The seasonal variation of the newtonian constant of gravitation and its relationship with the “classical tests” of general relativity

The ratio between the Earth's perihelion advance (Δθ) _E and the solar gravitational red shift (GRS) (Δø _s e)a ₀/c ² has been rewritten using the assumption that the Newtonian constant of gravitationG varies seasonally and is given by the relationship, first found by Gasanalizade (1992b) for an aphelion-perihelion difference of (ΔG)_a?p . It is concluded that $$\begin{gathered} (\Delta \theta )_E = \frac{{3\pi }}{e}\frac{{(\Delta \phi _{sE} )_{A_0 } }}{{c^2 }}\frac{{(\Delta G)_{a - p} }}{{G_0 }} = 0.038388 \sec {\text{onds}} {\text{of}} {\text{arc}} {\text{per}} {\text{revolution,}} \hfill \\ \frac{{(\Delta G)_{a - p} }}{{G_0 }} = \frac{e}{{3\pi }}\frac{{(\Delta \theta )_E }}{{(\Delta \phi _{sE} )_{A_0 } /c^2 }} = 1.56116 \times 10^{ - 4} . \hfill \\ \end{gathered} $$ The results obtained here can be readily understood by using the Parametrized Post-Newtonian (PPN) formalism, which predicts an anisotropy in the “locally measured” value ofG, and without conflicting with the general relativity.     

Characteristic actions ħ(s) in the structure of the universe

It is suggested that gravitationally bound systems in the Universe can be characterized by a set of actions ?^(s). The actions $$\hbar ^{\left( s \right)} = \left( {{\hbar \mathord{\left/ {\vphantom {\hbar {\frac{1}{{2\pi }}\frac{{C^5 }}{{GH_0^2 }}}}} \right. \kern-\nulldelimiterspace} {\frac{1}{{2\pi }}\frac{{C^5 }}{{GH_0^2 }}}}} \right)^{s/6} \left( {\frac{1}{{2\pi }}\frac{{C^5 }}{{GH_0^2 }}} \right)$$ ,derived from general theoretical consideration, are only determined by the fundamental physical constants (Planck's action ?, the velocity of lightC, gravitational constantG, and Hubble's constantH ₀) and a scale parameters. It is shown thats=1, 2, and 3 correspond, respectively, to the scales of galaxies, stars, and larger asteroids. The spectra of the characteristic angular momenta and masses for gravitationally bound systems in the Universe are estimated byJ ^(s) andM ^(s) =(? ^(s) Cα/G)^1/2. Taken together, an angular momentum-mass relation is obtained,J ^(s)=A(M^(s))², where \(A = G/C\alpha ,{\text{ }}\alpha \simeq \tfrac{{\text{1}}}{{{\text{137}}}}\) , for the astronomical systems observed on every scale. ThisJ-M relation is consistent with Brosche's empirical relation (Brosche, 1974).     

On pulsar-driven isothermal blast-wave models of supernova remanants

We investigate the ‘equilibrium’ and stability of spherically-symmetric self-similar isothermal blast waves with a continuous post-shock flow velocity expanding into medium whose density varies asr ^?ω ahead of the blast wave, and which are powered by a central source (a pulsar) whose power output varies with time ast ^ω?3. We show that:

1. for ω<0, no physically acceptable self-similar solution exists;
2. for ω>3, no solution exists since the mass swept up by the blast wave is infinite;
3. ? must exceed zero in order that the blast wave expand with time, but ?<2 in order that the central source injects a finite total energy into the blast wave;
4. for 3>ω_min(?)>ω>ω_max(?)>0, where $$\begin{gathered} \omega _{\min } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} + {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} + {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \omega _{\max } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} - {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} - {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \end{gathered} $$ two critical points exist in the flow velocity versus position plane. The physically acceptable solution must pass through the origin with zero flow speed and through the blast wave. It must also pass throughboth critical points if \(\varphi > \tfrac{5}{3}\) , while if \(\varphi< \tfrac{5}{3}\) it must by-pass both critical points. It is shown that such a solution exists but a proper connection at the lower critical point (for ?>5/3) (through whichall solutions pass with thesame slope) has not been established;
5. for 3>ω>ω_min(?) it is shown that the two critical points of (iv) disappear. However a new pair of critical points form. The physically acceptable solution passing with zero flow velocity through the origin and also passing through the blast wave mustby-pass both of the new critical points. It is shown that the solution does indeed do so;
6. for 3>ω_min(?)>ω_max(?)>ω it is shown that the dependence of the self-similar solution on either ω or ? is non-analytic and therefore, inferences drawn from any solutions obtained in ω>ω_max(?) (where the dependence of the solutionis analytic on ω and ?) are not valid when carried over into the domain 3>ω_min(?)>ω_max(?)>ω;
7. all of the physically acceptable self-similar solutions obtained in 3>ω>0 are unstable to short wavelength, small amplitude but nonself-similar radial velocity perturbations near the origin, with a growth which is a power law in time;
8. the physical self-similar solutions are globally unstable in a fully nonlinear sense to radial time-dependent flow patterns. In the limit of long times, the nonlinear growth is a power law in time for 5<ω+2?, logarithmic in time for 5>ω+2?, and the square of the logarithm in time for 5=ω+2?.

