Euler–Lagrange and Hamilton equations for non-holonomic systems in field theory

Krupková, Olga; Volný, Petr

doi:10.1088/0305-4470/38/40/015

Cited by 12 publications

(13 citation statements)

References 17 publications

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“…As W is admissible with respect to φ, we conclude that the right-hand side is zero, and this in turn implies the vanishing of the second term in (15). ✷…”

Section: Vertical Vector Fields Along the Projectionmentioning

confidence: 73%

Symmetry aspects of nonholonomic field theories

Vankerschaver¹,

Diego²

2008

J. Phys. A: Math. Theor.

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The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic symmetries are then introduced as symmetry generators whose virtual work is zero along the constraint submanifold, and we show that for every such symmetry, there exists a so-called momentum equation, describing the evolution of the associated component of the momentum map. Keeping up with the underlying geometric philosophy, a small modification of the derivation of the momentum lemma allows us to also treat generalized nonholonomic symmetries, which are vector fields along a projection. Such symmetries arise for example in practical examples of nonholonomic field theories such as the Cosserat rod, for which we recover both energy conservation (a previously known result) and a modified conservation law associated with spatial translations.

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“…As W is admissible with respect to φ, we conclude that the right-hand side is zero, and this in turn implies the vanishing of the second term in (15). ✷…”

Section: Vertical Vector Fields Along the Projectionmentioning

confidence: 73%

Symmetry aspects of nonholonomic field theories

Vankerschaver¹,

Diego²

2008

J. Phys. A: Math. Theor.

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“…where the first addendum is a (local) divergence, while the second one vanishes on the constraint. Thanks to the first variation (13) together with identity (14), the previous formula can be recasted into the identity (15), (16), (17) and (18) we find that the sum of the left hand sides vanishes if composed with a P S -critical section, while the current…”

Section: A Gauge-natural Example: Charged General Relativistic Fluidsmentioning

confidence: 86%

Gauge-Natural Parametrized Variational Problems, Vakonomic Field Theories and Relativistic Hydrodynamics of a Charged Fluid

Bibbona

Fatibene

Francaviglia

2006

Int. J. Geom. Methods Mod. Phys.

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Variational principles for field theories where variations of fields are restricted along a parametrization are considered. In particular, gauge-natural parametrized variational problems are defined as those in which both the Lagrangian and the parametrization are gauge covariant and some further conditions is satisfied in order to formulate a Nöther theorem that links horizontal and gauge symmetries to the relative conservation laws (generalizing what Fernández, García and Rodrigo did in some recent papers). The case of vakonomic constraints in field theory is also studied within the framework of parametrized variational problems, defining and comparing two different concepts of criticality of a section, one arising directly from the vakonomic schema, the other making use of an adapted parametrization. The general theory is then applied to the case of hydrodynamics of a charged fluid coupled with its gravitational and electromagnetic field. A variational formulation including conserved currents and superpotentials is given that turns out to be computationally much easier than the standard one.

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“…-adapted constraints and corresponding Lagrangian and Hamiltonian constrained systems are studied in [28].…”

Section: -Adapted Constraintsmentioning

confidence: 99%

Partial differential equations with differential constraints

Krupková¹

2006

Journal of Differential Equations

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A geometric setting for constrained exterior differential systems on fibered manifolds with n-dimensional bases is proposed. Constraints given as submanifolds of jet bundles (locally defined by systems of first-order partial differential equations) are shown to carry a natural geometric structure, called the canonical distribution. Systems of second-order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds. As an important particular case, Lagrangian systems subjected to first-order differential constraints are considered. Different kinds of constraints are introduced and investigated (Lagrangian constraints, constraints adapted to the fibered structure, constraints arising from a (co)distribution, semi-holonomic constraints, holonomic constraints).

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Euler–Lagrange and Hamilton equations for non-holonomic systems in field theory

Cited by 12 publications

References 17 publications

Symmetry aspects of nonholonomic field theories

Symmetry aspects of nonholonomic field theories

Gauge-Natural Parametrized Variational Problems, Vakonomic Field Theories and Relativistic Hydrodynamics of a Charged Fluid

Partial differential equations with differential constraints

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