − Zt1 " ✓ ◆ # 15.2 Functional Derivatives A functional derivative is a functional d δG[f][h]= G[f + ✏h] (15.3) d✏ ✏=0 of a functional. For instance, if Gn[f] is the functional n Gn[f]= dx f (x) (15.4) Z 626 Functional Derivatives then its functional derivative is the functional that maps the pair of functions f,h to the number d δG [f][h]= G [f + ✏h] n d✏ n ✏=0 d = dx (f(x)+ ✏h(x))n d✏ Z ✏=0 n 1 = dx nf − (x)h(x). (15.5) Z Physicists often use the less elaborate notation δG[f] = δG[f][δ ] (15.6) δf(y) y in which the function h(x)isδ (x)=δ(x y). Thus in the preceding y − example δG[f] n 1 n 1 = dx nf − (x)δ(x y)=nf − (y).
    [Show full text]
  • Instantaneous Ionization Rate As a Functional Derivative
    ARTICLE DOI: 10.1038/s42005-018-0085-5 OPEN Instantaneous ionization rate as a functional derivative I.A. Ivanov1,2, C. Hofmann 2, L. Ortmann2, A.S. Landsman2,3, Chang Hee Nam1,4 & Kyung Taec Kim1,4 1234567890():,; The notion of the instantaneous ionization rate (IIR) is often employed in the literature for understanding the process of strong field ionization of atoms and molecules. This notion is based on the idea of the ionization event occurring at a given moment of time, which is difficult to reconcile with the conventional quantum mechanics. We describe an approach defining instantaneous ionization rate as a functional derivative of the total ionization probability. The definition is based on physical quantities, such as the total ionization probability and the waveform of an ionizing pulse, which are directly measurable. The defi- nition is, therefore, unambiguous and does not suffer from gauge non-invariance. We com- pute IIR by numerically solving the time-dependent Schrödinger equation for the hydrogen atom in a strong laser field. In agreement with some previous results using attoclock methodology, the IIR we define does not show measurable delay in strong field tunnel ionization. 1 Center for Relativistic Laser Science, Institute for Basic Science, Gwangju 61005, Korea. 2 Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, 01187 Dresden, Germany. 3 Max Planck Center for Attosecond Science/Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea. 4 Department of Physics and Photon Science, GIST, Gwangju 61005, Korea. Correspondence and requests for materials should be addressed to I.A.I.
    [Show full text]
  • Functional Ito Calculus and Functional Kolmogorov Equations∗
    Functional Ito calculus and functional Kolmogorov equations∗ Rama CONT Lecture Notes of the Barcelona Summer School on Stochastic Analysis Centre de Recerca Matem`atica,July 2012. Abstract The Functional Ito calculus is a non-anticipative functional calculus which extends the Ito calculus to path-dependent functionals of stochas- tic processes. We present an overview of the foundations and key re- sults of this calculus, first using a pathwise approach, based on a notion of directional derivative introduced by Dupire, then using a probabilis- tic approach, which leads to a weak, Sobolev-type, functional calculus for square-integrable functionals of semimartingales, without any smoothness assumption on the functionals. The latter construction is shown to have connections with the Malliavin calculus and leads to computable mar- tingale representation formulae. Finally, we show how the Functional Ito Calculus may be used to extend the connections between diffusion processes and parabolic equations to a non-Markovian, path-dependent setting: this leads to a new class of 'path-dependent' partial differential equations, Functional Kolmogorov equations, of which we study the key properties. Keywords: stochastic calculus, functional Ito calculus, Malliavin calculus, change of variable formula, functional calculus, martingale, Kolmogorov equations, path-dependent PDE. Published as: Functional It^ocalculus and functional Kolmogorov equations, in: V. Bally, L Caramellino, R Cont: Stochastic Integration by Parts and Functional It^oCal- culus, Advanced Courses in Mathematics CRM Barcelona, pages 123{216. ∗These notes are based on a series of lectures given at various summer schools: Tokyo (Tokyo Metropolitan University, July 2010), Munich (Ludwig Maximilians University, Oct 2010), the Spring School "Stochastic Models in Finance and Insurance" (Jena, 2011), the In- ternational Congress for Industrial and Applied Mathematics (Vancouver 2011), the Barcelona Summer School on Stochastic Analysis (Barcelona, July 2012).
    [Show full text]