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Groningen The Netherlands, Our long term goal is to develop geometric structure. preserving numerical discretizations that systematically ad y. dresses the issue of gauge symmetries Eventually we wish. to study discretizations of general relativity that address. the issue of general covariance Towards this end we. will consider multi Dirac mechanics based on a Hamilton. Pontryagin variational principle for field theories 2 that is. well adapted to degenerate field theories The issue of general. covariance also leads us to avoid using a tensor product. discretization that presupposes a slicing of space time rather. we will consider 4 simplicial complexes in space time More. generally we will need to study discretizations that are. invariant with respect to some discrete analogue of the gauge x. symmetry group Fig 1 A section of the configuration bundle the horizontal axes represent. spacetime and the vertical axis represent dependent field variables The. II M ULTI D IRAC F ORMULATION OF F IELD T HEORIES section gives the value of the field variables at every point of spacetime. The Dirac 3 4 and multi Dirac formulation 2 of me, chanics and field theories can be viewed as a generalization. of the Lagrangian and multi symplectic formulation to the fiber over t x which is 1 t x This is illustrated in. case whether the Lagrangian is degenerate i e the Legendre Figure 1. transformation is not onto This approach is critical to gauge The multisymplectic analogue of the tangent bundle is. field theories as the gauge symmetries naturally lead to the first jet bundle J 1 Y which is a fiber bundle over X. degenerate Lagrangians that consists of the configuration bundle Y and the first. partial derivatives v a y a x of the field variables. A Hamilton Pontryagin Principle for Mechanics with respect to the independent variables Given a section. Consider a configuration manifold Q with associated tan X Y x0 xn x0 xn y 1 y m its. gent bundle T Q and phase space T Q Dirac mechanics is first jet extension j 1 X J 1 Y is a section of J 1 Y over. described on the Pontryagin bundle T Q T Q which has X given by. position velocity and momentum q v p as local coordi. j 1 x0 xn x0 xn y 1 y m y 1 0 y m n, nates The dynamics on the Pontryagin bundle is described. by the Hamilton Pontryagin variational principle where the. Lagrange multiplier and momentum p imposes the second The dual jet bundle J 1 Y is affine with fiber coordi. order condition v q nates p p a corresponding to the affine map v a 7 p. Z t2 p a v a dn 1 x where dn 1 x dx1 dxn dx0,L q v p q v dt 0 1. t1 C Hamilton Pontryagin Principle for Classical Fields. It provides a variational description of both Lagrangian The first order Lagrangian density is a map L. and Hamiltonian mechanics and yields the implicit Euler J 1Y X and let L j 1 L j 1 dV. Lagrange equations a a,L x y v dV where X is the space of alternat.

L L ing n 1 forms over X and L j 1 is a scalar function. q v on J 1 Y For field theories the analogue of the Pontryagin. The last equation is the Legendre transform FL q q 7 bundle is J 1 Y Y J 1 Y and the first jet condition x a. q L replaces v q so the Hamilton Pontryagin principle is. q This is important for degenerate systems as it, enforces the primary constraints that arise when the Legendre. 0 S y a y a p a,transform is not onto Z a,B Multisymplectic Geometry pa v L x y v dn 1 x 3. The geometric setting for Lagrangian PDEs is multisym. plectic geometry 5 6 The base space X consists of Taking variations with respect to y a v a and p a where y a. independent variables denoted by x0 xn t x vanishes on the boundary U yields the implicit Euler. where x0 t is time and x1 xn x are space Lagrange equations. variables The dependent field variables y 1 y m y p a L L y a. form a fiber over each space time basepoint The independent p a and v a 4. x y a v a x, and field variables form the configuration bundle Y. X The configuration of the system is specified by a section which generalizes 2 to the case of field theories see 7. of Y over X which is a continuous map X Y such for more details As the jet bundle is an affine bundle the. that 1X i e for every t x X t x is in the duality pairing used implicitly in 3 is more complicated. Groningen The Netherlands, The second equation of 4 yields the covariant Legendre Type I generating function of a symplectic map which is. transform FL J 1 Y J 1 Y intended to approximate the exact discrete Lagrangian. p a p L v 5 Lexact q q L q0 1 t q 0 1 t dt,v a v a d 0 1.

