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117
Discrete mechanics and variational integrators
 Acta Numer
, 2001
"... This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the disc ..."
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Cited by 284 (34 self)
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This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic
Asynchronous Variational Integrators
 ARCH. RATIONAL MECH. ANAL.
, 2003
"... We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation t ..."
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Cited by 66 (11 self)
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We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system. A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.
Canonical structure of classical field theory in the polymomentum phase space
 Rep. Math. Phys
, 1998
"... Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder–Weyl (DW). The bivertical (n + 1)form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given i ..."
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Cited by 66 (10 self)
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Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder–Weyl (DW). The bivertical (n + 1)form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given in intrinsic geometric terms differently than a certain coset it gives rise to an invariantly defined map between horizontal forms playing the role of dynamical variables and the socalled vertical multivectors generalizing Hamiltonian vector fields. The analogue of the Poisson bracket on forms is defined which leads to the structure of Zgraded Lie algebra on the socalled Hamiltonian forms for which the map above exists. A generalized Poisson structure appears in the form of what we call a “higherorder ” and a right Gerstenhaber algebra. The equations of motion of forms are formulated in terms of the Poisson bracket with the DW Hamiltonian nform H ˜ vol ( ˜ vol is the spacetime volume form, H is the DW Hamiltonian function) which is found to be related to the operation of the total exterior differentiation of forms. A few applications and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. 1
Categorified symplectic geometry and the classical string
, 2008
"... A Lie 2algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poi ..."
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Cited by 48 (10 self)
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A Lie 2algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an ndimensional field theory using a phase space that is an ‘nplectic manifold’: a finitedimensional manifold equipped with a closed nondegenerate (n + 1)form. Here we consider the case n = 2. For any 2plectic manifold, we construct a Lie 2algebra of observables. We then explain how this Lie 2algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2plectic structure for the string.
Geometry of Multisymplectic Hamiltonian Firstorder Field Theories
 J. Math. Phys
, 2000
"... In the jet bundle description of Field Theories (multisymplectic models, in particular), there are several choices for the multimomentum bundle where the covariant Hamiltonian formalism takes place. As a consequence, several proposals for this formalism can be stated, and, on each one of them, the d ..."
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Cited by 36 (10 self)
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In the jet bundle description of Field Theories (multisymplectic models, in particular), there are several choices for the multimomentum bundle where the covariant Hamiltonian formalism takes place. As a consequence, several proposals for this formalism can be stated, and, on each one of them, the differentiable structures needed for setting the formalism are obtained in different ways. In this work we make an accurate study of some of these Hamiltonian formalisms, showing their equivalence. In particular, the geometrical structures (canonical or not) needed for the Hamiltonian formalism, are introduced and compared, and the derivation of Hamiltonian field equations from the corresponding variational principle is shown in detail. Furthermore, the Hamiltonian formalism of systems described by Lagrangians is performed, both for the hyperregular and almostregular cases. Finally, the role of connections in the construction of Hamiltonian Field theories is clarified.
Multivector field formulation of Hamiltonian field theories: Equations and symmetries
, 1999
"... We state the intrinsic form of the Hamiltonian equations of firstorder Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are u ..."
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Cited by 31 (13 self)
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We state the intrinsic form of the Hamiltonian equations of firstorder Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given. Then, using multivector fields, we study several aspects of these equations, such as the existence and nonuniqueness of solutions, and the integrability problem. In particular, these problems are analyzed for the case of Hamiltonian systems defined in a submanifold of the multimomentum bundle. Furthermore, the existence of first integrals of these Hamiltonian equations is considered, and the relation between CartanNoether symmetries and general symmetries of the system is discussed. Noether’s theorem is also stated in this context, both the “classical” version and its generalization to include higherorder CartanNoether symmetries. Finally, the equivalence between the Lagrangian and Hamiltonian formalisms is also discussed.
CLASSICAL FIELD THEORY ON LIE ALGEBROIDS: MULTISYMPLECTIC FORMALISM
, 2004
"... The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in ..."
