The equations of motion for consistency of a post-Newtonian Lagrangian formulation

TL;DR

This paper introduces a coherent set of equations of motion derived from the Euler-Lagrange equations for post-Newtonian systems, which better conserve constants of motion and are more accurate than traditional truncated equations.

Contribution

It demonstrates that equations of motion based on generalized momenta provide a coherent, non-truncated alternative to velocity-based equations in post-Newtonian Lagrangian formulations.

Findings

Coherent equations conserve constants of motion more accurately.

Numerical evaluations confirm high precision with appropriate integrators.

Differences in dynamical behavior, including chaos, are analyzed.

Abstract

Equations of motion for a general relativistic post-Newtonian Lagrangian approach mainly refer to acceleration equations, i.e. differential equations of velocities. They are directly from the Euler-Lagrangian equations, and usually have higher-order terms truncated when they remain at the same post-Newtonian order of the Lagrangian. In this sense, they are incoherent equations of the Lagrangian and approximately conserve constants of motion in this system. In this paper, we show that the Euler-Lagrangian equations can also yield the equations of motion for consistency of the Lagrangian in the general case. The coherent equations are the \emph{differential} equations of generalized momenta rather than those of the velocities, and have no terms truncated. The velocities are not integration variables, but they can be solved from the \emph{algebraic} equations of the generalized momenta with an iterative method. Taking weak relativistic fields in the Solar System and strong relativistic fields of compact objects as examples, we numerically evaluate the accuracies of the constants of motion in the two sets of equations of motion. It is confirmed that these accuracies well satisfy the theoretical need if the chosen integrator can provide a high enough precision. The differences in the dynamical behavior of order and chaos between the two sets of equations are also compared. Unlike the incoherent post-Newtonian Lagrangian equations of motion, the coherent ones can theoretically, strictly conserve all integrals in some post-Newtonian Lagrangian problems, and therefore are worth recommending.

Tables

Table 1. Table 1: In the nonspinning compact binary system, the separation between the solution ( x c ​ e , y c ​ e ) subscript 𝑥 𝑐 𝑒 subscript 𝑦 𝑐 𝑒 (x_{ce},y_{ce}) in the coherent equations and the solution ( x i ​ e , y i ​ e ) subscript 𝑥 𝑖 𝑒 subscript 𝑦 𝑖 𝑒 (x_{ie},y_{ie}) in the incoherent ones, D = ( x c ​ e − x i ​ e ) 2 + ( y c ​ e − y i ​ e ) 2 𝐷 superscript subscript 𝑥 𝑐 𝑒 subscript 𝑥 𝑖 𝑒 2 superscript subscript 𝑦 𝑐 𝑒 subscript 𝑦 𝑖 𝑒 2 D=\sqrt{(x_{ce}-x_{ie})^{2}+(y_{ce}-y_{ie})^{2}} .

$t$	$x_{ie}$	$x_{ce}$	$y_{ie}$	$y_{ce}$	$D$
1	17.03913	17.03910	10.09351	10.09349	0.000036
10	16.9548	16.95137	10.89020	10.88785	0.0042
100	9.39192	9.03218	13.34784	13.03775	0.47
1000	2.97387	0.67666	-2.62049	7.82169	10.7
10000	15.05357	-14.83949	-4.46476	8.48086	32.6

Equations

$$\mathbf{p}=\frac{\partial \mathcal{L}}{\partial \mathbf{v}}.$$

$$\frac{d\mathbf{p}}{dt}=\frac{\partial \mathcal{L}}{\partial \mathbf{r}}.$$

$$\frac{d\mathbf{r}}{dt}=\mathbf{v}.$$

$$E=\mathbf{v}\cdot \mathbf{p}-\mathcal{L}.$$

$$\frac{d\mathbf{v}}{dt}={\mathbf{a}}_N+{\mathbf{a}}_{1PN}+\cdots +{\mathbf{a}}_{mPN},$$

