On the Legendre transformation for a class of nonregular higher-order Lagrangian field theories (original) (raw)

1990, Journal of Mathematical Physics

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On the Construction of K Operators in Field Theories as Sections Along Legendre Maps

Acta Applicandae Mathematicae, 2003

The ‘time-evolution K-operator’ (or ‘relative Hamiltonian vector field’) in mechanics is a powerful tool which can be geometrically defined as a vector field along the Legendre map. It has been extensively used by several authors for studying the structure and properties of the dynamical systems (mainly the nonregular ones), such as the relation between the Lagrangian and Hamiltonian formalisms, constraints, and higher-order mechanics. This paper is devoted to defining a generalization of this operator for field theories, in a covariant formulation. In order to do this, we use sections along maps, in particular multivector fields (skew-symmetric contravariant tensor fields of order greater than 1), jet fields and connection forms along the Legendre map. As a relevant result, we use these geometrical objects to obtain the solutions of the Lagrangian and Hamiltonian field equations, and the equivalence among them (specially for nonregular field theories).

On the multimomentum bundles and the Legendre maps in field theories

Reports on Mathematical Physics, 2000

We study the geometrical background of the Hamiltonian formalism of first-order Classical Field Theories. In particular, different proposals of multimomentum bundles existing in the usual literature (including their canonical structures) are analyzed and compared. The corresponding Legendre maps are introduced. As a consequence, the definition of regular and almost-regular Lagrangian systems is reviewed and extended from different but equivalent ways.

A nonlinear Hamiltonian formalism for singular Lagrangian theories

We introduce a "nonlinear" version of the Hamiltonian formalism which allows a self-consistent description of theories with degenerate Lagrangian. A generalization of the Legendre transform to the case when the Hessian is zero is done using the mixed (envelope/general) solutions of the multidimensional Clairaut equation. The corresponding system of equations of motion is equivalent to the Lagrange equations, but contains "nondynamical" momenta and unresolved velocities. This system is reduced to the physical phase space and presented in the Hamiltonian form by introducing a new (non-Lie) bracket.

A new Hamiltonian formalism for singular Lagrangian theories

2009

Ju l 2 01 0 Modified Hamiltonian formalism for higher-derivative theories

The alternative version of Hamiltonian formalism for higher-derivative theories is proposed. As compared with the standard Ostrogradski approach it has the following advantages: (i) the Lagrangian, when expressed in terms of new variables yields proper equations of motion; no additional Lagrange multipliers are necessary (ii) the Legendre transformation can be performed in a straightforward way provided the Lagrangian is nonsingular in Ostrogradski sense. The generalizations to singular Lagrangians as well as field theory are presented.

Legendre transformations and Clairaut-type equations

Physics Letters B, 2016

It is noted that the Legendre transformations in the standard formulation of quantum field theory have the form of functional Clairaut-type equations. It is shown that in presence of composite fields the Clairaut-type form holds after loop corrections are taken into account. A new solution to the functional Clairaut-type equation appearing in field theories with composite fields is found.

A new Lagrangian dynamic reduction in field theory

Annales de l’institut Fourier, 2010

L'accès aux articles de la revue « Annales de l'institut Fourier » (http://aif.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://aif.cedram.org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

Modified Hamiltonian formalism for higher-derivative theories

Physical Review D, 2010

Gauge transformations for higher-order Lagrangians

Journal of Physics A: Mathematical and General, 1995

Noether's symmetry transformations for higher-order lagrangians are studied. A characterization of these transformations is presented, which is useful to find gauge transformations for higher-order singular lagrangians. The case of secondorder lagrangians is studied in detail. Some examples that illustrate our results are given; in particular, for the lagrangian of a relativistic particle with curvature, lagrangian gauge transformations are obtained, though there are no hamiltonian gauge generators for them.

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Singular Lagrangians: some geometric structures along the Legendre map

Journal of Physics A: Mathematical and General, 2001

New geometric structures that relate the lagrangian and hamiltonian formalisms defined upon a singular lagrangian are presented. Several vector fields are constructed in velocity space that give new and precise answers to several topics like the projectability of a vector field to a hamiltonian vector field, the computation of the kernel of the presymplectic form of lagrangian formalism, the construction of the lagrangian dynamical vector fields, and the characterisation of dynamical symmetries.

Classical duals, Legendre transforms and the Vainshtein mechanism

Journal of High Energy Physics, 2012

We show how to generalize the classical duals found by Gabadadze et al to a very large class of self-interacting theories. This enables one to adopt a perturbative description beyond the scale at which classical perturbation theory breaks down in the original theory. This is particularly relevant if we want to test modified gravity scenarios that exhibit Vainshtein screening on solar system scales. We recognise the duals as being related to the Legendre transform of the original Lagrangian, and present a practical method for finding the dual in general; our methods can also be applied to self-interacting theories with a hierarchy of strong coupling scales, and with multiple fields. We find the classical dual of the full quintic galileon theory as an example.