The results of (vii) and (viii) imply that the memory of the system to initial and boundary values does not decay as time progresses and so the system does not tend to a self-similar form. These results strongly suggest that the evolution of supernova remnants is not according to the self-similar form.     

Dimensional approach to mass loss: O- and B-type stars

An attempt has been made to obtain an expression for the rate of stellar mass loss using dimensional analysis. The best expression for O and B stars is of the form: $$\dot M = A'{\text{ }}\left( {\frac{1}{{G^{1/2} c^4 }}} \right){\text{ }}L^{\text{2}} {\text{ (}}R/M)^{{\text{3/2}}} .$$ It is also found thatA′ increases as one goes from B→O stars and from O→O(f)→O(f)), but is not sensitive to luminosity.     

Gaussian variational equations for osculating elements of an arbitrary separable reference orbit

If a satellite orbit is described by means of osculating Jacobi α's and β's of a separable problem, the paper shows that a perturbing forceF makes them vary according to $$\dot \alpha _\kappa = {\text{F}} \cdot \partial {\text{r/}}\partial \beta _k {\text{ }}\dot \beta _k = {\text{ - F}} \cdot \partial {\text{r/}}\partial \alpha _k ,{\text{ (}}k = 1,2,3).{\text{ (A1)}}$$ Herer is the position vector of the satellite andF is any perturbing force, conservative or non-conservative. There are two special cases of (A1) that have been previously derived rigorously. If the reference orbit is Keplerian, equations equivalent to (A1), withF arbitrary, were derived by Brouwer and Clemence (1961), by Danby (1962), and by Battin (1964). IfF=?gradV ₁(t), whereV ₁ may or may not depend explicitly on the time, Equations (A1) reduce to the well known forms (e.g. Garfinkel, 1966) $$\dot \alpha _\kappa = {\text{ - }}\partial V_1 {\text{/}}\partial \beta _k {\text{ }}\dot \beta _k = \partial V_1 {\text{/}}\partial \alpha _k ,{\text{ (}}k = 1,2,3).{\text{ (A2)}}$$ holding for all separable reference orbits. Equations (A1) can of course be guessed from Equations (A2), if one assumes that \(\dot \alpha _k (t)\) and \(\dot \beta _k (t)\) depend only onF(t) and thatF(t) can always be modeled instantaneously as a potential gradient. The main point of the present paper is the rigorous derivation of (A1), without resort to any such modeling procedure. Applications to the Keplerian and spheroidal reference orbits are indicated.     