$$\frac{d\mathbf{r}}{dt}$$

$$\frac{d\mathbf{p}}{dt}$$

$$\mathcal{L}={\mathcal{L}}_0+\frac{1}{c^2}({\mathcal{L}}_1+{\mathcal{L}}_2),$$

$${\mathcal{L}}_0$$

$$r_1$$

$$r_2$$

$${\mathcal{L}}_1={\omega }_1(x\dot{y}-\dot{x}y+x^2+y^2),$$

$${\omega }_1=[(1-\mu )\mu -3]/(2a).$$

$${\mathcal{L}}_2$$

$$p_x$$

$$p_y$$

$$\frac{d{p}_x}{dt}$$

$$\frac{d{p}_y}{dt}$$

$$\dot{x}$$

$$\dot{y}$$

$$v_x$$

$$v_y$$

$$E$$

$$C_J$$

$$\ddot{x}$$

$$\ddot{y}$$

$$\ell ={\ell }_n+{\ell }_{1pn},$$

$${\ell }_n=\frac{v^2}{2}+\frac{1}{r},$$

$${\ell }_{1pn}$$

$$\mathbf{p}$$

$$\frac{d\mathbf{p}}{dt}$$

$$\mathbf{v}$$

$$\mathcal{E}$$

$$\mathbf{L}=\mathbf{r}\times \mathbf{p}=[1+\frac{1}{c^2}(\frac{1-3\eta }{2}{v}^2+\frac{3+\eta }{r})]\mathbf{r}\times \mathbf{v}.$$

$$\ddot{\mathbf{r}}={\mathbf{a}}_n+{\mathbf{a}}_{1pn},$$

Full text

I Introduction

The post-Newtonian (PN) Lagrangian approximations and the PN Hamiltonian approximations are often visible in relativistic astrophysics. They are widely applied to the description of the equations of motion of generic black hole binaries at a certain high PN order [1-3], including spin corrections. In this way, high-precision theoretical templates of gravitational waveforms can be provided. The PN approximations are also used to treat the equations of motion for the relativistic restricted three-body problem [4] and those for the relativistic $N$-body gravitational problem [5]. When one of the two PN formalisms is converted to another at the same order, their physical equivalence was shown in Refs. [6-8]. Due to higher-order terms truncated, the two formulations have somewhat differences and are not exactly equal [9-11]. For the Solar System as a weak gravitational field, the differences are too small to affect their equivalence, that is, the solutions of the two formulations should have no typical differences for the regular case. Some famous PN effects like perihelion or periastron advances for spinless binaries or the geodetic, the Lense-Thirring and the Schiff precessions for spinning bodies are the same in the two formulations. However, the differences lead to the two formulations having different solutions during a long integration time for the ordered case in a strong gravitational field of compact objects. Sometimes the differences even would make the two formulations have different dynamical properties of integrability and nonintegrability, or order and chaos.

It is well known that the paths for deriving the PN Hamiltonian equations of motion and the PN Lagrangian equations of motion are different. The Hamiltonian equations of motion, i.e. the canonical equations of a Hamiltonian formulation, have no terms truncated. Therefore, the equations of motion are consistent with this PN Hamiltonian. In this case, the constants of motion such as the energy integral are strictly conserved by the PN Hamiltonian equations of motion. The PN Lagrangian equations of motion are associated to the acceleration equations that result from the Euler-Lagrangian equations of a PN Lagrangian formulation. If the generalized momenta given by the PN Lagrangian system are nonlinear functions of the velocities, then the accelerations appear as the PN terms of the Euler-Lagrangian equations. When the total accelerations are required to remain at the same PN order of the Lagrangian, the accelerations in the PN terms of the Euler-Lagrangian equations must be replaced with the lower-order equations of motion. The higher-order PN terms truncated in the Euler-Lagrangian equations make the acceleration equations incoherent. That is, the acceleration equations are inconsistent with this PN Lagrangian formulation. Thus, the constants of motion are not strictly conserved by the incoherent acceleration equations. If the Lagrangian equations of motion can be given coherently, then they can naturally, strictly conserve the constants of motion in the PN Lagrangian formulation. Without doubt, both the incoherent Lagrangian equations of motion and the coherent ones have somewhat differences. When these differences are properly large, the two sets of equations of motion may exhibit distinct dynamical behaviors.

There is a question of whether the coherent Lagrangian equations of motion can be written. Recently, the authors of [12] suggested that coherent implicit acceleration equations, derived from the Lagrangian of a PN circular restricted three-body problem [4, 13], should be integrated by using implicit numerical integrators. Unlike them, we provide a simple method to construct the coherent Lagrangian equations of motion that strictly conserve the constants of motion. In our method, the differential equations of the generalized momenta rather than those of the velocities (i.e. the acceleration equations) directly result from the Euler-Lagrangian equations. They are still solved with an explicit numerical integrator. However, the velocities as not integration variables must be solved from the algebraic equations of the generalized momenta by means of an iterative method.

The paper is organized as follows. In Sect. 2, we provide two sets of PN equations of motion for a given PN Lagrangian. To clearly show the difference between them, we list two examples in detail. The PN circular restricted three-body problems are considered in Sect. 3. Then, PN Lagrangian systems of compact binaries are tested in Sect. 4. Finally, our main results are concluded in Sect. 5.

