arXiv:math.DG/0207039 v1 3 Jul 2002 Exterior Differential Systems and Euler-Lagrange Partial Differential Equations Robert Bryant Phillip Griffiths Daniel Grossman July 3, 2002 ii Contents Preface v Introduction vii 1 Lagrangians and Poincar´e-Cartan Forms 1 1.1 Lagrangians and Contact Geometry . . . . . . . . . . . . . . . . 1 1.2 The Euler-Lagrange System . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Variation of a Legendre Submanifold . . . . . . . . . . . . 7 1.2.2 Calculation of the Euler-Lagrange System . . . . . . . . . 8 1.2.3 The Inverse Problem . . . . . . . . . . . . . . . . . . . . . 10 1.3 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Hypersurfaces in Euclidean Space . . . . . . . . . . . . . . . . . . 21 1.4.1 The Contact Manifold over E n+1 . . . . . . . . . . . . . . 21 1.4.2 Euclidean-invariant Euler-Lagrange Systems . . . . . . . . 24 1.4.3 Conservation Laws for Minimal Hypersurfaces . . . . . . . 27 2 The Geometry of Poincar´e-Cartan Forms 37 2.1 The Equivalence Problem for n = 2 . . . . . . . . . . . . . . . . . 39 2.2 Neo-Classical Poincar´e-Cartan Forms . . . . . . . . . . . . . . . . 52 2.3 Digression on Affine Geometry of Hypersurfaces . . . . . . . . . . 58 2.4 The Equivalence Problem for n ≥ 3 . . . . . . . . . . . . . . . . . 65 2.5 The Prescribed Mean Curvature System . . . . . . . . . . . . . . 74 3 Conformally Invariant Systems 79 3.1 Background Material on Conformal Geometry . . . . . . . . . . . 80 3.1.1 Flat Conformal Space . . . . . . . . . . . . . . . . . . . . 80 3.1.2 The Conformal Equivalence Problem . . . . . . . . . . . . 85 3.1.3 The Conformal Laplacian . . . . . . . . . . . . . . . . . . 93 3.2 Conformally Invariant Poincar´e-Cartan Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.3 The Conformal Branch of the Equivalence Problem . . . . . . . . 102 3.4 Conservation Laws for ∆u = Cu n+2 n−2 . . . . . . . . . . . . . . . . 110 3.4.1 The Lie Algebra of Infinitesimal Symmetries . . . . . . . 111 3.4.2 Calculation of Conservation Laws . . . . . . . . . . . . . . 114 iii iv CONTENTS 3.5 Conservation Laws for Wave Equations . . . . . . . . . . . . . . . 118 3.5.1 Energy Density . . . . . . . . . . . . . . . . . . . . . . . . 122 3.5.2 The Conformally Invariant Wave Equation . . . . . . . . 123 3.5.3 Energy in Three Space Dimensions . . . . . . . . . . . . . 127 4 Additional Topics 133 4.1 The Second Variation . . . . . . . . . . . . . . . . . . . . . . . . 133 4.1.1 A Formula for the Second Variation . . . . . . . . . . . . 133 4.1.2 Relative Conformal Geometry . . . . . . . . . . . . . . . . 136 4.1.3 Intrinsic Integration by Parts . . . . . . . . . . . . . . . . 139 4.1.4 Prescribed Mean Curvature, Revisited . . . . . . . . . . . 141 4.1.5 Conditions for a Local Minimum . . . . . . . . . . . . . . 145 4.2 Euler-Lagrange PDE Systems . . . . . . . . . . . . . . . . . . . . 150 4.2.1 Multi-contact Geometry . . . . . . . . . . . . . . . . . . . 151 4.2.2 Functionals on Submanifolds of Higher Codimension . . . 155 4.2.3 The Betounes and Poincar´e-Cartan Forms . . . . . . . . . 158 4.2.4 Harmonic Maps of Riemannian Manifolds . . . . . . . . . 164 4.3 Higher-Order Conservation Laws . . . . . . . . . . . . . . . . . . 168 4.3.1 The Infinite Prolongation . . . . . . . . . . . . . . . . . . 168 4.3.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . 172 4.3.3 The K = −1 Surface System . . . . . . . . . . . . . . . . 182 4.3.4 Two B¨acklund Transformations . . . . . . . . . . . . . . . 