Gravitational Field as a Pressure Force from Logarithmic Lagrangians and Non-Standard Hamiltonians:The Case of Stellar Halo of Milky Way

Rami Ahmad El-Nabulsi

Athens Institute for Education and Research,Mathematics and Physics Divisions,8 Valaoritou Street,Kolonaki,10671 Athens,Greece

1 Introduction

Recently,the notion of non-standard Lagrangians(NSL)was extensively explored in literature in an attempt to explore the inverse variational problem of some familiar nonlinear differential equations.[1]A number of motivating results were obtained in the theory of dissipative dynamical systems,[2?6]geometric and nonlinear dynamics,[7?18]plasma physics,[19]quantum theory,[20?22]relativity,[23?25]and electromagnetic theory.[26]Despite the wide range of applications of NSL,the topic still requires more attention and deserves more concentration and studies.It is noteworthy that NSL are more appreciated by mathematicians than physicists and we already know the main reason.In physics,the meaning of NSL is still unclear,no physical foundation was studied carefully and accordingly they serve mathematicians more than physicists.This is an open dilemma that deserves a serious analysis.In applied mathematics and mainly in the framework of dynamical systems,NSL may increase the number of the initial data required to fix the classical trajectory and generate dynamical equations that go beyond the standard Newton’s law paradigm.NSL came in a large number of mathematical forms.[6]One interesting form introduced in literature is the Logarithmic Lagrangian(LL),[27]which possesses some interesting properties related to the Li′enard-type and Emden nonlinear differential equations.In fact,such forms of Lagrangians were introduced in order to study the root of reciprocal Lagrangians in the theory of inverse variational problem.Motivated by LL,the main aim of this paper is to introduce the corresponding non-standard Hamiltonians and to discuss some of its implications in stellar dynamics.We have selected this topic due to its double impacts in applied mathematics and physics.In reality,the stellar dynamics and the motion of stars within galaxies are getting more and more recognized to be important both in applied mathematics and practical applications as they connect the microscopic and macroscopic kinetic theory of gases and fl uids.In fact,the collisionless Boltzmann equation,which is the subject of this paper,is the most classical but fundamental equation in the kinetic theory used to describe the motion of stars within galaxies,which allows the numbers of stars to be calculated as a function of position and velocity in the galaxy.However,it is well-known that the collisionless Boltzmann equation can be derived using Hamiltonian’s equations.As the LL approach is expected to modify the Hamilton’s equations of motion,we expect accordingly that the Boltzmann equation is modi fied as well.We are curious to know the implications of the modi fied Boltzmann equation in stellar dynamics.The paper is organized as follows:in Sec.2,we introduce the basic setups of logarithmic Lagrangians and their corresponding non-standard Hamiltonians;in Sec.3 we discuss their implications in stellar dynamics and finally conclusions are given in Sec.4.

2 Logarithmic Lagrangians and Their Corresponding Non-Standard Hamiltonians

We start by introducing the basic setups of the LL approach.

Let timet∈R1be the base manifold,Mthendimensional con figuration space ontwith coordinatesqk(t),(k=1,2,...,n),TM the tangent bundle ofMwith coordinates(qk(t),˙qk(t)).The LL of the system is denoted by lnL(qk(t),˙qk(t),t)∈C2([a,b]×Rn×Rn;R)with(qk(t),˙qk(t),t)→L(qk(t),˙qk(t),t)assumed to be aC2function with respect to all its arguments.The action functional along a curveq(t)inMwith endpointsaandbis de fined by:

ReplacingF→ ξlnLgives effortlessly Eq.(2).

In what follows,we suppose for simplicity that the Lagrangian does not depend on time.The action principle can also be carried out on the phase space in the Hamiltonian formalism.We introduce a family of conjugate momenta from the family of Lagrangianpk=?lnL/?˙qkand take a Legendre transformation to get the Hamiltonian in the familyH(qk,pk)=ξ(pk˙qk?lnL(˙qk,qk))and the family of the action functionals is expressed as:

It is interesting to obtain conservation of energy in the LL approach and standard forms of the canonical equations.Remark 2It is notable that the non-standard Hamiltonians are exactly equivalent to the Logarithmic Lagrangians because no truncations occur when the Hamiltonians are derived from the Lagrangians.However,post-Newtonian Lagrangians and Hamiltonians at the same order are non-equivalent in general due to the truncations in the transformation between the Lagrangians and Hamiltonians.[29?31]Even the differences between the Lagrangians and Hamiltonians caused by the truncations make the Lagrangians and Hamiltonians have different dynamics.[32?34]

Equations(1)–(10)are general descriptions of the theory and method.We will use now these equations in the next section to derive the corresponding Boltzmann equation and discuss some of its implications in stellar dynamics.

3 Applications in Stellar Dynamics

In the framework of the semiclassical transport theory,the Boltzmann equation governs the spatio-temporal evolution of the particle gas.In fact,galaxies are collisionless systems as their relaxation time is very long.Hence,the motion of stars within galaxies can be described by the collisionless Boltzmann equation. This equation can be derived using Hamiltonian mechanics.In this section,the MCBE will be derived from the modified Hamiltonian equations(8)and(9).To do this,we let:be the distribution function of stars in a certain galaxy inN-particle phase space where?x∈R3and?p∈R3are position and momentum respectively.This is in fact the probability density in the 6-dimensional phase space of position and velocity at a given time known as the space phase density.[35]We shall assume in this work that the number of stars does not change.In fact,we can represent the fl ow of collisionless stars by:[36]

Remark 3It is noteworthy for elucidation that Eq.(11)is the well-known Liouville theorem(expressed in a differential form)which states that the phase space density of a certain element as it moves in phase space is fixed.However,the standard Hamiltonian is replaced by the non-standard one.In other words,the non-standard Lagrangians and its corresponding non-standard Hamiltonian are implemented in Eq.(11).

