Geometry of Lagrangian First-order Classical Field Theories

We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a first-order jet bundle and, using them, we develop the lagrangian formalism, defining the canonical forms associated with a lagrangian density and the density of lagrangian energy, obtaining the Euler-Lagrange equations in two equivalent ways: as the result of a variational problem and developing the jet field formalism (which is a formulation more similar to the case of mechanical systems). A statement and proof of Noether's theorem is also given, using the latter formalism.     

The Multimomentum hamiltonian formalism in gauge theory

We extend the jet bundle machinery of gauge theory to the multimomentum Hamiltonian formalism. This enables us to manipulate finite-dimensional momentum spaces of fields. In the framework of this formalism, time and spatial coordinates are regarded on the same footing, and a preliminary (3 + 1) splitting of a world manifold is not required. We get the canonical splitting of a multimomentum Hamiltonian form into a connection part and a Hamiltonian density.     

Parameter Invariance in Field Theory and the Hamiltonian Formalism

The deparametrization problem for parameter‐invariant Lagrangian densities defined over J¹(N, F), is solved in terms of a projection onto a suitable jet bundle. The Hamilton‐Cartan formalism for such Lagrangians is then introduced and the pre‐symplectic structure of such variational problems is proved to be projectable through the aforementioned projection. Specific examples with physical meaning are also analyzed. 1998 PACS codes. 02.20.Tw Infinite‐dimensional Lie groups, 02.30.Wd Calculus of variations and optimal control, 02.40.Ky Riemannian geometries, 02.40.Ma Global differential geometry, 02.40.Vh Global analysis and analysis on manifolds, 04.20.Fy Canonical formalism, Lagrangians, and variational principles, 11.10.Ef Lagrangian and Hamiltonian approach, 11.10.Kk Field theories in dimensions other than four, 11.25.Sq Nonperturbative techniques; string field theory. 1991 Mathematics Subject Classification. Primary: 58E30 Variational principles; Secondary: 53B20 Local Riemannian geometry, 58A20 Jets, 58E12 Applications to minimal surfaces (problems in two independent variables), 58G35 Invariance and symmetry properties, 81S10 Geometric quantization, symplectic methods, 83E30 String and superstring theories.     

Plane rotations and Hamilton—Dirac mechanics

Canonical formalism for SO(2) is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretations are given. The Euler-Lagrange and Hamiltonian canonical equations coincide with the Lie equations. It is shown that the constraints satisfy CCR. Consistency of the constraints is checked. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

The Hamilton-Cartan formalism for higher order conserved currents: 1. Regular first order Lagrangians

The Hamilton–Cartan formalism for regular first order Lagrangian field theories is extended to deal with conserved currents which depend on higher order derivatives of the field variables. These conserved currents are characterized. Exterior differential systems I^{(k + 1)} and I equivalent to the k-th and infinite prolongations of the Euler-Lagrange equations are defined. It is shown that to each conserved current is associated an equivalence class of infinitesimal symmetries of I. Conserved charges are defined and a Poisson bracket is constructed by analogy with the usual definition. The sine-Gordon equation is treated briefly as an application of the formalism.     

A Note on the Equivalence of Post-Newtonian Lagrangian and Hamiltonian Formulations

Recently,it has been generally claimed that a low order post-Newtonian(PN)Lagrangian formulation,whose Euler-Lagrange equations are up to an infinite PN order,can be identical to a PN Hamiltonian formulation at the infinite order from a theoretical point of view.In general,this result is difficult to check because the detailed expressions of the Euler-Lagrange equations and the equivalent Hamiltonian at the infinite order are clearly unknown.However,there is no difficulty in some cases.In fact,this claim is shown analytically by means of a special first-order post-Newtonian(1PN)Lagrangian formulation of relativistic circular restricted three-body problem,where both the Euler-Lagrange equations and the equivalent Hamiltonian are not only expanded to all PN orders,but have converged functions.It is also shown numerically that both the Euler-Lagrange equations of the low order Lagrangian and the Hamiltonian are equivalent only at high enough finite orders.     