II PN Lagrangian equations of motion

Let us consider a PN Lagrangian $\mathcal{L}(\mathbf{r},\mathbf{v})$ at $m$th order, where $\mathbf{r}$ is a position vector and $\mathbf{v}$ is a velocity vector. We have the generalized momentum

[TABLE]

In general, $\mathbf{p}$ is a nonlinear algebraic equation of $\mathbf{v}$. The Euler-Lagrangian equation is

[TABLE]

Obviously, this equation is a differential equation of the momentum $\mathbf{p}$ to time $t$ and remains at the same PN order of the PN Lagrangian $\mathcal{L}$. In the following discussions, we use two methods to derive the equations of motion from the PN Lagrangian.

II.1 Two sets of PN equations of motion

Besides Eq. (2), another equation is added by

[TABLE]

When we solve the differential equations (2) and (3), the pair $(\mathbf{r},\mathbf{p})$ is viewed as a set of integration variables, but the velocity $\mathbf{v}$ is not. In spite of this, the velocity can be solved from the generalized momentum algebraic equation (1) using an iterative method like the Newtonian iterative method. It is clear that no truncations occur when Eqs. (2) and (3) are obtained from the PN Lagrangian. Therefore, Eqs. (2) and (3) with Eq. (1) are called as coherent equations of motion for the PN Lagrangian $\mathcal{L}$. Without doubt, they strictly conserve constants of motion, such as the energy integral

[TABLE]

On the other hand, one usually substitutes Eq. (1) into Eq. (2) and obtains the acceleration

[TABLE]

where $\mathbf{a}_{N}$, $\mathbf{a}_{1PN}$, $\cdots$, $\mathbf{a}_{mPN}$ represent the accelerations from the Newtonian term, first post-Newtonian order contribution, $\cdots$, $m$th post-Newtonian order contribution. Since the total acceleration $d\mathbf{v}/dt$ remains at $m$th PN order, all terms higher than the order $m$ must be truncated. This shows that Eq. (5) is inconsistent with the PN Lagrangian $\mathcal{L}$. That is to say, Eqs. (3) and (5) use $(\mathbf{r},\mathbf{v})$ as a set of integration variables and are incoherent PN equations of motion for the Lagrangian $\mathcal{L}$. Naturally, they approximately conserve the energy integral (4).

It is worth noting that the iterative accuracy of the velocity $\mathbf{v}$ in Eq. (1) reaches an order of $10^{-15}$, which almost approaches to the double precision of the machine, an order of $10^{-16}$. This accuracy is much higher than the accuracy of the PN terms truncated in strong gravitational problems. This is why Eqs. (1)-(3) are called as the coherent equations of motion, and Eqs. (3) and (5), the incoherent equations of motion. Additionally, the incoherent equations of motion and the coherent ones are not completely the same but are only approximately related two different dynamical systems.

II.2 Comparison with PN Hamilton’s equations

An $m$th PN order Hamiltonian $\mathcal{H}(\mathbf{r},\mathbf{p})$ (nonequivalent to $\mathcal{L}$) has canonical equations

[TABLE]

The canonical equations are coherent equations of motion for this PN Hamiltonian, and exactly conserve the Hamiltonian.

By comparing between the coherent PN Lagrangian equations (2) and (3) and the coherent PN Hamilton’s equations (6) and (7), one can easily find that they are very similar. They are the differential equations with respect to $\mathbf{r}$ and $\mathbf{p}$, and remain the same PN order of $\mathcal{L}$ or $\mathcal{H}$. However, there are some typical differences between them. That is, the velocity must be calculated via a certain iterative method in the coherent PN Lagrangian equations, but does not need such a calculation in the coherent PN Hamilton’s equations. In addition to this, $\mathbf{r}$ and $\mathbf{p}$ are not canonical in the coherent PN Lagrangian equations, but are in the coherent PN Hamilton’s equations.

Perhaps someone may think that the acceleration in the coherent PN Lagrangian equations should be higher than the order $m$ when Eq. (1) is substituted into Eq. (2). It is true. In fact, the acceleration, obtained from the coherent PN Hamilton’s equations by substituting $\mathbf{p}$ (given by Eq. (6)) into Eq. (7), is also higher than the order $m$.

It is clear that the coherent PN Lagrangian equations of motion and the incoherent ones exist some differences in higher-order terms. Do they show different dynamical behaviors? To answer this question, we compare them using numerical simulations of three problems.