191 Preface During the 1996-97 academic year, Phillip Griffiths and Robert Bryant con- ducted a seminar at the Institute for Advanced Study in Princeton, NJ, outlin- ing their recent work (with Lucas Hsu) on a geometric approach to the calculus of variations in several variables. The present work is an outgrowth of that project; it includes all of the material presented in the seminar, with numerous additional details and a few extra topics of interest. The material can be viewed as a chapter in the ongoing development of a theory of the geometry of differential equations. The relative importance among PDEs of second-order Euler-Lagrange equations suggests that their geometry should be particularly rich, as does the geometric character of their conservation laws, which we discuss at length. A second purpose for the present work is to give an exposition of certain aspects of the theory of exterior differential systems, which provides the lan- guage and the techniques for the entire study. Special emphasis is placed on the method of equivalence, which plays a central role in uncovering geometric properties of differential equations. The Euler-Lagrange PDEs of the calculus of variations have turned out to provide excellent illustrations of the general theory. v vi PREFACE Introduction In the classical calculus of variations, one studies functionals of the form F L (z) =  Ω L(x, z, ∇z) dx, Ω ⊂ R n , (1) where x = (x 1 , . . . , x n ), dx = dx 1 ∧ ··· ∧ dx n , z = z(x) ∈ C 1 ( ¯ Ω) (for ex- ample), and the Lagrangian L = L(x, z, p) is a smooth function of x, z, and p = (p 1 , . . . , p n ). Examples frequently encountered in physical field theories are Lagrangians of the form L = 1 2 ||p|| 2 + F (z), usually interpreted as a kind of energy. The Euler-Lagrange equation describ- ing functions z(x) that are stationary for such a functional is the second-order partial differential equation ∆z(x) = F  (z(x)). For another example, we may identify a function z(x) with its graph N ⊂ R n+1 , and take the Lagrangian L =  1 + ||p|| 2 , whose associated functional F L (z) equals the area of the graph, regarded as a hypersurface in Euclidean space. The Euler-Lagrange equation describing functions z(x) stationary for this functional is H = 0, where H is the mean curvature of the graph N. To study these Lagrangians and Euler-Lagrange equations geometrically, one has to choose a class of admissible coordinate changes, and there are four natural candidates. In increasing order of generality, they are: • Classical transformations, of the form x  = x  (x), z  = z  (z); in this situation, we think of (x, z, p) as coordinates on the space J 1 (R n , R) of 1-jets of maps R n → R. 1 • Gauge transformations, of the form x  = x  (x), z  = z  (x, z); here, we think of (x, z, p) as coordinates on the space of 1-jets of sections of a bundle R n+1 → R n , where x = (x 1 , . . . , x n ) are coordinates on the base R n and z ∈ R is a fiber coordinate. 1 A 1-jet is an equivalence class of functions having the same value and the same first derivatives at some designated point of the domain. vii viii INTRODUCTION • Point transformations, of the form x  = x  (x, z), z  = z  (x, z); here, we think of (x, z, p) as coordinates on the space of tangent hyperplanes {dz − p i dx i } ⊥ ⊂ T (x i ,z) (R n+1 ) of the manifold R n+1 with coordinates (x 1 , . . . , x n , z). • Contact transformations, of the form x  = x  (x, z, p), z  = z  (x, z, p), p  = p  (x, z, p), satisfying the equation of differential 1-forms dz  −  p  i dx i = f ·(dz −  p i dx i ) for some function f(x, z, p) = 0. We will be studying the geometry of functionals F L (z) subject to the class of contact transformations, which is strictly larger than the other three classes. The effects of this choice will become clear as we proceed. Although contact transformations were recognized classically, appearing most notably in studies of surface geometry, they do not seem to have been extensively utilized in the calculus of variations. Classical calculus of variations primarily concerns the following features of a functional F L . The first variation δF L (z) is analogous to the derivative of a function, where z = z(x) is thought of as an independent variable in an infinite-dimensional space of functions. The analog of the condition that a point be critical is the condition that z(x) be stationary for all fixed-boundary variations. Formally, one writes δF L (z) = 0, and as we shall explain, this gives a second-order scalar