Dynamics of High-Order Spin-Orbit Couplings about Linear Momenta in Compact Binary Systems

This paper relates to the post-Newtonian Hamiltonian dynamics of spinning compact binaries, consisting of the Newtonian Kepler problem and the leading, next-to-leading and next-to-next-to-leading order spin-orbit couplings as linear functions of spins and momenta. When this Hamiltonian form is transformed to a Lagrangian form, besides the terms corresponding to the same order terms in the Hamiltonian, several additional terms, third post-Newtonian(3 PN),4 PN, 5 PN, 6 PN and 7 PN order spin-spin coupling terms, yield in the Lagrangian. That means that the Hamiltonian is nonequivalent to the Lagrangian at the same PN order but is exactly equivalent to the full Lagrangian without any truncations. The full Lagrangian without the spin-spin couplings truncated is integrable and regular. Whereas it is non-integrable and becomes possibly chaotic when any one of the spin-spin terms is dropped. These results are also supported numerically.     

The Hamiltonian formulation of regular rth-order Lagrangian field theories

A Hamiltonian formulation of regular rth-order Lagrangian field theories over an m-dimensional manifold is presented in terms of the Hamilton-Cartan formalism. It is demonstrated that a uniquely determined Cartan m-form may be associated to an rth-order Lagrangian by imposing conditions of congruence modulo a suitably defined system of contact m-forms. A geometric regularity condition is given and it is shown that, for a regular Lagrangian, the momenta defined by the Hamilton-Cartan formalism, together with the coordinates on the (r−1)st-order jet bundle, are a minimal set of local coordinates needed to express the Euler-Lagrange equations. When r is greater than one, the number of variables required is strictly less than the dimension of the (2r−1)st order jet bundle. It is shown that, in these coordinates, the Euler-Lagrange equations take the first-order Hamiltonian form given by de Donder. It is also shown that the geometrically natural generalization of the Hamilton-Jacobi procedure for finding extremals is equivalent to de Donder's Hamilton-Jacobi equation. Research supported by the Natural Sciences and Engineering Research Council.     

General relativistic dynamics of compact binary systems

The equations of motion of compact binary systems have been derived in the post-Newtonian (PN) approximation of general relativity. The current level of accuracy is 3.5PN order. The conservative part of the equations of motion (neglecting the radiation reaction damping terms) is deducible from a generalized Lagrangian in harmonic coordinates, or equivalently from an ordinary Hamiltonian in ADM coordinates. As an application, we investigate the problem of the dynamical stability of circular binary orbits against gravitational perturbations up to the 3PN order. We find that there is no innermost stable circular orbit or ISCO at the 3PN order for equal masses. To cite this article: L. Blanchet, C. R. Physique 8 (2007).     

Difference Discrete Variational Principles,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures Ⅲ：Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II,we have studied the difference discrete variational principle and the Euler-Lagrange cohomology in the framework of multi-parameter differential approach.We have gotten the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the lagrangian and Hamiltonian formalisms.In this paper,we apply the difference discrete variational principle and Euler-Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms.We will show that either Hamiltonian schemes of Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler-Lagrange cohomological conditions are satisfied.     

The Effect of Spin-Orbit Coupling and Spin-Spin Coupling of Compact Binaries on Chaos

There are spin-orbit interaction and spin-spin interaction in a generic post-Newtonian Lagrangian formulation of comparable mass spinning compact binaries. The spin-orbit coupling or the spin-spin coupling plays a quite important role in changing the evolution of the system and may sometime cause chaotic behavior. How do the two types of couplings exert together any influences on chaos in this formulation? To answer it, we simply take the Lagrangian formulation of a special binary system, including the Newtonian term and the leading-order spin-orbit and spin-spin couplings. The key to this question can be found from a Hamiltonian formulation that is completely identical to the Lagrangian formulation. If the Lagrangian does not include the spin-spin coupling, its equivalent Hamiltonian has an additional term(i.e. the next-order spin-spin coupling) as well as those terms of the Lagrangian. The spin-spin coupling rather than the spin-orbit coupling makes the Hamiltonian typically nonintegrable and probably chaotic when two objects spin. When the leading-order spin-spin coupling is also added to the Lagrangian, it still appears in the Hamiltonian.In this sense, the total Hamiltonian contains the leading-order spin-spin coupling and the next-order spin-spin coupling,which have different signs. Therefore, the chaos resulting from the spin-spin interaction in the Legrangian formulations is somewhat weakened by the spin-orbit coupling.     

The Effect of Spin-Orbit Coupling and Spin-Spin Coupling of Compact Binaries on Chaos

There are spin-orbit interaction and spin-spin interaction in a generic post-Newtonian Lagrangian formu-lation of comparable mass spinning compact binaries. The spin-orbit coupling or the spin-spin coupling plays a quite important role in changing the evolution of the system and may sometime cause chaotic behavior. How do the two types of couplings exert together any influences on chaos in this formulation? To answer it, we simply take the Lagrangian formulation of a special binary system, including the Newtonian term and the leading-order spin-orbit and spin-spin couplings. The key to this question can be found from a Hamiltonian formulation that is completely identical to the Lagrangian formulation. If the Lagrangian does not include the spin-spin coupling, its equivalent Hamiltonian has an additional term (i.e. the next-order spin-spin coupling) as well as those terms of the Lagrangian. The spin-spin coupling rather than the spin-orbit coupling makes the Hamiltonian typically nonintegrable and probably chaotic when two objects spin. When the leading-order spin-spin coupling is also added to the Lagrangian, it still appears in the Hamiltonian. In this sense, the total Hamiltonian contains the leading-order spin-spin coupling and the next-order spin-spin coupling, which have different signs. Therefore, the chaos resulting from the spin-spin interaction in the Legrangian formulations is somewhat weakened by the spin-orbit coupling.     