Sur un modele explicitant les differents types morphologiques des galaxies

In the now classical Lindblad-Lin density-wave theory, the linearization of the collisionless Boltzmann equation is made by assuming the potential functionU expressed in the formU=U ₀ + \(\tilde U\) +... WhereU ₀ is the background axisymmetric potential and \(\tilde U<< U_0 \) . Then the corresponding density distribution is \(\rho = \rho _0 + \tilde \rho (\tilde \rho<< \rho _0 )\) and the linearized equation connecting \(\tilde U\) and the component \(\tilde f\) of the distribution function is given by $$\frac{{\partial \tilde f}}{{\partial t}} + \upsilon \frac{{\partial \tilde f}}{{\partial x}} - \frac{{\partial U_0 }}{{\partial x}} \cdot \frac{{\partial \tilde f}}{{\partial \upsilon }} = \frac{{\partial \tilde U}}{{\partial x}}\frac{{\partial f_0 }}{{\partial \upsilon }}.$$ One looks for spiral self-consistent solutions which also satisfy Poisson's equation $$\nabla ^2 \tilde U = 4\pi G\tilde \rho = 4\pi G\int {\tilde f d\upsilon .} $$ Lin and Shu (1964) have shown that such solutions exist in special cases. In the present work, we adopt anopposite proceeding. Poisson's equation contains two unknown quantities \(\tilde U\) and \(\tilde \rho \) . It could be completelysolved if a second independent equation connecting \(\tilde U\) and \(\tilde \rho \) was known. Such an equation is hopelesslyobtained by direct observational means; the only way is to postulate it in a mathematical form. In a previouswork, Louise (1981) has shown that Poisson's equation accounted for distances of planets in the solar system(following to the Titius-Bode's law revised by Balsano and Hughes (1979)) if the following relation wasassumed $$\rho ^2 = k\frac{{\tilde U}}{{r^2 }} (k = cte).$$ We now postulate again this relation in order to solve Poisson's equation. Then, $$\nabla ^2 \tilde U - \frac{{\alpha ^2 }}{{r^2 }}\tilde U = 0, (\alpha ^2 = 4\pi Gk).$$ The solution is found in a classical way to be of the form $$\tilde U = cte J_v (pr)e^{ - pz} e^{jn\theta } $$ wheren = integer,p =cte andJ _v (pr) = Bessel function with indexv (v ² =n ² + α²). By use of the Hankel function instead ofJ _v (pr) for large values ofr, the spiral structure is found to be given by $$\tilde U = cte e^{ - pz} e^{j[\Phi _v (r) + n\theta ]} , \Phi _v (r) = pr - \pi /2(v + \tfrac{1}{2}).$$ For small values ofr, \(\tilde U\) = 0: the center of a galaxy is not affected by the density wave which is onlyresponsible of the spiral structure. For various values ofp,n andv, other forms of galaxies can be taken into account: Ring, barred and spiral-barred shapes etc. In order to generalize previous calculations, we further postulateρ ₀ =kU ₀/r ², leading to Poisson'sequation which accounts for the disc population $$\nabla ^2 U_0 - \frac{{\alpha ^2 }}{{r^2 }}U_0 = 0.$$ AsU ₀ is assumed axisymmetrical, the obvious solution is of the form $$U_0 = \frac{{cte}}{{r^v }}e^{ - pz} , \rho _0 = \frac{{cte}}{{r^{2 + v} }}e^{ - pz} .$$ Finally, Poisson's equation is completely solvable under the assumptionρ =k(U/r ². The general solution,valid for both disc and spiral arm populations, becomes $$U = cte e^{ - pz} \left\{ {r^{ - v} + } \right.\left. {cte e^{j[\Phi _v (r) + n\theta ]} } \right\},$$ The density distribution along the O _z axis is supported by Burstein's (1979) observations.     

Nonlinear newtonian theory and the effect of time dilatation

The fact that the energy density ρ_g of a static spherically symmetric gravitational field acts as a source of gravity, gives us a harmonic function \(f\left( \varphi \right) = e^{\varphi /c^2 } \) , which is determined by the nonlinear differential equation $$\nabla ^2 \varphi = 4\pi k\rho _g = - \frac{1}{{c^2 }}\left( {\nabla \varphi } \right)^2 $$ Furthermore, we formulate the infinitesimal time-interval between a couple of events measured by two different inertial observers, one in a position with potential φ-i.e., dt _φ and the other in a position with potential φ=0-i.e., dt ₀, as $${\text{d}}t_\varphi = f{\text{d}}t_0 .$$ When the principle of equivalence is satisfied, we obtain the well-known effect of time dilatation.     

Contribution of Vacuum Field to Angular Deviation of Light Path and Radar Echo Delay

The discovery of ‘twin quasistellar objects’ arose interests among astronomers and astrophysicists to study gravitational lensing problems. The deviation of light from its straight line path is caused by two sources according to the general theory of relativity: (i) the presence of massive objects, i.e. the presence of gravitational field and (ii) the presence of a ‘vacuum field’ which arises because there is a non-zero cosmological vacuum energy. Recently, the research on the relationship between cosmological constant and gravitational lensing process is rather active (see reference [1, 2, 3]. According to the Kottler space time metric, we have deduced an explicit representation of the angular deviation of light path. The deviation term is found to be simply , where M is the mass of the ‘astronomical lens’, r_min is the distance between the point of nearest approach and the centre of M, other symbols have their usual meaning. The presence of this term may be meaningful to the study of cosmological constant using the concept of gravitational lensing; however more sophisticated analysis awaits. Consider a signal radar to be sent from one planet to another. We have found that the radar echo delay contributed by the existence of the cosmological constant Λ is expressible as This revised version was published online in July 2006 with corrections to the Cover Date.     