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Cited by 29 (3 self)
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The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan form canonically associated to the Lagrangian. The Hamiltonian formalism is also extended to this setting and we find the relation between the solutions of both formalism. When the first Lie algebroid is a tangent bundle we give a variational description of the equations of motion. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant EulerPoincaré and Lagrange Poincaré cases), variational problems for holomorphic maps, Sigma models or ChernSimons theories. One of the advantages of our theory is that it is based in the existence of a multisymplectic form on a Lie algebroid.
CATEGORIFIED SYMPLECTIC GEOMETRY AND THE STRING LIE 2ALGEBRA
"... Abstract. Multisymplectic geometry is a generalization of symplectic geometry suitable for ndimensional field theories, in which the nondegenerate 2form of symplectic geometry is replaced by a nondegenerate (n + 1)form. The case n = 2 is relevant to string theory: we call this ‘2plectic geometry ..."
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Cited by 28 (8 self)
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Abstract. Multisymplectic geometry is a generalization of symplectic geometry suitable for ndimensional field theories, in which the nondegenerate 2form of symplectic geometry is replaced by a nondegenerate (n + 1)form. The case n = 2 is relevant to string theory: we call this ‘2plectic geometry’. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the observables associated to a 2plectic manifold form a ‘Lie 2algebra’, which is a categorified version of a Lie algebra. Any compact simple Lie group G has a canonical 2plectic structure, so it is natural to wonder what Lie 2algebra this example yields. This Lie 2algebra is infinitedimensional, but we show here that the subLie2algebra of leftinvariant observables is finitedimensional, and isomorphic to the already known ‘string Lie 2algebra ’ associated to G. So, categorified symplectic geometry gives a geometric construction of the string Lie 2algebra. 1.
Stressenergymomentum tensors and the BelifanteResenfeld formula
, 2001
"... We present a new method of constructing a stressenergymomentum tensor for a classical field theory based on covariance considerations and Noether theory. The stressenergymomentum tensor T µ ν that we construct is defined using the (multi)momentum map associated to the spacetime diffeomorphism gr ..."
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Cited by 24 (6 self)
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We present a new method of constructing a stressenergymomentum tensor for a classical field theory based on covariance considerations and Noether theory. The stressenergymomentum tensor T µ ν that we construct is defined using the (multi)momentum map associated to the spacetime diffeomorphism group. The tensor T µ ν is uniquely determined as well as gaugecovariant, and depends only upon the divergence equivalence class of the Lagrangian. It satisfies a generalized version of the classical BelinfanteRosenfeld formula, and hence naturally incorporates both the canonical stressenergymomentum tensor and the “correction terms ” that are necessary to make the latter well behaved. Furthermore, in the presence of a metric on spacetime, our T µν coincides with the Hilbert tensor and hence is automatically symmetric.
On the Canonical Structure of the De DonderWeyl Covariant Hamiltonian Formulation of Field Theory I. Graded Poisson brackets and equations of motion, Aachen
 hepth/9312162; I.V. Kanatchikov, Canonical Structure of Classical Field Theory in the Polymomentum Phase Space
, 1993
"... As opposed to the conventional fieldtheoretical Hamiltonian formalism, which requires the space+time decomposition and leads to the picture of a field as a mechanical system with infinitely many degrees of freedom, the De DonderWeyl (DW) Hamiltonian canonical formulation of field theory (which is ..."
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Cited by 24 (5 self)
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As opposed to the conventional fieldtheoretical Hamiltonian formalism, which requires the space+time decomposition and leads to the picture of a field as a mechanical system with infinitely many degrees of freedom, the De DonderWeyl (DW) Hamiltonian canonical formulation of field theory (which is known for about 60 years) keeps the spacetime symmetry explicit, works in the finite dimensional analogue of the phase space and leads to the Hamiltonian and HamiltonJacobi formulations of field equations in terms of partial derivative equations. No field quantization procedure based on this ”finite dimensional” covariant canonical formalism is known. As a first step in this direction we consider the appropriate generalization of the Poisson bracket concept to the DW Hamiltonian formalism and the expression of the DW Hamiltonian form of field equations in terms of these generalized Poisson brackets. Starting from the PoincaréCartan form of the multidimensional variational calculus we argue that the analogue of the Poisson brackets is defined on forms