Fractional hamilton formalism within caputo’s derivative

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange formulations lead to the same set of equations.     

On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism

We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively.We also explore their certain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variational principle with the difference being regarded as an entire grometric object and the noncommutative differential calculus on regular lattice.In order to show that in all these cases the symplectic and multisymplectic preserving properties do not necessarily depend on the relevant Euler-Lagrange equations,the Euler-Lagrange cohomological concepts and content in the configuration space are employed.     

The neutrino energy-momentum tensor and the Weyl equations in curved space-time

In general, a first order Lagrangian gives rise to second order Euler-Lagrange equations. However, there are important examples where the associated Euler-Lagrange equations are of first order only, the Weyl neutrino equations being of this type. In this paper we therefore consider first order spinor Lagrangians which give rise to firstorder Euler-Lagrange equations. Specifically, the most general first order spinor field equations of rank one in curved space-time which are derivable from a first order Lagrangian of the same type are explicitly constructed. Subject to a certain restriction, the Weyl neutrino equation is the only possibility. Furthermore, if the spinor field satisfies the Weyl neutrino equation, then the associated energy momentum tensor is the conventional neutrino energymomentum tensor.     

Difference Discrete Variational Principle,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures II：Euler—Lagrange Cohomology

In this second paper of a series of papers,we explore the difference discrete versions for the Euler-Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multiparameter differential approach.In terms of the difference discrete Euler-Lagrange cohomological concepts,we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler-Lagrange or canonical equations erived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler-Lagrange cohomological conditions are satisfied.     

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.     

On the two-body problem in general relativity

We consider the two-body problem in post-Newtonian approximations of general relativity. We report the recent results concerning the equations of motion, and the associated Lagrangian formulation, of compact binary systems, at the third post-Newtonian order (∼1/c⁶ beyond the Newtonian acceleration). These equations are necessary when constructing the theoretical templates for searching and analyzing the gravitational-wave signals from inspiralling compact binaries in VIRGO and LISA type experiments.     

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.     

The Dual Action of Fractional Multi Time Hamilton Equations

The fractional multi time Lagrangian equations has been derived for dynamical systems within Riemann-Liouville derivatives. The fractional multi time Hamiltonian is introduced as Legendre transformation of multi time Lagrangian. The corresponding fractional Euler-Lagrange and the Hamilton equations are obtained and the fractional multi time constant of motion are discussed.     

Equations of motion of compact binaries at the third post-Newtonian order

The equations of motion of two point masses in harmonic coordinates are derived through the third post-Newtonian (3PN) approximation. The problem of selffield regularization (necessary for removing the divergent self-field of point particles) is dealt with in two separate steps. In the first step the extended Hadamard regularization is applied, resulting in equations of motion which are complete at the 3PN order, except for the occurrence of one and only one unknown parameter. In the second step the dimensional regularization (ind dimensions) is used as a powerful argument for fixing the value of this parameter, thereby completing the 3-dimensional Hadamard-regularization result. The complete equations of motion and associated energy at the 3PN order are given in the case of circular orbits.     

The Hamiltonian Canonical Form for Euler-Lagrange Equations

Based on the theory of calculus of variation, some sufficient conditions are given for some Euler-Lagrange equations to be equivalently represented by finite or even infinite many Hamiltonian canonical equations. Meanwhile, some further applications for equations such as the KdV equation, MKdV equation, the general linear Euler-Lagrange equation and the cylindric shell equations are given.     

Regular dynamics of canonical post-Newtonian Hamiltonian for spinning compact binaries with next-to-leading order spin-orbit interactions

This paper shows that a conservative canonical post-Newtonian Hamiltonian formulation of spinning compact binaries with a pure orbital part up to third post-Newtonian order and spin-orbit contributions at the next-to-leading post-Newtonian order is explicitly integrable and regular because there are 5 independent exact isolating integrals in the 10-dimensional phase space. With the help of symplectic integrators and the fast Lyapunov indicators of two nearby trajectories, numerical investigations also support the absence of chaos.     

General Nth-order differential spectral problem: General structure of the integrable equations,nonuniqueness of recursion operator and gauge invariance

The general scalar Gelfand-Dikij-Zakharov-Shabat spectral problem of arbitrary order is considered within the framework of the AKNS method. The general form of the integrable equations is found. Uncertainties which appear in the construction of recursion operator and transformation properties of the integrable equations under the gauge transformations are considered. The manifestly gauge-invariant formulation of the integrable equations is given. It is shown that the intergrable equations under consideration are Hamiltonian ones with respect to the infinite family of Hamiltonian structures.