Noether’s Two Theorems !! Adventures in Integration by Parts !! Peter J. Olver University of Minnesota http://www.math.umn.edu/ ∼ olver Convergence, Perimeter Institute, June, 2015 References E. Noether, Invariante Variationsprobleme, Nachr. K¨onig. Gesell. Wissen. G¨ottingen, Math.–Phys. Kl. (1918) 235–257. P.J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, Springer–Verlag, New York, 1993. Y. Kosmann-Schwarzbach, The Noether Theorems. Invariance and Conservation Laws in the Twentieth Century, Springer, New York, 2011. Noether’s Three Fundamental Contributions to Analysis and Physics First Theorem. There is a one-to-one correspondence between symmetry groups of a variational problem and conservation laws of its Euler–Lagrange equations. Second Theorem. An infinite-dimensional variational symmetry group depending upon an arbitrary function corresponds to a nontrivial differential relation among its Euler–Lagrange equations. Introduction of higher order generalized symmetries. =⇒ later (1960’s) to play a fundamental role in the discovery and classification of integrable systems and solitons. The Noether Triumvirate ! Variational Principle ! Symmetry ! Conservation Law Symmetry Symmetry Groups of Differential Equations =⇒ Sophus Lie (1842–1899). System of differential equations ∆(x, u(n))=0 G — Lie group or Lie pseudo-group acting on the space of independent and dependent variables: (x, u)=g · (x, u) ! ! G acts on functions by transforming their graphs: u = f(x)˜u = f˜(˜x) g !−→ Definition. G is a symmetry group of the system ∆= 0 if f = g · f is a solution whenever f is. ! Variational Symmetries Definition. A variational symmetry is a transformation of space/time and the field variables (x, u)=g · (x, u) that leaves the variational problem invariant: ! ! L(x, u(n)) dx = L(x, u(n)) dx "Ω "Ω ! ! ! ! Theorem. Every symmetry of the variational problem is a symmetry of the Euler–Lagrange equations. (but not conversely) One–Parameter Groups ALiegroupwhosetransformationsdependuponasingle parameter ε ∈ R is called a one-parameter group. Translations in a single direction: (x, y, z) $−→ (x + ε,y+2ε,z− ε) Rotations around a fixed axis: (x, y, z) $−→ (x cos ε − z sin ε,y,xsin ε + z cos ε) Screw motions: (x, y, z) $−→ (x cos ε − y sin ε,xsin ε + y cos ε,z+ ε) Scaling transformations: − (x, y, z) $−→ (λ x, λ y, λ 1 z) Infinitesimal Generators Every one-parameter group can be viewed as the flow of avectorfieldv,knownasitsinfinitesimal generator. In other words, the one-parameter group is realized as the solution to the system of ordinary differential equations governing the vector field’s flow: dz = v(z) dε Equivalently, if one expands the group transformations in powers of the group parameter ε,theinfinitesimalgenerator comes from the linear terms: z(ε)=z + ε v(z)+ ··· Translations in a single direction: (x, y, z) $−→ (x + ε,y+2ε,z− ε) v =(1, 2, −1) Rotations around a fixed axis: (x, y, z) $−→ (x cos ε − z sin ε,y,xsin ε + z cos ε) v =(−z,0,x) Screw motions: (x, y, z) $−→ (x cos ε − y sin ε,xsin ε + y cos ε,z+ ε) v =(−y, x,1) Scaling transformations: − (x, y, z) $−→ (λ x, λ y, λ 1 z) λ = eε v =(x, y, −z) Infinitesimal Generators = Vector Fields In differential geometry, it has proven to be very useful to identify a vector field with a first order differential operator (or derivation). In local coordinates (...xi ...uα ...), the vector field v =( ... ξi(x, u) ... ϕα(x, u) ... ) that generates the one-parameter group (flow) dxi duα = ξi(x, u) = ϕα(x, u) dε dε is identified with the differential operator p ∂ q ∂ v = ξi(x, u) + ϕα(x, u) ∂xi ∂uα i#=1 α#=1 Invariance AfunctionF : M → R is invariant if it is not affected by the group transformations: F (g · z)=F (z) for all g ∈ G and z ∈ M. Infinitesimal Invariance Theorem. (Lie) A function is invariant under a one-parameter group with infinitesimal generator v (viewed as a differen- tial operator) if and only if v(F )=0 Translations: ∂ ∂ ∂ v =(1, 2, −1) $−→ +2 − ∂x ∂y ∂z invariance : F (x + ε,y+2ε,z− ε)=F (x, y, z) ∂F ∂F ∂F ⇐⇒ 0=v(F )= +2 − ∂x ∂y ∂z Rotations: ∂ ∂ v =(−z,0,x) $−→ −z + x ∂x ∂z invariance : F (x cos ε − z sin ε,y,xsin ε + z cos ε)=F (x, y, z) ∂F ∂F ⇐⇒ 0=v(F )=−z + x ∂x ∂z Prolongation Since G acts on functions, it acts on their derivatives u(n), leading to the prolonged group action: (x, u(n))=pr(n) g · (x, u(n)) =⇒ formulas provided! by! implicit differentiation Prolonged vector field or infinitesimal generator: α (n) ∂ pr v = v + ϕJ (x, u ) α ∂uJ α#,J The Prolongation Formula The coefficients of the prolonged vector field are given by the explicit prolongation formula: p α α i α ϕJ = DJ Q + ξ uJ,i i#=1 p ∂uα where Qα(x, u(1))=ϕα − ξi ∂xi i#=1 Q =(Q1,...,Qq)—characteristic of v ! Invariant functions are solutions to Q(x, u(1))=0. Example. The