Non-uniqueness of solutions of Percival's Euler-Lagrange equation

Percival [5,6] introduced a Langrangian and an Euler-Lagrange equation for finding quasi-periodic orbits. In [3], we studied area preserving twist homeomorphisms of the annulus, using Percival's formalism. We showed that Percival's Lagrangian has a maximum on a suitable function space, and that a point where it takes its maximum is a solution of Percival's Euler-Lagrange equation. Moreover, in the rigorous interpretation of Percival's formalism which we gave in [3], the solutions of Percival's Euler-Lagrange equation correspond bijectively to a certain class of minimal sets. (We will prove this in Sect. 2.) In [4], we showed that Percival's Lagrangian takes its maximum at only one point. In this paper, we show that there existC area preserving twist diffeomorphisms of the annulus, for which there exists at least one solution of Percival's Euler-Lagrange equation where Percival's Lagrangian does not take its maximum. In other words, solutions of Percival's Euler-Lagrange equation need not be unique.Supported by NSF Grant No. MCS 79-02017     

A Note on Symplectic Algorithm

We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler-Lagrange cohomological concepts. We also show that the trapezoidal integrator is symplectic in certain sense.     

A Note on Symplectic Algorithm

We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler-Lagrange cohomological concepts. We also show that the trapezoidal integrator is symplectic in certain sense.``     

Ambient space formulations and statistical mechanics of holonomically constrained Langevin systems

The most classic approach to the dynamics of an n-dimensional mechanical system constrained by d independent holonomic constraints is to pick explicitly a new set of (n − d) curvilinear coordinatesparametrizingthe manifold of configurations satisfying the constraints, and to compute the Lagrangian generating the unconstrained dynamics in these (n − d) configuration coordinates. Starting from this Lagrangian an unconstrained Hamiltonian H(q,p) on 2(n−d) dimensional phase space can then typically be defined in the standard way via a Legendre transform. Furthermore, if the system is in contact with a heat bath, the associated Langevin and Fokker-Planck equations can be introduced. Provided that an appropriate fluctuation-dissipation condition is satisfied, there will be a canonical equilibrium distribution of the Gibbs form exp(−βH) with respect to the flat measure dqdp in these 2(n − d) dimensional curvilinear phase space coordinates. The existence of (n − d) coordinates satisfying the constraints is often guaranteed locally by an implicit function theorem. Nevertheless in many examples these coordinates cannot be constructed in any tractable form, even locally, so that other approaches are of interest. In ambient space formulations the dynamics are defined in the full original n-dimensional configuration space, and associated 2n-dimensional phase space, with some version of Lagrange multipliers introduced so that the 2(n − d) dimensional sub-manifold of phase space implied by the holonomic constraints and their time derivative, is invariant under the dynamics. In this article we review ambient space formulations, and explain that for constrained dynamics there is in fact considerable freedom in how a Hamiltonian form of the dynamics can be constructed. We then discuss and contrast the Langevin and Fokker-Planck equations and their equilibrium distributions for the different forms of ambient space dynamics.     

The Canonical Path Integral Quantization of CP1 Model

The Hamilton-Jacobi method of quantizing singular systems is discussed. The equations of motion are obtained as total differential equations in many variables. It is shown that if the system is integrable, then one can obtain the canonical phase space coordinates and the set of the canonical Hamilton-Jacobi partial differential equations without any need to introduce unphysical auxiliary fields. As an example we quantize the CP¹ model using the canonical path integral quantization formalism to obtain the path integral as an integration over the canonical phase-space coordinates.     

Jet methods in time-dependent Lagrangian biomechanics

In this paper we propose the time-dependent generalization of an ‘ordinary’ autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds associated to the extended musculo-skeletal configuration manifold, called the configuration bundle. We start with an ordinary configuration manifold of human body motion, given as a set of its all active degrees of freedom (DOF) for a particular movement. This is a Riemannian manifold with a material metric tensor given by the total mass-inertia matrix of the human body segments. This is the base manifold for standard autonomous biomechanics. To make its time-dependent generalization, we need to extend it with a real time axis. By this extension, using techniques from fibre bundles, we defined the biomechanical configuration bundle. On the biomechanical bundle we define vector-fields, differential forms and affine connections, as well as the associated jet manifolds. Using the formalism of jet manifolds of velocities and accelerations, we develop the time-dependent Lagrangian biomechanics. Its underlying geometric evolution is given by the Ricci flow equation.     

Solutions of Two-Mode Bosonic and Transformed Hamiltonians

We have constructed the quasi-exactly-solvable two-mode bosonic realization of SU(2). Two-mode boson Hamiltonian is defined through a differential equation which is solved by quantum Hamilton-Jacobi formalism. The squeezed states of two-mode boson systems are characterized through canonical transformation. The illustrated concept of squeezed boson systems has been applied two-mode bosonic Hamiltonian which is a squeezed one and is determined through a differential equation. This differential equation is solved and energy eigenvalues are found approximately.     

