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Massimo Pauri - One of the best experts on this subject based on the ideXlab platform.
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standard and generalized newtonian gravities as gauge theories of the extended galilei group i the standard theory
Classical and Quantum Gravity, 1995Co-Authors: R De Pietri, Luca Lusanna, Massimo PauriAbstract:Newton's standard theory of gravitation is reformulated in terms of a generally Galilei-covariant Action Principle as a gauge theory of the extended Galilei group. A suitable modification of Utiyama's method for gauging the projective realization of the Galilei group associated with the free mass-point, together with the connections between the consequent 11 external gauge fields and known facts about Galilean and Newtonian geometrical structures, are discussed from a unified point of view. Then the problem of the existence of an Action Principle for the dynamical evolution of the gauge fields is analysed. Since it is not known how to extend Utiyama's method from the case of `invariance' to the case of `quasi-invariance, modulo the equations of motion', which turns out to be the key factor in the Galilean case, an Action Principle is derived, starting from a suitable power expansion of the 4-metric tensor, as a contrAction, for , of the ADM-De-Witt Action of general relativity. The Galilean Action depends on 27 fields (i.e. it contains 16 auxiliary fields besides the 11 gauge fields) and is indeed quasi-invariant, modulo the equations of motion under general Galilean coordinate transformations. The physical equivalence of this theory and Newton's theory of gravity is shown explicitly, by analysing its first- and second-class constraints. Finally, we discuss the feasibility of a symplectic reduction of the 27-fields theory to a minimal theory depending on only the 11 gauge fields, in the sense of Utiyama's method.
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standard and generalized newtonian gravities as gauge theories of the extended galilei group i the standard theory
arXiv: General Relativity and Quantum Cosmology, 1994Co-Authors: R De Pietri, Luca Lusanna, Massimo PauriAbstract:Newton's standard theory of gravitation is reformulated as a {\it gauge} theory of the {\it extended} Galilei Group. The Action Principle is obtained by matching the {\it gauge} technique and a suitable limiting procedure from the ADM-De Witt Action of general relativity coupled to a relativistic mass-point.
R De Pietri - One of the best experts on this subject based on the ideXlab platform.
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standard and generalized newtonian gravities as gauge theories of the extended galilei group i the standard theory
Classical and Quantum Gravity, 1995Co-Authors: R De Pietri, Luca Lusanna, Massimo PauriAbstract:Newton's standard theory of gravitation is reformulated in terms of a generally Galilei-covariant Action Principle as a gauge theory of the extended Galilei group. A suitable modification of Utiyama's method for gauging the projective realization of the Galilei group associated with the free mass-point, together with the connections between the consequent 11 external gauge fields and known facts about Galilean and Newtonian geometrical structures, are discussed from a unified point of view. Then the problem of the existence of an Action Principle for the dynamical evolution of the gauge fields is analysed. Since it is not known how to extend Utiyama's method from the case of `invariance' to the case of `quasi-invariance, modulo the equations of motion', which turns out to be the key factor in the Galilean case, an Action Principle is derived, starting from a suitable power expansion of the 4-metric tensor, as a contrAction, for , of the ADM-De-Witt Action of general relativity. The Galilean Action depends on 27 fields (i.e. it contains 16 auxiliary fields besides the 11 gauge fields) and is indeed quasi-invariant, modulo the equations of motion under general Galilean coordinate transformations. The physical equivalence of this theory and Newton's theory of gravity is shown explicitly, by analysing its first- and second-class constraints. Finally, we discuss the feasibility of a symplectic reduction of the 27-fields theory to a minimal theory depending on only the 11 gauge fields, in the sense of Utiyama's method.
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standard and generalized newtonian gravities as gauge theories of the extended galilei group i the standard theory
arXiv: General Relativity and Quantum Cosmology, 1994Co-Authors: R De Pietri, Luca Lusanna, Massimo PauriAbstract:Newton's standard theory of gravitation is reformulated as a {\it gauge} theory of the {\it extended} Galilei Group. The Action Principle is obtained by matching the {\it gauge} technique and a suitable limiting procedure from the ADM-De Witt Action of general relativity coupled to a relativistic mass-point.
Luca Lusanna - One of the best experts on this subject based on the ideXlab platform.
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standard and generalized newtonian gravities as gauge theories of the extended galilei group i the standard theory
Classical and Quantum Gravity, 1995Co-Authors: R De Pietri, Luca Lusanna, Massimo PauriAbstract:Newton's standard theory of gravitation is reformulated in terms of a generally Galilei-covariant Action Principle as a gauge theory of the extended Galilei group. A suitable modification of Utiyama's method for gauging the projective realization of the Galilei group associated with the free mass-point, together with the connections between the consequent 11 external gauge fields and known facts about Galilean and Newtonian geometrical structures, are discussed from a unified point of view. Then the problem of the existence of an Action Principle for the dynamical evolution of the gauge fields is analysed. Since it is not known how to extend Utiyama's method from the case of `invariance' to the case of `quasi-invariance, modulo the equations of motion', which turns out to be the key factor in the Galilean case, an Action Principle is derived, starting from a suitable power expansion of the 4-metric tensor, as a contrAction, for , of the ADM-De-Witt Action of general relativity. The Galilean Action depends on 27 fields (i.e. it contains 16 auxiliary fields besides the 11 gauge fields) and is indeed quasi-invariant, modulo the equations of motion under general Galilean coordinate transformations. The physical equivalence of this theory and Newton's theory of gravity is shown explicitly, by analysing its first- and second-class constraints. Finally, we discuss the feasibility of a symplectic reduction of the 27-fields theory to a minimal theory depending on only the 11 gauge fields, in the sense of Utiyama's method.