III PN circular restricted three-body problems

In a planar circular restricted three-body problem, two primary bodies with masses $M_{1}$ and $M_{2}$ always have an invariant separation $a$ because they are restricted to moving in circular orbits around their center of mass under their gravities. A third body has such a small mass $m$ that it does not exert any influence on the circular motions of the binaries. Let $M=M_{1}+M_{2}$ be the total mass of the primary bodies, and take the gravitational constant as a geometrized unit $G=1$. When the distances (e.g. $a$) and time $t$ are measured in terms of $M$, the two primaries have dimensionless masses $\mu_{1}=M_{1}/M=1-\mu$ and $\mu_{2}=M_{2}/M=\mu$. If we further adopt the unit system of [13], that is, the distances and time are respectively measured in terms of $a$ and $1/\omega_{0}$ with the Newtonian angular speed of the primaries $\omega_{0}=a^{-3/2}$, then the distance of the binaries and the Newtonian angular speed of the circular orbits of the binaries become one geometric unit. In this case, the two central bodies are fixed at points $(-\mu,0)$ and $(1-\mu,0)$ in the rotating frame. Under the gravities of the binaries, the motion of the third body with coordinate $\textbf{r}=(x,y)$ and velocity $\textbf{v}=(\dot{x},\dot{y})$ is described by the following dimensionless 1PN Lagrangian formalism

[TABLE]

where $c$ is the velocity of light. The first part is the Newtonian circular restricted three-body problem

[TABLE]

The second part is an indirect 1PN relativistic effect to the circular orbits of the two central bodies, which affects the motion of the third body. It is expressed as

[TABLE]

where $\omega_{1}$ is the PN angular velocity of the primaries

[TABLE]

The third part is a direct 1PN relativistic effect to the third body. For our purpose, we choose a part of the complete expression on the relativistic effect [4, 13]:

[TABLE]

Noting Eq. (1), we have the generalized momenta

[TABLE]

Based on Eq. (2), the Euler-Lagrangian equations are

[TABLE]

Eq. (3) corresponds to the following two equations

[TABLE]

Eqs. (15)-(18) are the above-mentioned coherent equations of motion for the Lagrangian (8). In terms of Eqs. (13) and (14), we use an iterative method to get the velocities

[TABLE]

As a suitable choice of the initial values of $(v_{x},v_{y})$ in the right sides of the above equations, $v_{x}=p_{x}$ and $v_{y}=p_{y}$ are suggested. It is clear that the denominators in Eqs. (19) and (20) are larger than 1. Therefore, the iterative solutions are convergent. Eqs. (15)-(18) with Eqs. (19) and (20) strictly conserve the energy

[TABLE]

Notice that $C_{J}$ is the Jacobi constant when the Newtonian problem $\mathcal{L}_{0}$ is considered only, but is not in the present PN problem.

If Eq. (5) is considered, we have the acceleration equations

[TABLE]

In the above equations, $\ddot{x}_{0}$ and $\ddot{y}_{0}$ denote the accelerations from the Newtonian term $\mathcal{L}_{0}$. In fact, $\ddot{x}_{0}$ and $\ddot{y}_{0}$ should be $\ddot{x}$ and $\ddot{y}$, respectively. In this way, there are implicit acceleration equations suggested in [12]. They are given coherently, and should be completely equivalent to Eqs. (15)-(20). When the accelerations are solved explicitly from the coherent implicit acceleration equations, they have long and complicated expressional forms. Fortunately, Eqs. (15)-(20) have simple explicit expressional forms. It is clear that Eqs. (23) and (24) are obtained via $\ddot{x}\rightarrow\ddot{x}_{0}$ and $\ddot{y}\rightarrow\ddot{y}_{0}$ in the right functions of the coherent implicit acceleration equations. In other words, the PN terms of the accelerations $\ddot{x}$ and $\ddot{y}$ in the right functions of the coherent implicit acceleration equations are truncated. In this sense, Eqs. (23) and (24), as 1PN equations of motion, are the incoherent equations of motion for the Lagrangian $\mathcal{L}$ in Eq. (8). Since this system is conservative, the energy should be a conserved quantity. However, we have no way to give the energy an exact expression when the incoherent equations of motion are used. In this sense, Eqs. (23) and (24) approximately conserve the energy (21).