This unifies the two aspects of the Legendre transform by where q0 1 t satisfies the Euler Lagrange boundary value. combining the definitions of the momenta and the Hamilto problem There are systematic methods of constructing. nian into a single covariant entity computable approximations of the discrete Lagrangian 9. 10 and it can be shown that if the computable discrete. D Multi Dirac formulation of Maxwell s equations Lagrangian approximates the exact discrete Lagrangian to. The electromagnetic Lagrangian density is given by a given order of accuracy then the resulting variational. 1 1 integrator exhibits the same order of accuracy 11 12. L A j 1 A dA dA F F 6 In the case of field theories the boundary Lagrangian 13. which is a scalar valued function on the space of boundary. where F A A d is the exterior derivative and data plays a similar role and the exact boundary Lagrangian. is the Minkowski Hodge star M M has the form,defined uniquely by the identity Z. hh k k iiv k k U, where hh ii is the Minkowski metric on differential forms where satisfies the boundary conditions U U. and v is the volume form For example for standard and satisfies the Euler Lagrange equation in the interior. Minkowski spacetime with metric signature the of U As with the case for Lagrangian ODEs a computable. Hodge star acts on 2 forms as follows approximation of the boundary Lagrangian can be obtained. by replacing the space time integral with a quadrature rule. dt dx dz dy dy dz dt dx and considering a finite element approximation of the con. dt dy dx dz dz dx dt dy figuration bundle As we will see in the case of Maxwell s. equations this involves the use of Whitney forms as the. dt dz dy dx dx dy dt dz,finite dimensional configuration bundle. For a more in depth discussion see for example page 411. III S PACE TIME W HITNEY F ORMS, The Hamilton Pontryagin action principle is given in co The Whitney k forms are a finite dimensional subspace. ordinates by of k differential forms and they are dual to k simplices via. Z integration pairing They were introduced by Whitney in. A 1 14 and they are typically expressed in terms of barycentric. S p A F F d4 x, U x 4 coordinates The barycentric coordinates i are defined on a.

where U is an open subset of X The implicit Euler k simplex with vertex vectors v0 v1 vk as functions of. Lagrange equations are given by the position vector x such that. A p i vi x i 1,p F A 0 i 0 i 0,Then on a k simplex v0 v1 vk the Whitney k. and by eliminating p lead to Maxwell s equations,form k w is. Note that the gauge symmetry of the action given in 6 Xk. w k 1 i i d 0 d 1 d ci d k, is more general than what is typically considered in the i 0. standard formulation of electromagnetism Since the action where the hat indicates an omitted term and the superscript. only depends on dA then the Lagrangian density is invariant k is usually dropped when the order of the form is clear. under shifts of A by any closed 1 form in other words 1 Whitney forms are a crucial ingredient in Finite Element. forms such that d 0 Contrast this with the standard Exterior Calculus FEEC a finite element framework that. formulation shown in section I which implies that only exact encompasses all standard and mixed finite element formula. 1 forms df leave the dynamics invariant tions through the use of the de Rham complex formed by. the exterior derivative d and the Koszul operator The. E Discrete Multi Dirac Variational Integrators framework is described in terms of the Pr k spaces which. The theory of variational integrators provide a way of are the spaces of order r polynomials on differential k forms. discretizing Lagrangian mechanical systems so as to ob Indeed Whitney forms characterize Pr k Pr k For. tain numerical integration schemes that are automatically more details see 15 16. symplectic They are based on the concept of a discrete The problem with the representation of Whitney forms. Lagrangian Ld Q Q R which can be viewed as a in terms of barycentric coordinates is that the Hodge star. Groningen The Netherlands, of a differential form is significantly easier to compute in is given by. space time adapted coordinates and the Hodge star shows k. up in the Lagrangian density for electromagnetism Even w Wk sgn. though this does not present much of an obstacle for vacuum DV. electromagnetism an explicit characterization of the Hodge i 1 v i v 0 vj v j x W k. star simplifies the calculations when matter sources and Vn Vn. h i 1 vi v0 i 1 vi v0 i, material properties are added into the dynamics This is Then one can show that the characterization of Whitney.