vector field ∂ ∂ ∂ ∂ v = ξ + ϕ = −u + x ∂x ∂u ∂x ∂u generates the rotation group (x, u) $−→ =(x cos ε − u sin ε, x sin ε + u cos ε) The prolonged action is (implicit differentiation) sin ε + ux cos ε ux $−→ cos ε − ux sin ε u u xx xx $−→ 3 (cos ε − ux sin ε) (cos ε − u sin ε) u − 3u2 sin ε u x xxx xx xxx $−→ 5 (cos ε − ux sin ε) . ∂ ∂ ∂ ∂ v = ξ + ϕ = −u + x ∂x ∂u ∂x ∂u Characteristic: Q(x, u, ux)=ϕ − ux ξ = x + uux By the prolongation formula, the infinitesimal generator is ∂ ∂ 2 ∂ ∂ pr v = −u + x +(1+ux) +3uxuxx + ··· ∂x ∂u ∂ux ∂uxx ! The solutions to the characteristic equation Q(x, u, ux)=x + uux =0 are circular arcs — rotationally invariant curves. Lie’s Infinitesimal Symmetry Criterion for Differential Equations Theorem. AconnectedgroupoftransformationsG is a symmetry group of a nondegenerate system of differential equations ∆= 0 if and only if pr v(∆) = 0 whenever ∆= 0 for every infinitesimal generator v of G. Calculation of Symmetries pr v(∆) = 0 whenever ∆= 0 These are the determining equations of the symmetry group to ∆=0.Theyformanoverdeterminedsystemofelementary partial differential equations for the coefficients ξi,ϕα of v that can (usually) be explicitly solved — there are even Maple and Mathematica packages that do this automat- ically — thereby producing the most general infinitesimal symmetry and hence the (continuous) symmetry group of the system of partial differential equations. ! For systems arising in applications, many symmetries are evident from physical intuition, but there are significant examples where the Lie method produces new symmetries. The Calculus of Variations The First Derivative Test Aminimumofafunctionofseveralvariablesf(x1,...,xn)isa place where the gradient vanishes: ∇f =0. This condition also holds at maxima as well as saddle points. ! Distinguishing minima from maxima from saddle points requires the second derivative test —notusedhere! Variational Problems Avariationalproblemrequiresminimizingafunctional F [u]= L(x, u(n)) dx " The integrand is known as the Lagrangian. The Lagrangian L(x, u(n))candependuponthespace/time coordinates x,thefunction(s)orfield(s)u = f(x)andtheir derivatives up to some order n —typically,butnotalways n =1. Functionals Distance functional =arclengthofacurvey = u(x): b F [u]= 1+u$(x)2 dx, a " $ Boundary conditions: u(a)=αu(b)=β Solutions: geodesics (straight lines) Surface area functional: ∂u 2 ∂u 2 F [u]= %1+ + dx dy. Ω & ( ∂x ) ( ∂y ) "" & Minimize subject to Dirichlet' boundary conditions u(x, y)=g(x, y)for(x, y) ∈ ∂Ω. Solutions: minimal surfaces The Euler–Lagrange Equations The minimum of the functional F [u]= L(x, u(n)) dx " must occur where the functional gradient vanishes: δF[u]=0 This is a system of differential equations ∆=E(L)=0 known as the Euler–Lagrange equations. E —Euleroperator(variationalderivative): α δL J ∂L E (L)= α = (−D) α =0 δu ∂uJ #J The (smooth) minimizers u(x)ofthefunctionalaresolutionsto the Euler–Lagrange equations — as are any maximizers and, in general, all “critical functions”. Functional Gradient Functional F [u]= L(x, u(n)) dx " Variation v = δu: F [u + v ]=F [u]+) δF ; v * +h.o.t. ∂L ∂L ∂L = L(u, ut,utt,...) dt + v + vt + vtt + ··· dt + ··· " " ( ∂u ∂ut ∂utt ) Integration by parts: ∂L ∂L ∂L ∂L ∂L 2 ∂L v + vt + vtt + ··· dt = − Dt + Dt −··· vdt " ( ∂u ∂ut ∂utt ) " ( ∂u ∂ut ∂utt ) Euler–Lagrange equations: ∂L ∂L 2 ∂L δF = E(L)= − Dt + Dt −···=0 ∂u ∂ut ∂utt Variational Symmetries Definition. Astrictvariational symmetry is a transformation (x, u)=g · (x, u)whichleavesthevariationalproblem invariant: ! ! L(x, u(n)) dx = L(x, u(n)) dx "Ω "Ω Infinitesimal invariance! criterion:! ! ! pr v(L)+L Div ξ =0 Divergence symmetry (Bessel–Hagen): pr v(L)+L Div ξ =DivB =⇒ Every divergence symmetry has an equivalent strict variational symmetry Conservation Laws Conservation Laws Conservation Laws A conservation law of a discrete dynamical system of ordinary differential equations is a function T (t, u, ut,...) depending on the time t,thefieldvariablesu,andtheirderiva- tives, that is constant on solutions, or, equivalently, DtT =0 on all solutions to the field equations. Conservation Laws — Dynamics In continua, a conservation law states that the temporal rate of change of a quantity T in a region of space D is governed by the associated flux through its boundary: ∂ Tdx= X ∂t "D *∂D or, in differential form, DtT =DivX • In particular, if the flux X vanishes on the boundary ∂D, then the total density Tdxis conserved — constant. "D Conservation Laws — Statics In statics, a conservation law corresponds to a path- or surface- independent integral X =0 —indifferentialform, *C Div X =0 Thus, in fracture mechanics, one can measure the conserved quantity near the tip of a crack by evaluating the integral at a safe distance.