In short, the 1PN Lagrangian $\mathcal{L}$ in Eq. (8) has two descriptions of the equations of motion, namely, the coherent equations of motion (i.e. the differential equations of the generalized momenta) and the incoherent ones (namely, the acceleration equations or the differential equations of the velocities). What results are caused by the two distinct treatments to the equations of motion? Its answer awaits detailed numerical comparisons.

III.1 Solar System

First, we consider only the Newtonian circular restricted three-body problem $\mathcal{L}_{0}$ in Eq. (9) in the Solar System. An eighth- and ninth-order Runge-Kutta-Fehlberg explicit integration algorithm [RKF8(9)] of a variable time step is chosen as a numerical integrator. We take the dimensionless mass parameter $\mu=0.001$, which approaches to the ratio of Jupiter to the Sun. The Jacobi constant in Eq. (22) is given by $C_{J}=3.07$. Two orbits with the initial values of $x$=0.55 and 0.56 are chosen. They have the same initial values $y=\dot{x}=0$, but their initial values $\dot{y}$ $(>0)$ can be solved from Eq. (22) and are different. It is shown via the Poincaré surface of section in Fig. 1(a) that orbit 1 with the initial value $x=0.55$ forms three islands and therefore is regular. It is clear that orbit 2 with the initial value $x=0.56$ has many random points in a large region and thus exhibits a chaotic behavior. Although RKF8(9) is not an energy-preserving method, it gives such a high accuracy in the magnitude of an order $10^{-13}$ to the energies of the two orbits in Fig. 1(b).

Then, we focus on the PN circular restricted three-body problem $\mathcal{L}$ in Eq. (8) in the Solar System. As claimed in [14], $a=1$ and $c=10^{4}$ are recommended. $1/c^{2}=10^{-8}$ is near to the 1PN relativistic effect of the Sun and Jupiter. The PN terms $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are so small that they have negligible effects on the structure of phase space of orbits 1 and 2 in the Newtonian problem when the incoherent Eqs. (23) and (24) are used. Because of this, the accuracy of energies of the two orbits in the present case is almost the same as that in the Newtonian problem although the energy $E$ in Eq. (21) is approximately conserved by Eqs. (23) and (24). These results are not typically different when the incoherent Eqs. (23) and (24) give place to the coherent Eqs. (15)-(20). That is to say, for such weak relativistic effects in the Solar System, the incoherent equations and the coherent ones provide almost the same accuracy of energy. As a point to illustrate, the conservation of the Jacobi integral in the PN circular restricted three-body problem [14] is still based on the use of the incoherent equations. In addition, there is no explicit difference in the structure of phase space of orbits on the Poincaré surface of section between the two cases. Namely, an orbit is not chaotic in the coherent equations if it is ordered in the incoherent ones; inversely, an orbit is not regular in the incoherent equations if it is chaotic in the coherent ones.

Although the orbital dynamical features of order and chaos are absolutely dominated by the Newtonian term $\mathcal{L}_{0}$ and are not typically affected by the weak relativistic effects, the incoherent equations and the coherent ones still lead to somewhat differences in the positions and velocities of orbits. Even there are large differences in some cases. For the ordered orbit 1 in Fig. 2, the separation between the positions in the two cases is small and grows nearly linearly with time. However, this separation becomes typically large and increases exponentially for the chaotic orbit 2. These results should be reasonable. The difference between the incoherent equations and the coherent ones is the 2PN relativistic effect having an order of $10^{-16}$. The chaoticity of orbit 2 makes the difference larger and larger with the integration time increasing. As a noticeable point, the separation remains invariant at the value 1 after $t=1168$. This is due to the saturation of chaotic orbits in a bounded region. This problem also occurs in computations of Lyapunov exponents of two nearby orbits [15, 16]. The renormalization must be considered. As a result, the saturation of orbits is avoided and the Lyapunov exponents of two nearby orbits can be computed in a long enough time.

III.2 Compact objects

Now, assume the three bodies consisting of compact objects, such as neutron stars and/or black holes. Here, $c=1$ is given as in the usual relativistic issue. In this case, $1/a$ in the PN parts $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ plays book-keeping for the 1PN relativistic effect and thus $a\gg 1$ is required. Setting $a=1345$, the 1PN relativistic effect in an order of $10^{-3}$ in the case of compact objects is much stronger than that in an order of $10^{-8}$ in the Solar System. We take only orbit 1 as an example and compare the related differences between the incoherent equations and the coherent ones.