A few observations of and remarks on V1500 Cygni

The development of the post-nova light curve of V1500 Cyg inUBV andHβ, for 15 nights in September and October 1975 are presented. We confirm previous reports that superimposed on the steady decline of the light curve are small amplitude cyclic variations. The times of maxima and minima are determined. These together with other published values yield the following ephemerides from JD 2 442 661 to JD 2 442 674: $$\begin{gathered} {\text{From}} 17 {\text{points:}} {\text{JD}}_{ \odot \min } = 2 442 661.4881 + 0_{^. }^{\text{d}} 140 91{\text{n}} \hfill \\ \pm 0.0027 \pm 0.000 05 \hfill \\ {\text{From}} 15 {\text{points:}} {\text{JD}}_{ \odot \max } = 2 442 661.5480 + 0_{^. }^{\text{d}} 140 89{\text{n}} \hfill \\ \pm 0.0046 \pm 0.0001 \hfill \\ \end{gathered} $$ with standard errors of the fits of ±0 _. ^d 0052 for the minima and ±0 _. ^d 0091 for the maxima. Assuming V1500 Cyg is similar to novae in M31, we foundr=750 pc and a pre-nova absolute photographic magnitude greater than 9.68.     

The diffuse cosmic X-ray background spectrum in the 2–10 keV energy range

We report a measurement of the background spectrum based on 10000 counts observed in the energy range 2–10 keV. The rocketborne detector system was optimised for cosmic ray noise rejection. A best fit power law spectrum $$\frac{{dN}}{{dE}} = 16E^{ - 1.8} photons{\text{ }}cm^{ - 2} s^{ - 1} sr^{ - 1} keV^{ - 1} .$$ resulted from the analysis. At 10 keV this result is consistent with recently assessed higher energy data. We show therefore that the change in spectral index between 5 and 50 ke V is approximately ?0.2.     

Effect of oblateness on the non-linear stability of L 4 in the restricted three-body problem

Non-linear stability of the libration point L ₄ of the restricted three-body problem is studied when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion, Moser's conditions are utilised in this study by employing the iterative scheme of Henrard for transforming the Hamiltonian to the Birkhoff's normal form with the help of double D'Alembert's series. It is found that L ₄ is stable for all mass ratios in the range of linear stability except for the three mass ratios: $$\begin{gathered} \mu _{c1} = 0.0242{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.1790{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c2} = 0.0135{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0993{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c3} = 0.0109{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0294{\text{ }}...{\text{ }}A_1 . \hfill \\ \end{gathered} $$      

The cosmogonic shadow effect

We consider the Alfvén-Arrhenius fall-down mechanism and describe an approximate model for the infall, capture and distribution of dust particles on a given magnetic field line and their possible neutralization at the ‘2’/3 points, the points at which the field aligned compnents of the gravitational and centrifugal forces are equal and opposite. We find that a small fraction (<10%) of an incoming particle distribution will actually contribute to the above ‘2’/3 fall-down process. We also show that if at the 2/3 points, the ratio of dust to plasma density is $$\frac{{n_D \left( {\tfrac{2}{3}} \right)}}{{n_p \left( {\tfrac{2}{3}} \right)}} > \frac{{10^{ - 3} }}{{r_{g_\mu } T_{eV} }}$$ . (r _gμ=radius of a grain in microns,T=plasma temperature in eV), then the dust particles will lose their charge, decouple from the field line and follow Keplerian orbits in accordance with the Alfvén-Arrhenius mechanism. We then determine the limits on the plasma parameters in order that rotation of a quasi-neutral plasma in thermal equilibrium be possible in the gravitational and dipole field of a rotating central body. The constraints imposed by the above conditions are rather weak, and the plasma parameters can have a wide range of values. For a plasma corotating with an angular velocity Ω～10^?4s^?1, we show that the plasma temperature and density must satisfy $$10^{ - 1}<< T_{(eV)}<< 10^2 ,10T_{eV}^2<< n^p \left( {cm^3 } \right)<< 10^6 $$ .     

