Higher Order Lagrangians Research Articles

Geometric Formulation of Higher-Order Lagrangian Systems in Supermechanics

Using supervector fields along a morphism of graded manifolds, we study both the geometry of the tangent supermanifolds of an arbitrary order and k th-order ordinary differential superequations, in order to develop the higher-order Lagrangian formalism of supermechanics. In particular, we obtain, in this formalism, a generalization of Noether's theorem and its converse.

Canonical Quantal Symmetry and Conserved Charges for a System with Higher-Order Lagrangian11The project supported by National Natural Science Foundation of China under grant 19275009

Based on the phase space generating functional of Green function for a system with regular higher-order Lagrangian, the conserved charge is deduced at quantum level if the canonical action is invariant under the global symmetry transformation in extended phase space. The generalization of our results to a system with singular higher-order Lagrangian is also studied. In general the quantal conserved charges are different from classical ones.

Canonical global symmetry and quantal conservation laws for a system with singular higher order Lagrangian

Starting from the phase-space generating functional of the Green function for a system with singular higher order Lagrangian, the generalized canonical Ward identities under the global symmetry transformation in phase space is deduced. The local transformation connected with this global symmetry transformation is studied, and the quantal conservation laws are obtained for such a system. We give a preliminary application to higher derivative Yang-Mills theory; a generalized quantal BRS conserved quantity is found.

Higher-order gauge invariant Lagrangians on

Higher-order Lagrangians on T*(M) invariant under the natural representation of gauge fields of M x U(1) on the cotangent bundle are determined.

Gauge transformations for higher-order Lagrangians

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 second-order 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.

The Second Noether Theorem in the formalism of jet-bundles: Symmetries and degeneration

By means of the jet-bundle formalism, the Second Noether Theorem is formulated for a general first-order Lagrangian field theory with infinitesimal local symmetries. These symmetries are implemented by a linear differential operator acting between the sections of a vector bundle and vector fields on the configuration bundle. The problem of the degeneration of the Lagrangian system is examined from a covariant and an instantaneous (i.e. space+time split) viewpoint. It is shown that in the instantaneous approach the presence of infinitesimal local symmetries leads to degeneration of the theory. Vertical local symmetries are shown to imply degeneration also in the covariant formalism. These results can be extended to higher-order Lagrangians as well.

Generalized Noether identities for nonlocal transformations

Starting from the transformation properties of an action integral of a system under local and nonlocal transformations, we derive the generalized Noether identities for a variant system under those transformations. The applications of the theory to the Yang-Mills field with higher order Lagrangian is presented under the Coulomb gauge condition, a new conserved PBRS charge is found which differs from the BRS conserved charge, and another conserved charge connected with nonlocal transformation is also obtained.

Noether theorem in higher-order Lagrangian mechanics

We study higher-order Lagrangian mechanics on thek-velocity manifold. The variational problem gives rise to new concepts, such as main invariants, Zermelo conditions, higher-order energies, and new conservation laws. A theorem of Noether type is proved for higher-order Lagrangians. The invariants to the infinitesimal symmetries are explicitly written. All this construction is a natural extension of classical Lagrangian mechanics.

Symmetry in phase space for a system with a singular higher-order Lagrangian.

Symmetry in phase space for a system with a singular higher-order Lagrangian.

Transformation properties of a constrained hamiltonian system and PBRST charge

We derive a generalized first Noether theorem for weakly quasi-invariant systems with singular higher-order Lagrangians, subject to the extra constraints and generalized Noether identities for a variant system in phase space. The strong and weak conservation laws for variant systems are also deduced. Some preliminary applications to field theories are given. In certain cases a variant system is also a constrained Hamiltonian system. A PBRST (weak) conserved charge is obtained that differs from the usual BRST charge.

Fermion coupling to Chern-Simons theories with higher-order Lagrangian in (2+1) dimensions

The classical and quantum formalism for the constrained Hamiltonian system with singular higher-order Lagrangian describing the fermion coupling to Chern-Simons theories in (2+1) dimensions is constructed. We perform the canonical and path-integral quantizations for the Abelian case. The non-Abelian case is also discussed.

SYMMETRY IN FIELD THEORY FOR SINGULAR LAGRANGIAN WITH DERIVATIVES OF HIGHER ORDER

We have derived the first Noether theorem and Noether identities in canonical formalism for field theory with higher-order singular Lagrangian, which is a powerful tool to analyse Dirac constraint for such system. A gauge-variant system in canonical variables formalism must has Dirac constraint. For a system with first class constraint (FCC), we have developed an algorithm for construction of the gauge generator of such system. An application to the Podolsky generalized electromagnetic field was given.