partial differential equa- tion for the unknown function z(x) of the form ∂L ∂z −  d dx i  ∂L ∂p i  = 0. This is the Euler-Lagrange equation of the Lagrangian L(x, z, p), and we will study it in an invariant, geometric setting. This seems especially promising in light of the fact that, although it is not obvious, the process by which we asso- ciate an Euler-Lagrange equation to a Lagrangian is invariant under the large class of contact transformations. Also, note that the Lagrangian L determines the functional F L , but not vice versa. To see this, observe that if we add to L(x, z, p) a “divergence term” and consider L  (x, z, p) = L(x, z, p) +   ∂K i (x, z) ∂x i + ∂K i (x, z) ∂z p i  for functions K i (x, z), then by Green’s theorem, the functionals F L and F L  differ by a constant depending only on values of z on ∂Ω. For many purposes, ix such functionals should be considered equivalent; in particular, L and L  have the same Euler-Lagrange equations. Second, there is a relationship between symmetries of a Lagrangian L and conservation laws for the corresponding Euler-Lagrange equations, described by a classical theorem of Noether. A subtlety here is that the group of symmetries of an equivalence class of Lagrangians may be strictly larger than the group of symmetries of any particular representative. We will investigate how this discrepancy is reflected in the space of conservation laws, in a manner that involves global topological issues. Third, one considers the second variation δ 2 F L , analogous to the Hessian of a smooth function, usually with the goal of identifying local minima of the functional. There has been a great deal of analytic work done in this area for classical variational problems, reducing the problem of local minimization to understanding the behavior of certain Jacobi operators, but the geometric theory is not as well-developed as that of the first variation and the Euler-Lagrange equations. We will consider these issues and several others in a geometric setting as suggested above, using various methods from the subject of exterior differential systems, to be explained along the way. Chapter 1 begins with an introduc- tion to contact manifolds, which provide the geometric setting for the study of first-order functionals (1) subject to contact transformations. We then con- struct an object that is central to the entire theory: the Poincar´e-Cartan form, an explicitly computable differential form that is associated to the equivalence class of any Lagrangian, where the notion of equivalence includes that alluded to above for classical Lagrangians. We then carry out a calculation using the Poincar´e-Cartan form to associate to any Lagrangian on a contact manifold an exterior differential system—the Euler-Lagrange system—whose integral man- ifolds are stationary for the associated functional; in the classical case, these correspond to solutions of the Euler-Lagrange equation. The Poincar´e-Cartan form also makes it quite easy to state and prove Noether’s theorem, which gives an isomorphism between a space of symmetries of a Lagrangian and a space of conservation laws for the Euler-Lagrange equation; exterior differential systems provides a particularly natural setting for studying the latter objects. We illus- trate all of this theory in the case of minimal hypersurfaces in Euclidean space E n , and in the case of more general linear Weingarten surfaces in E 3 , providing intuitive and computationally simple proofs of known results. In Chapter 2, we consider the geometry of Poincar´e-Cartan forms more closely. The main tool for this is ´ E. Cartan’s method of equivalence, by which one develops an algorithm for associating to certain geometric structures their differential invariants under a specified class of equivalences. We explain the various steps of this method while illustrating them in several major cases. First, we apply the method to hyperbolic Monge-Ampere systems in two inde- pendent variables; these exterior