Etude variationnelle des decalages spectraux

In a static gravitational field the paths of light are curved, as noticed by H. Weyl. This property can bea priori stated for aV ₃ Riemannian manifold: through any two points ofV ₃ it is possible to draw two families of curves, the straight lines of Euclidean geometry and the photon trajectoriesz. We can perform a fibration of the Galilean space-time in an original way, by taking thez-trajectories of the photons as the base, the isochronic surfaces as fibres, and ‘the equal length time on az trajectory to reach a given point’ as the equivalence relation. The straight lines of Euclidean geometry can then carry the classical mechanics timet, and thez trajectories can carry the optics time t. These times are related by dt=F(x,t) dt. If we class the Universe as a pseudo-Riemannian manifold of normal hyperbolic typeC ^∞, the time t determined above can be taken as the time coordinate inV ₄. Under these conditions we have \(d\overline s ^2 \) =F ² \(d\overline s ^2 \) , where \(d\overline s ^2 \) is the metric of the Riemannian manifold, conforming to the metric ds ² and allowing t as the cosmic time. We can then use the results previously achieved by the author (Peton, 1979) and write: 1 +Z _G =F(A _s,t _s,)/F(Ao_s,t _o) wherez _G denotes the shift of the spectral lines due to the metric. In the case of relative motion betweenO andS, we have $${\text{1 + z' = (1 + }}z_{\text{G}} {\text{)(1 + }}\beta _{\text{r}} {\text{)(1 }} - {\text{ }}\beta ^2 {\text{)}}^{ - 1/2} $$ The Doppler-Fizeau effect therefore appears as a result of the application of the Fermat principle.     

Chaos in cuspy triaxial galaxies with a supermassive black hole: a simple toy model

This paper summarises an investigation of chaos in a toy potential which mimics much of the behaviour observed for the more realistic triaxial generalisations of the Dehnen potentials, which have been used to model cuspy triaxial galaxies both with and without a supermassive black hole. The potential is the sum of an anisotropic harmonic oscillator potential, ${\text{V}}_{\text{0}} = \frac{1}{2}\left( {a^2 x^2 + b^2 y^2 + c^2 z^2 } \right)$ , and aspherical Plummer potential, ${\text{V}}_{\text{P}} = M_{BH} /\sqrt {r^2 + \varepsilon ^2 } $ , with $r^2 = x^2 + y^2 + z^2$ . Attention focuses on three issues related tothe properties of ensembles of chaotic orbits which impact on chaotic mixing and the possibility of constructing self-consistent equilibria:(1) What fraction of the orbits are chaotic? (2) How sensitive are the chaotic orbits, that is, how large are their largest (short time) Lyapunov exponents? (3) To what extent is the motion of chaotic orbits impeded by Arnold webs, that is, how 'sticky' are the chaotic orbits? These questions are explored as functions of the axis ratio a: b: c, black hole mass M _BH, softening length ε, and energy E with the aims of understanding how the manifestations of chaos depend onthe shape of the system and why the black hole generates chaos. The simplicity of the model makes it amenable to a perturbative analysis. That it mimics the behaviour of more complicated potentials suggests that much of this behaviour should be generic.     

A note on theH-functions of transfer problems in multiplying media

Some useful results and remodelled representations ofH-functions corresponding to the dispersion function $$T\left( z \right) = 1 - 2z^2 \sum\limits_1^n {\int_0^{\lambda r} {Y_r } \left( x \right){\text{d}}x/\left( {z^2 - x^2 } \right)} $$ are derived, suitable to the case of a multiplying medium characterized by $$\gamma _0 = \sum\limits_1^n {\int_0^{\lambda r} {Y_r } \left( x \right){\text{d}}x > \tfrac{1}{2} \Rightarrow \xi = 1 - 2\gamma _0< 0} $$      

Static spherically-symmetric scalar-field theory in general relativity

A spherically-symmetric static scalar field in general relativity is considered. The field equations are defined by $$\begin{gathered} R_{ik} = - \mu \varphi _i \varphi _k ,\varphi _i = \frac{{\partial \varphi }}{{\partial x^i }}, \varphi ^i = g^{ik} \varphi _k , \hfill \\ \hfill \\ \end{gathered} $$ where ?=?(r,t) is a scalar field. In the past, the same problem was considered by Bergmann and Leipnik (1957) and Buchdahl (1959) with the assumption that ?=?(r) be independent oft and recently by Wyman (1981) with the assumption ?=?(r, t). The object of this paper is to give explicit results with a different approach and under a more general condition $$\phi _{;i}^i = ( - g)^{ - 1/2} \frac{\partial }{{\partial x^i }}\left[ {( - g)^{1/2} g^{ik} \frac{\partial }{{\partial x^k }}} \right] = - 4\pi ( -g )^{ - 1/2} \rho $$ where ?=?(r, t) is the mass or the charge density of the sources of the field.     