It can be seen clearly from Fig. 3 (a) and (b) that orbit 1 on the Poincaré surface of section becomes chaotic in the incoherent equations, whereas it is still regular in the coherent equations. Therefore, it is not unexpected that the difference between the incoherent equations and the coherent ones may lead to the two sets of equations having different dynamical behaviors of order and chaos. The energy accuracy with an order of $10^{-5}$ is poor in the incoherent equations in Fig. 3(c). If the distance between the binaries gets smaller and smaller, e.g. $a=100,50,10,\cdots$, then the energy accuracy becomes poorer and poorer. Particularly for $a=1$, the energy cannot be conserved numerically at all, as claimed in [14]. These results completely satisfy the theoretical need because the energy is not an integral of motion that can be conserved exactly by the incoherent equations. On the contrary, the energy accuracy is still high enough and reaches an order of $10^{-13}$ in the coherent equations in Fig. 3(d). The extremely high accuracy of energy does not depend on the value of $a$. This is owing to the energy as an exact integral of the coherent equations.

IV PN Lagrangian systems of compact binaries

In a spinless compact binary system, the binaries have masses $M_{1}$ and $M_{2}$. We take the total mass $M=M_{1}+M_{2}$, the reduced mass $\mu=M_{1}M_{2}/M$, the mass ratio $\beta=M_{2}/M_{1}$ and the dimensionless mass parameter $\eta=\mu/M=\beta(1+\beta)^{-2}$. The position and velocity of body 1 relative to body 2 are $\mathbf{r}=(x,y,z)$ and $\mathbf{v}=(\dot{x},\dot{y},\dot{z})$, respectively. $r=|\mathbf{r}|$ represents the relative distance of the binaries, and $v=|\mathbf{v}|$ is the magnitude of the relative velocity. Scale transformations are given to the coordinate $\mathbf{r}$ and time $t$ as follows: $\mathbf{r}\rightarrow M\mathbf{r}$ and $t\rightarrow Mt$.

According to two cases, the dynamical differences off the binaries between the incoherent equations of motion and the coherent ones are compared.

IV.1 Two bodies nonspinning

The evolution of the binaries is described by the dimensionless PN Lagrangian formulation

[TABLE]

where the first term is the Newtonian two-body problem

[TABLE]

and the second term is the 1PN contribution [17]

[TABLE]

The generalized momenta satisfy the following algebraic equation

[TABLE]

As was mentioned in Sect. 2, the Lagrangian $\ell$ has a coherent 1PN generalized momentum differential equation

[TABLE]

Eq. (3) is also included in the coherent equations of motion. Using Eq. (28), we have the velocity

[TABLE]

Because $1-3\eta=(\beta^{2}-\beta+1)/(\beta+1)^{2}>0$, the denominator in Eq. (30) is always larger than 1. This shows that the iterative solution of $\mathbf{v}$ is convergent. Eqs. (3), (29) and (30) exactly conserve the energy

[TABLE]

They also exactly conserve the orbital angular momentum vector

[TABLE]

The 1PN relative acceleration is written as

[TABLE]

where the two parts are

[TABLE]

[TABLE]

Eq. (33) approximately conserves the energy (31) or the 1PN Hamiltonian quantity

[TABLE]

The symbol $``\approx"$ means that $\mathcal{E}$ is not exactly equal to $\mathcal{H}$ because the higher-order (2PN, 3PN, $\cdots$) terms are truncated in $\mathcal{H}$ but such truncations do not occur in $\mathcal{E}$. In addition, Eq. (33) approximately conserves the orbital angular momentum vector (32).

Setting the speed of light $c=1$, the mass ratio $\beta=5/4$ and the initial conditions ($x=17.04$, $y=10$, $\dot{y}=0.094$, $z=\dot{x}=\dot{z}=0$), we apply RKF8(9) to numerically integrate the coherent Eqs. (3) and (29) or the incoherent Eq. (33). In Fig. 4 (a) and (b), the energy error is very large in the magnitude of an order of 0.1 in the incoherent equations, but it is so small that it arrives at an order of $10^{-12}$ in the coherent equations when the integration time $t=10^{6}$. In Fig. 4 (c) and (d), the errors of the orbital angular momentum in the two sets of equations of motion are also similar to those of the energy. These results strongly support the preference of the coherent equations over the incoherent equations in the conservation of the energy and the angular momentum.

It should be emphasized that the chosen orbit is limited to the plane $z=0$ due to the conserved angular momentum (32) with the initial angular momentum $\mathbf{L}=(0,0,L_{z})$. Additionally, the system $\ell$ is integrable and regular because its formal equivalent Hamiltonian (that is not $\mathcal{H}$ and is not easily written in detail) has four independent integrals of motion in the six-dimensional phase space [9]. That means that any orbit in the system $\ell$ is always nonchaotic regardless of the choice of the incoherent equations and the coherent ones. In spite of this, the separation between the solution of the coherent equations and that of the incoherent equations will become rather large as the integration time is long enough in Table 1.