From Hamiltonians and Lagrangians to Legendrians

2006

The aim of the paper is to define Legendrians and their dual objects, Legendriens ⁄ , as generalizations of Lagrangians and ane Finsler and Lagrange spaces of higher order were studied in (9, 4) using the bundles of accelerations T k M ! M. Related to T k M, we consider in this paper the ane bundle T k M ! T ki1 M. A dual theory of higher order Hamilton spaces was recently studied in (5, 6) using the dual bundles T ki1 M £M T ⁄ M = T k⁄ M ! M. Related to T k M and T k⁄ M, we consider in this paper the ane exact Legendrian ⁄ of order k on M. The forms of closed Lagrangians and Lagrangians ⁄ of order k ‚ 1 are given by Propositions 1.1 and 1.5 respectively. The top components of a Legendrian or of a Legendrian ⁄ are particular cases of a top Legendrian and of a top Legendrian ⁄ respectively. If these are non-degenerated, the Legendrian, respectively the Legendrian ⁄ is called regular. We prove that the regular Legendrians and Legendrians ⁄ are in duality by Legendre transformations (T...

2 Lagrangian formalism 2 . 1 The setting for classical field theories

2007

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.

A note on the Hamiltonian formalism for higher-derivative theories

An alternative version of Hamiltonian formalism for higher-derivative theories is presented. It is related to the standard Ostrogradski approach by a canonical transformation. The advantage of the approach presented is that the Lagrangian is nonsingular and the Legendre transformation is performed in a straightforward way.

Infinities of Nonlinear and Lagrangian Theories

Physical Review, 1969

The technique for summing perturbation contributions introduced by Efimov and Fradkin is extended and applied to nonlinear (chiral) Lagrangian theories. It is shown that the only likely infinites in these theories are those associated with self-mass and self-charge. 1 W)NE of the significant recent advances in particle theory has been the formulation of chirally in- variant Lagrangian theories. ' These theories have so far been used with reasonable success for predicting low- energy (soft-meson) amplitudes in the following way: The interaction Lagrangianan exponential or rational function of the spin-zero meson fields p'is expanded as an infinite power series in y' and then used to eval- uate tree-diagram' contributions to the amplitudes. Clearly, at the next level of sophistication one is inter- ested in the closed-loop contributions, at which stage two related problems arise: (i) Since the Lagrangian itself is expressed as an infinite power series, 2;"&= P a g"y" (By)', the number of perturbation diagrams in each order n increases (typically) as fast or faster than n!. On any reasonable estimate, the perturbation expansion must be a diver- gent series. For respectable theories like quantum electrodynamics, with Lagrangians which are poly- nomials in the field variables, one has always suspected' that the perturbation expansion provides an asymptotic series in e')Iic; here, with Lagrangians which are them- selves infinite series, this behavior appears to be a virtual certainty. (ii) Each of the terms in the expansion of the Lagran- gian [terms like y"(By)2; n~&1j represents a nonre- normalizable interaction in the conventional sense. The ultraviolet infinities of the perturbation expansion there- fore get progressively more virulent. On the face of it, this is rather surprising, since it is well known that every nonlinear theory can be reformulated as a theory of linear group representations4 with polynomial Lagrangians together with a certain number of constraints on the fields q '. Before the imposition of the constraint,

Bound-State Problem in a Model Nonpolynomial-Lagrangian Theory

Physical Review D, 1973

We have shown that the bound-state problem in a nonpolynomial-Lagrangian theory can be solved as in the usual polynomial theory assuming that a Wick rotation is admissible. For definiteness we have assumed the interaction of two scalar fields interacting via the exchange of a superfield to be of the form U (x) = expig Ip(x)], where g denotes the minor coupling constant and Ip(x) is a massless neutral scalar field. The major coupling constant is introduced through the ladder diagrams in the Bethe-Salpeter formalism. We find that the Wick-rotated Bethe-Salpeter equation reduces to a standard Fredholm equation with a modified kernel corresponding to the exchange of the superfield U (x). To study the physical content in the theory we have investigated the equation in the instantaneous approximation. The resulting nonrelativistic equation is projected onto the surface of a four-dimensional sphere by using Fock's transformation variables. The bound-state eigenvalue problem is solved approximately in the weak-binding limit, using Hecke's theorem, leading to a Balmer-type formula. Finally, the fully relativistic equation at E = 0 is considered by transforming it onto the surface of a five-dimensional Euclidean sphere. The approximate-symmetry property of the equation is studied, and the eigenvalue problem is solved in terms of the coupling constants of the theory.