perhaps best seen through the permittivity and permeability k forms presented in Proposition 2 and Theorem 1 are equiv. tensors any variation in their values mimic the effects of alent Since the Hodge star applied twice is the identity map. a varying metric tensor In particular the electric field E up to a sign the coordinate independent characterization. and electric displacement field D are related by the Hodge of Whitney forms given in Theorem 1 provide an explicit. star with respect to a metric induced by the permittivity and characterization of the Hodge dual of the space of Whitney. the magnetic induction B and the magnetic intensity H are k forms. related by the the Hodge star with respect to a metric induced. by the permeability B Space time Whitney forms in Electromagnetism. We now apply our space time FEEC discretization to the. vacuum electromagnetic action,A Space time Whitney Forms 1. To simplify the task of using Whitney forms in space time 4 M. multi Dirac discretizations of Maxwell s equations we intro Assume the manifold M has a simplicial triangulation into. duce a characterization of Whitney forms that is coordinate simplices p Discretizing the vector potential A to linear. order yields Ap i j apij wij bpij d i jP, independent The proofs of these results can be found in 17 on the edges. In the following v represents the co vector associated with of a given simplex p This implies dAp k i j apij wijk. the vector v on the faces of the simplex Applying our discretization to. Theorem 1 Let v0 v1 vn an ordered set of the Maxwell action gives. vertex vectors represent an oriented n simplex on a flat n 1 X X. S apij apkl hwijk wlmn ivol p, dimensional manifold with position vector x Let be 4. p k n p i j l m,a k subsimplex and r be the ordered complement. Let s now investigate the issue of gauge invariance at this. of in The Whitney k form over can be written as, linear order of approximation We d like to shift our potential.

Ap by a closed 1 form p However at the simplicial level. sgn k all closed forms are exact so we can take p df Pp where fp. w x vj x 7, is a 0 form Taking fp to quadratic order fp ij cpij i j. which yields the gauge shifted potential,with vol n 1. i 1 vi v0 the volume form of A0p Ap dfp apij wij bpij cpij d i j. Note that the term outside the parenthesis is simply a. This represents all allowed gauge transformations to linear. normalization factor and the expression inside the paren. order The full gauge group can be better approximated as. thesis can be understood in terms of the observation that the. the order of approximation increases as well In fact at. Whitney form vanishes on the complementary subsimplex. linear order Ap automatically satisfies the Lorenz gauge. and the vj x terms ensure that the proposed differential. Ap d Ap 0 where is the codifferential defined, form vanishes on each of the generators of the complemen. as 1 nk n 1 s d with s as the signature of the, tary subsimplex and by linearity it vanishes on the entire. metric The Lorenz gauge is automatically satisfied since the. complementary subsimplex Since there are n k vertices in. codifferential of any Whitney form is zero as can be seen. the complementary subsimplex one then V uses the Hodge. from 7 If we now calculate dA0p we find that, star to convert the n k differential form vj vj x X X p.

to a k form The overall effect of the above formula is dA0p aij wijk dAp. a covector field that rotates about the complementary. simplex The proof of the correct normalization is more thus our space time FEEC discretization automatically sat. involved and the most direct proof involves introducing an isfies gauge invariance at linear order and in particular. equivalent characterization of Whitney forms in terms of dF 0 is automatically satisfied Therefore integrating the. vector proxies which is given in the following Proposition associated discrete Noether current jp over an arbitrary 1. chain loop yields,Proposition 2 Let v0 v1 vn an ordered set of Z Z. vertex vectors represent an oriented n simplex on a flat n jp Fp dfp 0. dimensional manifold Let be a k subsimplex and, r be the ordered complement of in The WhitneyVk implying Noether s first theorem is upheld within this frame. Space Time Finite Element Exterior Calculus and Variational Discretizations of Gauge Field Theories Joe Salamon 1 John Moody 2 and Melvin Leok 3 Abstract Many gauge eld theories can be described using a multisymplectic Lagrangian formulation where the Lagrangian density involves space time differential forms While there has

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