IV.2 Two bodies spinning

Now, let us consider that the binaries in Eq. (25) are spinning, and their spin effects are restricted to spin-orbit coupling interaction with a 1.5 PN accuracy

[TABLE]

where $\gamma_{1}=2+3\beta/2$, $\gamma_{2}=2+3/(2\beta)$, and $\mathbf{S}_{1}$, $\mathbf{S}_{2}$ are measured in terms of $\mu M$. The spin magnitudes are $S_{1}=|\mathbf{S}_{1}|=\chi_{1}M^{2}_{1}/(\mu M)$ and $S_{2}=|\mathbf{S}_{2}|=\chi_{2}M^{2}_{2}/(\mu M)$, where $0\leq\chi_{1}\leq 1$ and $0\leq\chi_{2}\leq 1$ are dimensionless spin parameters. Under the Newton-Wigner-Pryce spin supplementary condition [18], the spin-orbit coupling term does not depend on accelerations. When the spin-orbit term is included in Eq. (25), the PN Lagrangian system becomes

[TABLE]

The generalized momentum of $\mathbb{L}$ is

[TABLE]

The PN Lagrangian $\mathbb{L}$ has a coherent 1.5PN differential equation of the generalized momentum

[TABLE]

The spin precession equations are given by

[TABLE]

The velocity has an iterative form

[TABLE]

Eqs. (3), (40)-(43) are the coherent equations of motion with respect to the PN Lagrangian $\mathbb{L}$. The energy (31) is exactly conserved by these equations of motion. There is the conserved total angular momentum vector

[TABLE]

The spin magnitudes $S_{1}$ and $S_{2}$ are exact constants of motion, too.

The 1.5PN acceleration equation is expressed as

[TABLE]

where the third term is

[TABLE]

Eq. (45) is approximately provided because of the higher-order terms truncated. Eqs. (41), (42) and (45) approximately conserve the energy (31) or the 1.5PN Hamiltonian

[TABLE]

They also approximately conserve the total angular momentum vector (44), but exactly conserve the spin magnitudes $S_{1}$ and $S_{2}$.

Given $\chi_{1}=\chi_{2}=1$, the initial spin vectors are chosen as $\mathbf{S}_{1}=S_{1}\mathbf{\hat{S}}_{1}$ and $\mathbf{S}_{2}=S_{2}\mathbf{\hat{S}}_{2}$, where $\mathbf{\hat{S}}_{1}$ and $\mathbf{\hat{S}}_{2}$ are two unit vectors

[TABLE]

The mass ratio and the orbit in Sect. 3 are still used. The accuracies of the energy and the total angular momentum in the coherent equations of motion are much better than those in the incoherent ones for the two bodies spinning, as shown in Fig. 5. This result is very similar to that for the two bodies nonspinning. However, unlike in the case of the two bodies nonspinning, the spin-orbit effects lead to the orbit precessing in the direction $z$. Fig. 6 clearly shows that the precession values of $z$ in the coherent equations are very inconsistent with those in the incoherent ones after a short time.

To know whether the solutions between the coherent equations and the incoherent ones have the same chaotic behavior, we rely on the largest Lyapunov exponent of two nearby orbits [15]

[TABLE]

where $d(t)$ and $d(0)$ denote the separations between the two nearby orbits at times $t$ and 0. The Lyapunov exponent of the orbit in the incoherent equations tends to zero in Fig. 7, and therefore the solutions of the incoherent equations are regular. However, the bounded solutions of the coherent equations should be chaotic due to the presence of a positive Lyapunov exponent. It is clear that the time for the Lyapunov exponent tending to zero or a stabilizing positive value is long enough, e.g. $t=10^{7}$. By contrast, a fast Lyapunov indicator can distinguish between the two cases of order and chaos with less computational cost. The fast Lyapunov indicator of two nearby orbits [16] is calculated by

[TABLE]

This indicator $\Lambda$ increasing in a power law with time $\log_{10}t$ indicates the regularity of the solutions in the incoherent equations, whereas the indicator having an exponential growth with time shows the chaoticity of the solutions in the coherent equations. Only when the time arrives at $3\times 10^{4}$ in Fig. 8, can the two cases of order and chaos clearly be distinguished.