differential systems include many important Euler-Lagrange systems that arise from classical problems, and among other results, we find a characterization of those PDEs that are contact-equivalent x INTRODUCTION to the homogeneous linear wave equation. We then turn to the case of n ≥ 3 independent variables, and carry out several steps of the equivalence method for Poincar´e-Cartan forms, after isolating those of the algebraic type arising from classical problems. Associated to such a neo-classical form is a field of hypersur- faces in the fibers of a vector bundle, well-defined up to affine transformations. This motivates a digression on the affine geometry of hypersurfaces, conducted using Cartan’s method of moving frames, which we will illustrate but not dis- cuss in any generality. After identifying a number of differential invariants for Poincar´e-Cartan forms in this manner, we show that they are sufficient for char- acterizing those Poincar´e-Cartan forms associated to the PDE for hypersurfaces having prescribed mean curvature. A particularly interesting branch of the equivalence problem for neo-classical Poincar´e-Cartan forms includes some highly symmetric Poincar´e-Cartan forms corresponding to Poisson equations, discussed in Chapter 3. Some of these equations have good invariance properties under the group of conformal trans- formations of the n-sphere, and we find that the corresponding branch of the equivalence problem reproduces a construction that is familiar in conformal ge- ometry. We will discuss the relevant aspects of conformal geometry in some detail; these include another application of the equivalence method, in which the important conceptual step of prolongation of G-structures appears for the first time. This point of view allows us to apply Noether’s theorem in a partic- ularly simple way to the most symmetric of non-linear Poisson equations, the one with the critical exponent: ∆u = Cu n+2 n−2 . Having calculated the conservation laws for this equation, we also consider the case of wave equations, and in particular the very symmetric example: z = Cz n+3 n−1 . Here, conformal geometry with Lorentz signature is the appropriate background, and we present the conservation laws corresponding to the associated symmetry group, along with a few elementary applications. The final chapter addresses certain matters which are thus far not so well- developed. First, we consider the second variation of a functional, with the goal of understanding which integral manifolds of an Euler-Lagrange system are local minima. We give an interesting geometric formula for the second variation, in which conformal geometry makes another appearance (unrelated to that in the preceding chapter). Specifially, we find that the critical submanifolds for certain variational problems inherit a canonical conformal structure, and the second variation can be expressed in terms of this structure and an additional scalar curvature invariant. This interpretation does not seem to appear in the classical literature. Circumstances under which one can carry out in an invariant manner the usual “integration by parts” in the second-variation formula, which is crucial for the study of local minimization, turn out to be somewhat limited. [...]... Lagrangian functionals and their Euler-Lagrange systems Definition 1.1 A contact manifold (M, I) is a smooth manifold M of dimension 2n + 1 (n ∈ Z+ ), with a distinguished line sub-bundle I ⊂ T ∗ M of the cotangent bundle which is non-degenerate in the sense that for any local 1-form θ generating I, θ ∧ (dθ)n = 0 1 2 ´ CHAPTER 1 LAGRANGIANS AND POINCARE-CARTAN FORMS Note that the non-degeneracy criterion... both of G-structures and of differential systems, and a use of the notion of integrable extension to clarify a confusing issue Of course, the study of Euler-Lagrange equations by means of exterior differential forms and the method of equivalence is not new In fact, much of the 19th century material in this area