A simplified form for the Hamiltonian and Lagrangian of the spin-independent gravitational two-body system

In general, the gravitational two-body Hamiltonian, to orderc ^–2, containsGP ²,G (P · r)², andG ² terms. We have previously shown [4–6] that a proper choice of coordinate system enables one to eliminate theG (P · r)² term. We now show that, making use of energy conservation, and coordinate transformations, we can eliminate either of the remaining two terms. In particular, we are able to write down a Hamiltonian and a Lagrangian that contain no mixed potential and kinetic terms.Laboratoire associé au Centre National de la Recherche Scientifique.     

Quantization of Holonomic Systems Using WKB Approximation

The Lagrange multipliers for holonomic systems are introduced as generalized coordinates, then, the system is enlarged to be singular system. The Hamilton-Jacobi function is obtained. This function is used to determine the solution of the equations of motion for holonomic systems and to quantize these systems using the WKB approximation. Two examples are considered to demonstrate the application of our formalism. The solution of the two examples are found to be in exact agreement with the Euler-Lagrange equations.     

Filed Approach to Classical Mechanics

We show that in classical mechanics the momentum may depend only on the coordinates and can thus be considered as a field. We formulate a special Lagrangian formalism as a result of which the momenta satisfy differential equations which depend only on the coordinates. The solutions correspond to all possible trajectories. As a bonus the Hamilton-Jacobi equation results in a very simple way.     

Integrable Fredholm Operators and Dual Isomonodromic Deformations

The Fredholm determinants of a special class of integrable integral operators K supported on the union of m curve segments in the complex λ-plane are shown to be the τ-functions of an isomonodromic family of meromorphic covariant derivative operators , having regular singular points at the 2m endpoints of the curve segments, and a singular point of Poincaré index 1 at infinity. The rank r of the corresponding vector bundle over the Riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel. The matrix Riemann–Hilbert problem method is used to deduce an identification of the Fredholm determinant as a τ-function in the sense of Segal–Wilson and Sato, i.e., in terms of abelian group actions on the determinant line bundle over a loop space Grassmannian. An associated dual isomonodromic family of covariant derivative operators , having rank n= 2m, and r finite regular singular points located at the values of the exponents defining the kernel of K is derived. The deformation equations for this family are shown to follow from an associated dual set of Riemann–Hilbert data, in which the r?les of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged. The operators are analogously associated to an integral operator whose Fredholm determinant is equal to that of K. Received: 10 June 1997 / Received revised: 16 February 2001 / Accepted: 27 November 2001     

Perturbation theory with higher derivative couplings

The formalism of partial differential equations with respect to coupling constants is used to develop a covariant perturbation theory for the interpolating fields and theS matrix when the coupling terms in the Larangian density involve arbitrary (first and higher) derivatives. Through the notion of pure noncovariant contractions, the free-fieldT and the (covariant)T ^* products can be related to each other, allowing us to avoid the Hamiltonian density altogether when dealing with theS matrix. The important ingredients in our approach are (1) the adiabatic switching on and off of the interactions in the infinite past and future, respectively, and (2) the vanishing of four-dimensional delta functions and their derivatives at zero space-time points. The latter ingredient is a prerequisite that our formalism and the canonical formalism be consistent with each other, and on the other hand, it is supported by the dimensional regularization. Corresponding to any Lagrangian, the generalized interaction Hamiltonian density is defined from the covariantS matrix with the help of the pure noncovariant contractions. This interaction Hamiltonian density reduces to the usual one when the Lagrangian density depends on just first derivatives and when the usual canonical formalism can be applied.     

Minimum dissipative relaxed states in toroidal plasmas

Relaxation of toroidal discharges is described by the principle of minimum energy dissipation together with the constraint of conserved global helicity. The resulting Euler-Lagrange equation is solved in toroidal coordinates for an axisymmetric torus by expressing the solutions in terms of Chandrasekhar-Kendall (C-K) eigenfunctions analytically continued in the complex domain. The C-K eigenfunctions are obtained as hypergeometric functions that are solutions of scalar Helmholtz equation in toroidal coordinates in the large aspect-ratio approximation. Equilibria are constructed by assuming the current to vanish at the edge of plasma. For the m=0, n=0 (m and n are the poloidal and toroidal mode numbers respectively) relaxed states, the magnetic field, current, q (safety factor) and pressure profiles are calculated for a given value of aspect-ratio of the torus and for different values of the eigenvalue λ _{r 0}. The new feature of the present model is that solutions allow for both tokamak as well as RFP-like behaviour with increase in the values of λ _{r 0}, which is related directly to volt-sec in the experiment.