Normality condition in the ideal resonance problem

The Ideal Resonance Problem, defined by the Hamiltonian $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x,\mu \ll 1,$$ has been solved in Garfinkelet al. (1971). As a perturbed simple pendulum, this solution furnishes a convenient and accurate reference orbit for the study of resonance. In order to preserve the penduloid character of the motion, the solution is subject to thenormality condition, which boundsAB" andB' away from zero indeep and inshallow resonance, respectively. For a first-order solution, the paper derives the normality condition in the form $$pi \leqslant max(|\alpha /\alpha _1 |,|\alpha /\alpha _1 |^{2i} ),i = 1,2.$$ Herep _i are known functions of the constant ‘mean element’y', α is the resonance parameter defined by $$\alpha \equiv - {\rm B}'/|4AB\prime \prime |^{1/2} \mu ,$$ and $$\alpha _1 \equiv \mu ^{ - 1/2}$$ defines the conventionaldemarcation point separating the deep and the shallow resonance regions. The results are applied to the problem of the critical inclination of a satellite of an oblate planet. There the normality condition takes the form $$\Lambda _1 (\lambda ) \leqslant e \leqslant \Lambda _2 (\lambda )if|i - tan^{ - 1} 2| \leqslant \lambda e/2(1 + e)$$ withΛ ₁, andΛ ₂ known functions of λ, defined by $$\begin{gathered} \lambda \equiv |\tfrac{1}{5}(J_2 + J_4 /J_2 )|^{1/4} /q, \hfill \\ q \equiv a(1 - e). \hfill \\ \end{gathered}$$      

On the number of isolating integrals in systems with three degrees of freedom

Dynamical systems with three degrees of freedom can be reduced to the study of a fourdimensional mapping. We consider here, as a model problem, the mapping given by the following equations: $$\left\{ \begin{gathered} x_1 = x_0 + a_1 {\text{ sin (}}x_0 {\text{ + }}y_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{)}} \hfill \\ y_1 = x_0 {\text{ + }}y_0 \hfill \\ z_1 = z_0 + a_2 {\text{ sin (}}z_0 {\text{ + }}t_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{) (mod 2}}\pi {\text{)}} \hfill \\ t_1 = z_0 {\text{ + }}t_0 \hfill \\ \end{gathered} \right.$$ We have found that as soon asb≠0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).     

A Check on the Validity of Magnetic Field Reconstructions

We investigate a method to test whether a numerically computed model coronal magnetic field \({\boldsymbol {B}}\) departs from the divergence-free condition (also known as the solenoidality condition). The test requires a potential field \({\boldsymbol {B}}_{0}\) to be calculated, subject to Neumann boundary conditions, given by the normal components of the model field \({\boldsymbol {B}}\) at the boundaries. The free energy of the model field may be calculated using \(\frac{1}{2\mu _{0}}\int ({\boldsymbol {B}}-{\boldsymbol {B}}_{0})^{2}\mathrm{d}V\), where the integral is over the computational volume of the model field. A second estimate of the free energy is provided by calculating \(\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}^{2}\,\mathrm{d}V-\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}_{0}^{2}\,\mathrm{d}V\). If \({\boldsymbol {B}}\) is divergence free, the two estimates of the free energy should be the same. A difference between the two estimates indicates a departure from \(\nabla \cdot {\boldsymbol {B}}=0\) in the volume. The test is an implementation of a procedure proposed by Moraitis et al. (Solar Phys.289, 4453, 2014) and is a simpler version of the Helmholtz decomposition procedure presented by Valori et al. (Astron. Astrophys.553, A38, 2013). We demonstrate the test in application to previously published nonlinear force-free model fields, and also investigate the influence on the results of the test of a departure from flux balance over the boundaries of the model field. Our results underline the fact that, to make meaningful statements about magnetic free energy in the corona, it is necessary to have model magnetic fields that satisfy the divergence-free condition to a good approximation.