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Appendix 1 Variational Derivation of Differential Equations Appendix 1 Variational Derivation of Differential Equations Let us recall the definition of a critical point for a function f belonging to the class C 2 ..a; b/ RI R/. Critical points are furnished by the roots of the equation f 0.x/ WD df.x/ D 0 in the interval .a; b/. Local (or relative) extrema are usually dx 00 0 00 given by a critical point x0 such that f .x0/ ¤ 0. In case f .x0/ D 0 and f .x0/> 0, the function f has a local (or relative) minimum. Similarly, for a critical point 00 x0 with f .x0/<0, f has a local (or relative) maximum. In case a critical point is neither a maximum nor a minimum, it is called a stationary point (or a point of inflection). We will now provide some simple examples. (1) Consider f.x/ WD 1 for x 2 .1; 1/. Every point in the interval .1; 1/ is a critical point. (2) Consider f.x/ WD x in the open interval .2;2/.Thereexistno critical points in the open interval. However, if we extend the function into the closed interval Œ2;2,there exists an extremum at the end points (which are not critical points). (3) The function f.x/ WD .x/3,forx 2 R, has a stationary point (or a point of inflection) at the origin. (4) The periodic function f.x/ WD sin x; x 2 R, has denumerably infinite number of local extrema, but no stationary points. (See, e.g., [32].) Now, we generalize these concepts to a function of N real variables x Á 1 N 2 N .x ; :::; x /. We assume that f 2 C .D R I R/. A local minimum x0 is defined by the property f.x/ f.x0/ for all x in the neighborhood Nı.x0/. Similarly, a local maximum is defined by the property f.x/ Ä f.x0/ for all x in the neighborhood. Consider f as a 0-form (see Sect. 1.2). The critical points are provided by the roots of the 1-form equation: df.x/ D 0Q.x/; @i f D 0; i 2f1;2;:::;Ng: (A1.1) A local (strong) minimum at x0 is implied by the inequality Ä XN XN @2f.x/ xi xi xj xj ˇ >0: (A1.2) 0 0 @xi @xj ˇ iD1 j D1 x0 A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 569 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 570 Appendix 1 Variational Derivation of Differential Equations x^ χ^ x χ t1 t t2 N Fig. A1.1 Two twice-differentiable parametrized curves into RN Similarly, a local (strong) maximum is guaranteed by Ä XN XN @2f.x/ xi xi xj xj ˇ <0: (A1.3) 0 0 @xi @xj ˇ iD1 j D1 x0 P P N N i j Example A1.1. Consider the quadratic polynomial f.x/ WD iD1 j D1 ıij x x N for x 2 R . There exists one critical point at the origin x0 WD .0;0;:::;0/.Now, Ä XN XN @2f.x/ XN XN xi xj ˇ D 2 ı xi xj >0: @xi @xj ˇ ij iD1 j D1 .0;0;:::;0/ iD1 j D1 Therefore, the origin is a local, as well as a global (strong), minimum. 2 N Consider a parametrized curve X of class C Œt1;t2 RI R , as furnished in 0 R equation (1.19). A Lagrangian function L W Œt1;t2 DN DN ! is such that L.tI xI u/ is twice differentiable. The action function (or action functional) J 2 N is a mapping from the domain set C Œt1;t2 RI R into the range set R and is given by: Zt2 X i J. / WD ŒL.tI xI u/j i DX i i D dX .t/ dt: (A1.4) x .t/; u dt t1 We assume that the function J is of class C 2, thus, totally differentiable in the abstract sense. (See (1.13).) b 2 N We consider a “slightly varied differentiable curve” X 2 C Œt1;t2 RI R . (See Fig. A1.1.) Two nearby curves are specified by the equations Appendix 1 Variational Derivation of Differential Equations 571 xi D X i .t/; bxi D Xbi .t/ WD X i .t/ C "hi .t/; t 2 Œt1;t2 R: (A1.5) Here, j"j >0is a small positive number and hi .t/ is arbitrary twice-differentiable function. The “slight variation” of the action function J in (A1.4) is furnished by Á J.X / WD J Xb J.X / ( ) Zt2 Ä Ä @L./ dhi .t/ @L./ D " hi .t/ ˇ C ˇ dt C 0."2/ @xi ˇ:: dt @ui ˇ:: t1 ( ) ( ) ˇ Zt2 Ä Ä ˇt2 @L./ @L./ ˇ D " hi .t/ ˇ dt C " hi .t/ ˇ ˇ i i ˇ @x ˇ:: @u ˇ:: t1 t1 ( ) Zt2 Ä d @L./ " hi .t/ ˇ dt C 0."2/: (A1.6) dt @ui ˇ:: t1 The variational derivative of the function J is given by ıJ.X / J.X / WD lim ıX "!0 " ( ) Zt2 Ä Ä @L./ d @L./ D ˇ ˇ hi .t/ dt @xi ˇ:: dt @ui ˇ:: t1 ( ) ˇ Ä ˇt2 @L./ ˇ C ˇ hi .t/ ˇ C 0: (A1.7) i ˇ @u ˇ:: t1 The critical functions or critical curves X.0/ are defined by the solutions of the ıJ.X / equation ıX D 0. We can prove from (A1.7) (with some more work involving the Dubois-Reymond lemma [224]), that critical functions X.0/ must satisfy N Euler– Lagrange equations: ( Ä ) @L./ˇ d @L./ ˇ ˇ D 0: (A1.8) @xi :: dt @ui ˇ:: Moreover, we must impose variationally permissible boundary conditions (con- sisting partly of prescribed conditions and partly of variationally natural boundary conditions [159]) so that the equations 572 Appendix 1 Variational Derivation of Differential Equations ( ) ˇ ( ) Ä ˇt2 Ä @L./ ˇ @L./ ˇ hi .t/ ˇ D ˇ hi .t/ i ˇ i ˇ @u ˇ:: @u ˇ:: ˇ t1 t2 ( Ä ) @L./ i ˇ h .t/ ˇ D 0: (A1.9) @ui ˇ:: ˇ t1 must hold. Usually, 2N Dirichlet boundary conditions X i i X i i .t1/ D x.1/ D prescribed consts.; .t2/ D x.2/ D prescribed consts. i i are chosen. Such conditions imply, by (A1.5), that h .t1/ D h .t2/ Á 0. Therefore, (A1.9) will be validated. However, there are many other variationally permissible boundary conditions which imply (A1.9). Example A1.2. Consider the Newtonian mechanics of a free particle in a Cartesian coordinate chart. The Lagrangian is furnished by m L.v/ WD ı v˛vˇ;m>0; 2 ˛ˇ @L.v/ D mı vˇ; @v˛ ˛ˇ @2L.v/ D mı : (A1.10) @v˛ @vˇ ˛ˇ The Euler–Lagrange equations (A1.8), with a slight change of notation and fixed boundary conditions X .t1/ D x.1/ and X .t2/ D x.2/, yield the unique critical curve Ä x˛ x˛ X ˛ .2/ .1/ ˛ .0/.t/ D .t t1/ C x.1/: (A1.11) t2 t1 This is a portion of a straight line or geodesic. Substituting the above solution into the finite variation in (A1.6), we derive that Appendix 1 Variational Derivation of Differential Equations 573 F (r+s) y DN x DN π (r+s) N N(r+s) y (r+s) Fig. A1.2 The mappings corresponding to a tensor field y.rCs/ D .rCs/.x/ J X.0/ C "h J X.0/ " # Zt2 ˇ Zt2 dh˛.t/ dX .t/ 1 dh˛.t/ dhˇ.t/ D "m ı .0/ dt C "2m ı dt C 0 ˛ˇ dt dt 2 ˛ˇ dt dt t1 t1 Zt2 1 dh˛.t/ dhˇ.t/ D 0 C "2m ı dt>0: (A1.12) 2 ˛ˇ dt dt t1 Therefore, the critical curve X.0/, which is a part of a geodesic in the Euclidean space, constitutes a strong minimum compared to other neighboring smooth curves. Remarks. (i) Local, strong minima provide stable equilibrium configurations. (ii) Timelike geodesics in a pseudo-Riemannian space–time manifold constitute strong, local maxima ! Now we shall consider .r C s/th order, differentiable tensor fields in an N - dimensional manifold MN , as discussed in (1.30). (More sophisticated treatments of tensor fields employ the concepts of tensor bundles [38,56].) For the sake of brevity, we denote an .r Cs/th order tensor field by the symbol .rCs/.x/ (in this appendix). Various mappings, which are necessary for the variational formalism, are shown in Fig. A1.2. 574 Appendix 1 Variational Derivation of Differential Equations According to the mapping diagram, y.rCs/ D .rCs/ ı F.x/ DW .rCs/.x/; N x 2 DN R : (A1.13) C C C .N /r s .N /r s 1 The Lagrangian function L W DN R R ! R is such that L .rCs/ .rCs/ .xI y I y i / is twice differentiable. The action function (or functional) is defined by Z Á L .rCs/ .rCs/ ˇ N J.F/ WD xI y I y C d x: (A1.14) i ˇ .rCs/D .rCs/ .r s/D .rCs/ y .x/; y i @i DN A “slightly varied function” is defined by by .rCs/ D .rCs/ ı F.x/b D .rCs/.x/ C "h.rCs/.x/; (A1.15) where j"j is a small positive number. The variation in the totally differentiable action function is furnished by J.F / WD J Fb J.F/ 8 " # 9 Z < Ä = @L./ @L./ D " h.rCs/.x/ ˇ C@ hrCs dN x C 0."2/: : @y .rCs/ ˇ i rCs ˇ ; :: @y i ˇ:: DN (A1.16) Here, indices .r C s/ are to be automatically summed appropriately.
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