is so naturally formulated in terms of differential forms (cf the Hilbert form in the one-variable... difference to b(xi, z) and relabelling the result as b(xi, z), 1.2 THE EULER-LAGRANGE SYSTEM 13 we see that our criterion for the Monge-Ampere system to be Euler-Lagrange is that f(xi , z, pi) be of the form f(xi , z, pi) = 1 bz (x, z) 2 p2 + i bxi (x, z)pi + a(x, z) for some functions b(x, z), a(x, z) These describe exactly those Poisson equations that are locally contact-equivalent to Euler-Lagrange equations... Example 2 An example that is not quasi-linear is given by det( 2 z) − g(x, z, z) = 0 The n-form Ψ = dp − g(x, z, p)dx and the standard contact system generate a Monge-Ampere system whose transverse integral manifolds correspond to solutions of this equation A calculation similar to that in the preceding example shows that this Monge-Ampere system is Euler-Lagrange if and only if g(x, z, p) is of the form... where α is the u(1)-valued Hermitian connection form on M Note θ∧(dθ)2 = 0, because the 4-form (dθ)2 = ω2 is actually a volume form on M (by positivity) and θ is non-vanishing on fibers of M → X, unlike (dθ)2 Now we trivialize the canonical bundle of X with a holomorphic 2-form Φ = Ψ + iΣ, and take for our Monge-Ampere system E = {θ, dθ = ω, Ψ = Re(Φ)} We can see that E is locally Euler-Lagrange as follows... and applied in the context of CR geometry in [Rum90] 10 where ´ CHAPTER 1 LAGRANGIANS AND POINCARE-CARTAN FORMS d ∂ ∂ = + z xi + i i dx ∂x ∂z zxi xj j ∂ ∂pj is the total derivative This is the usual Euler-Lagrange equation, a secondorder, quasi-linear PDE for z(x1, , xn) having symbol Lpi pj It is an exercise to show that this symbol matrix is invertible at (xi, z, pi) if and only if Λ is non-degenerate... reasonable model for exterior differential systems of Euler-Lagrange type” Definition 1.4 A Monge-Ampere differential system (M, E) consists of a contact manifold (M, I) of dimension 2n + 1, together with a differential ideal E ⊂ Ω∗ (M ), generated locally by the contact ideal I and an n-form Ψ ∈ Ω n (M ) Note that in this definition, the contact line bundle I can be recovered from E as its degree-1 part We can... contact-equivalent to an Euler-Lagrange equation To apply our framework, we let M = J 1(Rn , R), θ = dz − pidxi so dθ = − dpi ∧ dxi, and set Ψ= dpi ∧ dx(i) − f(x, z, p)dx ∂z Restricted to a Legendre submanifold of the form N = {(xi, z(x), ∂xi (x)}, we find Ψ|N = (∆z − f(x, z, z))dx 12 ´ CHAPTER 1 LAGRANGIANS AND POINCARE-CARTAN FORMS Evidently Ψ is primitive modulo {I}, and E = {θ, dθ, Ψ} is a Monge-Ampere... a famous question Inverse Problem: When is a given Monge-Ampere system E on M equal to the Euler-Lagrange system EΛ of some Lagrangian Λ ∈ Ωn (M )? Note that if a given E does equal EΛ for some Λ, then for some local generators θ, Ψ of E we must have θ ∧ Ψ = Π, the Poincar´-Cartan form of Λ Indeed, e we can say that (M, E) is Euler-Lagrange if and only if there is an exact form Π ∈ Ωn+1 (M ), locally... be a globally defined exact n-form Π, locally of the form fθ∧Ψ with Ψ normalized as above This suggests the more accessible local inverse problem, which asks whether there is a closed n-form that is locally expressible as fθ ∧ Ψ It is for this local version that we give a criterion 1.2 THE EULER-LAGRANGE SYSTEM 11 We start with any candidate Poincar´-Cartan form Ξ = θ ∧ Ψ, and consider e the following . hyperbolic Monge-Ampere systems in two inde- pendent variables; these exterior differential systems include many important Euler-Lagrange systems that arise from classical problems, and among other results,. 2002 Exterior Differential Systems and Euler-Lagrange Partial Differential Equations Robert Bryant Phillip Griffiths Daniel Grossman July 3, 2002 ii Contents Preface v Introduction vii 1 Lagrangians and. use of prolonga- tion both of G-structures and of differential systems, and a use of the notion of integrable extension to clarify a confusing issue. Of course, the study of Euler-Lagrange equations