Legendre transformation and analytical mechanics: A geometric approach

Journal of Mathematical Physics, 2003

A revisitation of the Legendre transformation in the context of affine principal bundles is presented. The argument, merged with the gauge-theoretical considerations developed in [1], provides a unified representation of Lagrangian and Hamiltonian Mechanics, extending to arbitrary non-autonomous systems the symplectic approach of W.M. Tulczyjew.

Legendre transforms and rrr-particle irreducibility in quantum field theory : the formalism for fermions

Annales De L Institut Henri Poincare-physique Theorique, 1985

L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ p. 29

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Physical equivalence between nonlinear gravity theories and a general-relativistic self-gravitating scalar field

Physical Review D, 1994

We argue that in a nonlinear gravity theory (the Lagrangian being an arbitrary function of the curvature scalar R), which according to well-known results is dynamically equivalent to a self-gravitating scalar field in General Relativity, the true physical variables are exactly those which describe the equivalent general-relativistic model (these variables are known as Einstein frame). Whenever such variables cannot be defined, there are strong indications that the original theory is unphysical, in the sense that Minkowski space is unstable due to existence of negative-energy solutions close to it. To this aim we first clarify the global net of relationships between the nonlinear gravity theories, scalar-tensor theories and General Relativity, showing that in a sense these are "canonically conjugated" to each other. We stress that the isomorphisms are in most cases local; in the regions where these are defined, we explicitly show how to map, in the presence of matter, the Jordan frame to the Einstein one and backwards. We study energetics for asymptotically flat solutions for those Lagrangians which admit conformal rescaling to Einstein frame in the vicinity of flat space. This is based on the second-order dynamics obtained, without changing the metric, by the use of a Helmholtz Lagrangian. We prove for a large class of these Lagrangians that the ADM energy is positive for solutions close to flat space, and this is determined by the lowest-order terms, R + aR 2 , in the Lagrangian. The proof of this Positive Energy Theorem relies on the existence of the Einstein frame, since in the (Helmholtz-)Jordan frame the Dominant Energy Condition does not hold and the field variables are unrelated to the total energy of the system. This is why we regard the Jordan frame as unphysical, while the Einstein frame is physical and reveals the physical contents of the theory. The latter should hence be viewed as physically equivalent to a self-interacting general-relativistic scalar field.

Equivalence of black hole thermodynamics between a generalized theory of gravity and the Einstein theory

Physical Review D, 1998

We analyze black hole thermodynamics in a generalized theory of gravity whose Lagrangian is an arbitrary function of the metric, the Ricci tensor and a scalar field. We can convert the theory into the Einstein frame via a "Legendre" transformation or a conformal transformation. We calculate thermodynamical variables both in the original frame and in the Einstein frame, following the Iyer-Wald definition which satisfies the first law of thermodynamics. We show that all thermodynamical variables defined in the original frame are the same as those in the Einstein frame, if the spacetimes in both frames are asymptotically flat, regular and possess event horizons with non-zero temperatures. This result may be useful to study whether the second law is still valid in the generalized theory of gravity.

Conformal transformations in cosmology of modified gravity: the covariant approach perspective

General Relativity and Gravitation, 2010

The 1+3 covariant approach and the covariant gauge-invariant approach to perturbations are used to analyze in depth conformal transformations in cosmology. Such techniques allow us to obtain insights on the physical meaning of these transformations when applied to non-standard gravity. The results obtained lead to a number of general conclusions on the change of some key quantities describing any two conformally related cosmological models. For example, even if some of the geometrical properties of the cosmology are preserved (homogeneous and isotropic Universes are mapped into homogeneous and isotropic universes), it can happen that decelerating cosmologies can be mapped into accelerated ones. From the point of view of the cosmological perturbations it is shown how these fluctuation transform. We find that first-order vector and tensor perturbations equations are left unchanged in their structure by the conformal transformation, but this cannot be said of the scalar perturbations, which present differences in their evolutionary features. The results obtained are then explicitly interpreted and verified with the help of some clarifying examples based on f (R)-gravity cosmologies. Keywords conformal transformations • modified gravity • cosmology • 1+3 covariant approach • covariant gauge invariant theory of perturbations

Higher-derivative boson field theories and constrained second-order theories

Journal of Physics A: Mathematical and General, 2001

As an alternative to the covariant Ostrogradski method, we show that higherderivative relativistic Lagrangian field theories can be reduced to second differentialorder by writing them directly as covariant two-derivative theories involving Lagrange multipliers and new fields. Despite the intrinsic non-covariance of the Dirac's procedure used to deal with the constraints, the explicit Lorentz invariance is recovered at the end. We develop this new setting on the grounds of a simple scalar model and then its applications to generalized electrodynamics and higher-derivative gravity are worked out. For a wide class of field theories this method is better suited than Ostrogradski's for a generalization to 2n-derivative theories.

On the Legendre transformation for a class of nonregular higher-order Lagrangian field theories (original) (raw)

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