In a word, the methods of Lyapunov exponents and fast Lyapunov indicators have confirmed together that the orbit is ordered in the incoherent equations but chaotic in the coherent ones. Of course, it is possible that an orbit is chaotic in the incoherent equations but regular in the coherent ones when the dynamical parameters, the initial conditions and the initial spin vectors are altered. These results are due to the nonintegrability of the PN Lagrangian $\mathbb{L}$ and the nonequivalence between the two sets of equations. On one hand, the Lagrangian $\mathbb{L}$ has its formal exact equivalent Hamiltonian (nonequal to $\mathbb{H}$) [9], which includes higher-order spin-spin coupling terms and contains four independent integrals of the energy (31) and the total angular momentum vector (44). When the canonical, conjugate spin variables of [19] are adopted, this Hamiltonian has a ten-dimensional phase space and therefore is nonintegrable. Namely, the Lagrangian $\mathbb{L}$ is nonintegrable and may be chaotic. By contrast, the 1.5PN Hamiltonian $\mathbb{H}$ is integrable and nonchaotic because of the existence of a fifth integral, the length of the orbital angular momentum vector $\mathbf{r}\times\mathbf{P}$ [19]. On the other hand, the incoherent equations and the coherent ones exist the difference of 2PN order terms and are only approximately related two different systems. It is supported in dynamical systems theory that one system is chaotic but an approximately related system can be ordered. Therefore, it should be reasonable that the solutions of the incoherent equations and those of the coherent equations have different dynamical behaviors under a certain circumstance.

It is worth pointing out that the difference between the incoherent equations and the coherent ones are too small to affect the solutions of the two sets of equations for regular orbits in the PN circular restricted three-body problem in the Solar System. In other words, the two sets of equations are almost equivalent and have the same dynamical behavior. This result is also suitable for spinless binaries or spinning bodies in the Solar System. Thus, the famous PN effects like perihelion or periastron advances for spinless binaries or the geodetic, the Lense-Thirring and the Schiff precessions for spinning bodies are given the same results by the two sets of equations in the Solar System.

V Summary

Based on the construction of PN Lagrangian equations of motion in the above examples, several points can be concluded as follows.

There are two paths to obtain the equations of motion from a PN Lagrangian formalism. As one path, the total accelerations at the same PN order of the Lagrangian are derived from the Euler-Lagrangian equations of this Lagrangian, by truncating higher-order terms of the accelerations in the Euler-Lagrangian equations. They are the incoherent PN equations of motion of the Lagrangian system. Consequently, the constants of motion such as the energy integral are only approximately conserved in the incoherent equations. As another path, the differential equations with respect to the generalized momenta directly come from the Euler-Lagrangian equations, and also remain at the same PN order of the Lagrangian. Although the velocities are not integration variables, they can be solved from the generalized momentum algebraic equations with an iterative method. Such equations of motion exactly conserve the constants of motion in the PN Lagrangian formalism, as Hamilton’s equations exactly do in a PN Hamiltonian formulation. In this way, the PN Lagrangian equations of motion are coherent.

In fact, the incoherent equations of motion and the coherent ones have somewhat differences and belong to approximately related two different dynamical systems. When the differences associated to the relativistic effects are such small quantities in the Solar System, the accuracies of the constants of motion in the incoherent equations of motion are almost the same as those in the coherent ones if the adopted numerical integrator can achieve at a high enough precision. The solutions of the incoherent equations and those of the coherent equations have the same dynamical behaviors of order and chaos. In addition, the differences between the two sets of solutions are also small for the ordered case, but large in a long time for the chaotic case. Unlike in the Solar System, the differences between the two sets of equations of motion are somewhat large in the strong gravitational systems of compact objects. Because of this, the accuracies of the constants of motion are very poorer in the incoherent equations of motion even if the chosen integrator has a high precision, but very better in the coherent equations of motion. The dynamical behaviors may be different in the two cases. It is possible that an orbit is chaotic in the incoherent equations while regular in the coherent ones. It is also possible that an orbit is ordered in the incoherent equations whereas chaotic in the coherent ones. These results are supported in dynamical systems theory. Even for the regular case, the differences between the solutions in the two sets of equations get larger and larger with time increasing.

There are no truncations when the coherent equations of motion are derived from some PN Lagrangian problems. Therefore, the coherent equations are worth recommending in the study of the dynamics and gravitational waveforms of these PN Lagrangian systems.

Acknowledgments

This research has been supported by the National Natural Science Foundation of China under Grant No. 11533004 and the Natural Science Foundation Innovation Group of Guangxi under Grant No. 2018GXNSFGA281007.