Generalized Noether Theorem and Poincaré-Cartan Integral Invariant for Singular High-order Lagrangian in Fields Theories

A generalized first Noether theorem (GFNT) originating from the invariance under the finite continuous group for singular high-order Lagrangian and a generalized second Noether theorem (or generalized Noether identities (GNI)) for variant system under the infinite continuous group of field theory in canonical formalism are derived. The strong and weak conservation laws in canonical formalism are also obtained. It is pointed out that some variant systems also have Dirac constraint. Based on the canonical action, the generalized Poincare-Cartan integral invariant (GPCⅡ) for singular high-order Lagrangian in the field theory is deduced. Some confusions in literafure are clarified. The GPCⅡ connected with canonical equations and canonical transformation are discussed.

Global d-invariance in field theory

In a previous work the authors derived a remarkable local identity which allows the writing of any Lagrangian as a 'linear combination' of its field equations plus a divergence. Using this identity they were able to provide an alternative proof of the fact that a (higher-order) Lagrangian has identically vanishing field equations if and only if it is locally a divergence. The aim of this work is to investigate how far one can go in globalizing the previous results for (higher-order) Lagrangians. In the case of vector or affine bundles the previous results admit global generalizations in a natural way. The true obstacle is the topological structure of the fibre bundle (both of the basis manifold and of the fibres). As a general rule, it turns out that one can globalize in a non-unique way the previous results when the fibre bundle admits global sections and, moreover, it is contractible by fibred morphisms to one of its global sections. Uniqueness is lost at the level of affine bundles, and for 'non-trivial' topologies one loses the globality of the result.

A generalized geometric framework for constrained systems

A geometric framework for constrained dynamical systems is presented. It allows to describe in a unified way a general type of first order singular differential equations on a manifold; these equations can not be written in normal form since the derivatives appear multiplied by a linear operator, therefore we call them linearly constrained systems. The concepts of constraints and morphisms between linearly constrained systems are defined, and their relationships studied. Finally, a stabilization algorithm is devised and carefully discussed in order to solve the equation of motion. Our formalism includes the presymplectic and the lagrangian formalisms, as well as higher order lagrangians, and we give several applications of it; in particular, a stabilization algorithm for the lagrangian formalism is obtained.

Note on the representation of many-particle dynamics as higher-order single-particle dynamics, with a means to relativistic elevation of Newtonian dynamics

It is shown how (first in Newtonian physics), by processes of repeated differentiation of equations of motion and of algebraic elimination, the dynamics of many particles may be brought to another equivalent representation, that of a single higher-order equation of motion for a single particle. Here a higher-order Lagrangian followed by an Ostrogradsky Hamiltonian may be brought into play. (This leads parenthetically to the construction of a considerable class of canonically inequivalent specifications of one basic set of equations of motion.) The higher-order one-particle representation translates simply and directly (but not uniquely) into relativistic generalization (because only a single world line is being described), in which the Poincaré group is canonically represented in an Ostrogradsky Hamiltonian formulation. The examples of the Kepler problem and the harmonic oscillator are elaborated in detail.

Equivalence of higher-order Lagrangians. III. New invariant differential equations

For pt.I see J. Math. Appliquees, vol.70, p.369 (1991). The solution to the equivalence problem for a higher-order Lagrangian leads to new differential equations which are invariantly associated with the variational functional. The authors derive explicit expressions for these equations in the case of second-order particle Lagrangians on the line under fibre-preserving, point and contact transformations. A geometrical interpretation of these equations based on the Poincare-Cartan form is discussed.

Higher-order Lagrangian systems: Geometric structures, dynamics, and constraints

In order to study the connections between Lagrangian and Hamiltonian formalisms constructed from a—perhaps singular—higher-order Lagrangian, some geometric structures are constructed. Intermediate spaces between those of Lagrangian and Hamiltonian formalisms, partial Ostrogradskiĭ’s transformations and unambiguous evolution operators connecting these spaces are intrinsically defined, and some of their properties studied. Equations of motion, constraints, and arbitrary functions of Lagrangian and Hamiltonian formalisms are thoroughly studied. In particular, all the Lagrangian constraints are obtained from the Hamiltonian ones. Once the gauge transformations are taken into account, the true number of degrees of freedom is obtained, both in the Lagrangian and Hamiltonian formalisms, and also in all the ‘‘intermediate formalisms’’ herein defined.

Symmetry in a constrained Hamiltonian system with singular higher-order Lagrangian

For a canonical formalism with a higher-order derivative, the corresponding generalized first Noether theorem (GFNT) for a constrained Hamiltonian system and the generalized Noether identities (GNI) for a system with a non-invariant action integral are derived, which may be useful to analyse the Dirac constraint for such a system. Using the GFNT another example is given in which Dirac's conjecture fails; using the GNI the strong and weak conservation laws are deduced and it is pointed out that for certain variant systems there is also a Dirac constraint. Suppose that there are only first-class constraints (FCC) in a system, then an algorithm for the construction of a gauge generator is developed, once the Hamiltonian and the FCC of the system with a higher-order Lagrangian are given.

Quantization of a higher-order derivative spinning particle

The quantization of a higher-order derivative Lagrangian for a relativistic spinning particle is performed. The Dirac equation is recovered.