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standard and generalized newtonian gravities as gauge theories of the extended galilei group i the standard theory
arXiv: General Relativity and Quantum Cosmology, 1994Co-Authors: R De Pietri, Luca Lusanna, Massimo PauriAbstract:Newton's standard theory of gravitation is reformulated as a {\it gauge} theory of the {\it extended} Galilei Group. The Action Principle is obtained by matching the {\it gauge} technique and a suitable limiting procedure from the ADM-De Witt Action of general relativity coupled to a relativistic mass-point.
G. Rosi - One of the best experts on this subject based on the ideXlab platform.
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analytical continuum mechanics a la hamilton piola least Action Principle for second gradient continua and capillary fluids
Mathematics and Mechanics of Solids, 2015Co-Authors: Nicolas Auffray, V. Eremeyev, A. Madeo, Francesco Dellisola, G. RosiAbstract:In this paper a stationary Action Principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg–de Vries or Cahn–Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola–Toupin, Mindlin, Green–Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian Action is introduced in both the material and spatial descriptions and the corresponding Euler–Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C−1 and ∇C−1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materia...
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Least Action Principle for second gradient continua and capillary fluids: a Lagrangian approach following Piola's point of view
2014Co-Authors: Nicolas Auffray, F. Dell'isola, V. Eremeyev, A. Madeo, Luca Placidi, G. RosiAbstract:As Piola would have surely conjectured, the stationary Action Principle holds also for capillary uids, i.e. those fluids for which the deformation energy depends on spatial derivative of mass density (a modelling necessity which has been already remarked by Cahn and Hilliard [15, 16]). For capillary fluids it is indeed possible to de fine a Lagrangian density function whose corresponding Euler-Lagrange stationarity conditions once transported on the actual con guration, via a Piola's transformation, are exactly those obtained, with di fferent methods, in the literature. We recall that some particulat classes of second gradient fluids are sometimes also called Korteweg-de Vries or Cahn-Allen fluids. More generally those continua (which may be solid or fluid) whose deformation energy depends on the second gradient of placement are called second gradient (or Piola-Toupin or Mindlin or Green-Rivlin or Germain or second grade) continua. In the present work, following closely the procedure fi rst conceived by Piola and carefully presented in his works translated in the present volume, a material (Lagragian) description for second gradient continua is formulated. Subsequently a Lagrangian Action is introduced and by means of Piola's transformations this Action is calculated in both the material and spatial descriptions. Then the corresponding Euler-Lagrange equations and boundary conditions are calculated by using some kinematical relationships suitably established. Once an objective deformation energy volume density is assumed to depend on either C and ∇C or on C^(-1) and (where C is the Cauchy-Green deformation tensor) the particular form of aforementioned Euler-Lagrange conditions and boundary conditions are established. When further particularizing the treatment to those energies which characterize fl uid materials, the capillary fluid evolution conditions (see e.g. Casal [25] or Seppecher [142, 145] for an alternative deduction based on thermodynamic arguments) are recovered. Also a version of Bernoulli's law which is valid for capillary fluids is found and, in Appendix B, all the kinematic formulas which we have found useful for the present variational formulation are gathered. Many historical comments about Gabrio Piola's contribution to analytical continuum mechanics are also presented when it has been considered useful. In this context the reader is also referred to Capecchi and Ruta [17].
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Analytical continuum mechanics à la Hamilton-Piola: least Action Principle for second gradient continua and capillary fluids
Mathematics and Mechanics of Solids, 2013Co-Authors: Nicolas Auffray, F. Dell'isola, V. Eremeyev, A. Madeo, G. RosiAbstract:In this paper a stationary Action Principle is proven to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments, for instance by Cahn and Hilliard. Remark that these fluids are sometimes also called Korteweg-de Vries or Cahn-Allen. In general continua whose deformation energy depend on the second gradient of placement are called second gradient (or Piola-Toupin or Mindlin or Green-Rivlin or Germain or second gradient) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian Action is introduced in both material and spatial description and the corresponding Euler-Lagrange bulk and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and grad C or on C^-1 and grad C^-1 ; where C is the Cauchy-Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions (see e.g. Casal or Seppecher for an alternative deduction based on thermodynamic arguments) are recovered. A version of Bernoulli law valid for capillary fluids is found and, in the Appendix B, useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola's contribution to continuum analytical mechanics are also presented. In this context the reader is also referred to Capecchi and Ruta.
Pablo A Morales - One of the best experts on this subject based on the ideXlab platform.
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field theory of the eulerian perfect fluid
Classical and Quantum Gravity, 2018Co-Authors: Taketo Ariki, Pablo A MoralesAbstract:The Eulerian perfect-fluid theory is reformulated from its Action Principle in a pure field-theoretic manner. Conservation of the convective current is no longer imposed by Lin's constraints, but rather adopted as the central idea of the theory. Our formulation, for the first time, successfully reduces redundant degrees of freedom promoting one half of the Clebsch variables as the true dynamical fields. InterActions on these fields allow for the exchange of the convective current of quantities such as mass and charge, which are uniformly understood as the breaking of the underlying symmetry of the force-free fluid. The Clebsch fields play the essential role in the exchange of angular momentum with the force field producing vorticity.