For the case of a star of massmmoving in a gravitational field Φ,the Hamiltonian isUsually,the gravitational field obeys the Poisson equationwhereGis the gravitational constant andis the mass density.Then from Eq.(11),we find usingv

Using the fact thatp=?lnL/?v=mv/L,we get:

Equation(13)is the MCBE.From this equation more useful equations may be derived accordingly,mainly the Jeans equations,which relate number densities,mean velocities,velocity dispersions,and the gravitational potential.However,we may distinguish between two independent cases:weak and strong gravitational field.Although we know that when the gravitational field is strong,e.g.near the vicinity of black holes,general relativity must be used accordingly,and the evolution of the system is described by the general relativistic Vlasov-Einstein system,[37]we will explore some of the effects of LL approach classically for the case of dense systems.

Case 1Weak gravitational field:To derive the first of these equations,we consider the 0thmoment by integrating Eq.(11)over all velocities:

which is the standard continuity equation.To derive the 2ndof the Jeans equations,we consider the 1stmoment of the MCBE by multiplying by the velocity and integrating again over all velocities:and accordingly we find:

If the gravitational field is weak,one can simplify Eq.(20)to:

A comparison with the Euler equation in fl uid dynamics shows that the termhas a similar effect as a pressureP.As an application,let us search the conditions under,which stars can form out of the interstellar medium.For simplicity,we will deal with a one-dimensional problem.In that case the following relations hold:

n≡ρis the density of the gas assumed to be at rest with initial densityρ0.The gravitational potential in Eq.(29)obeys the equation:

In what follow we consider a small perturbation around the equilibrium position such that(n,P,v,Φ)=(n0+and that the gas is isothermal,is the isothermal sound speed,k,μandmuare respectively the Boltzmann constant,the mean molecular weight and the atomic mass unit.In fact,we have based our hypothesis on the fact that the energy exchange by radiation is very proficient for interstellar mater.[40]To solve the system of equations,we assume a harmonic perturbation such thatkis the wave vector andωis the corresponding frequency.Therefore,we obtain after simple algebra the following dispersion relation:

It is obvious that whenξ=1/4,we fall into the isothermal sound waves.Ifξ>1/4 then the system is stable and the perturbation varies periodically in time.Ifξ<1/4 andk?1 the system is stable whereas fork?1,the system is unstable.Notice that the critical wave-number now isthe critical wavelength isWe can deduce now the modi fied Jeans mass,which isHence we argue that a physical solution is obtained only ifξ<1/4.Any gas of cloud with mass larger that the modi fied Jeans mass is unstable and hence will collapse due to its own gravitational field.It is notable at the end that after settingit is easy to check that the perturbed part of the density obeys the sound wave equationwhere

Case 2Strong gravitational field:If,for instance,the gravitational field is strong,such as that in the proximity of the event horizon of a black hole,then Eq.(20)is replaced by:

The last term is proportional to the logarithmic of the gravitational field.This is interesting as logarithmic correction to the gravitational field was discussed in differential celestial problems(see the most recent work.[41])Further that for strong gravitational field the continuity equation is modi fied.To check this,we can write:

In Eq.(37)the pressureis coupled to the gravitational if eld and accordingly rending the system of equation coupled altogether and the Fourier decomposition can not be applied.Obviously whenEq.(37)is reduced to its standard form and the Fourier decomposition can be applied easily and stability may be obtained accordingly.It is noteworthy that the relationis interesting since it relates the pressure to the gravitational field.Such a relation has interesting consequences in general relativity and a theory of gravity as a pressure force was discussed in Refs.[42–43]and have interesting features in cosmology and theory of dark matter and dark energy.[44?45]Considering the previous perturbation ansatzs we obtain,which corresponds to isothermal sound waves.We may conclude that within the LL approach,a strong gravitational field is stable if it is subject to an extra-pressureand dominated by isothermal sound waves.As a simple numerical test,we consider the simple model of our Galaxy’s stellar halo assumed to be spherical and governed by the logarithmic potentiallnrwherev0is a constant(assuming the velocity components are isotropic).For the Milky Way’s halo,observations show that the velocity of gas on circular orbits isv0≈220 km·s?1(rotation is negligible)and thatThe extra-pressure is then given bylnr/2ξ.This pressure is tiny for largerbut signi ficant for smallrand vanishes atr=1(in unitsv0=1).We first plot in Fig.1 the variations ofˉPforξ=1/8 andξ=?1/8 after settingv0=1 for graphical illustration purpose:

Fig.1(Color online)Variations ofˉP for ξ=1/8(blue graph)and ξ= ?1/8(red graph).

Forξ=1/8 the global minimum(≈?8/7e)occurs atr=e2/7and forξ=?1/8 the global maximum(≈8/7e)occurs as well atIn Figs.2–5 we plot in 3D the variations oflnr/2ξfor different ranges ofξand their corresponding contour plot:

Fig.2(Color online)Variations ofˉP for?1/4<ξ<1/4.

Fig.3 (Color online)Contour plot of Fig.2.

Fig.4(Color online)Variations of for?10<ξ<0.

Fig.5 (Color online)Contour plot of Fig.4.

Fig.6(Color online)Variations of for ξ=1/8(blue graph)and ξ= ?1/8(red graph).

More generally,the dark halo potential of the Milky Way is characterized by the logarithmic potential Φ=withd=12 Kpc andvhalo=131.5 km·s?1.The density distribution is given by the Plummer pro-wherer0=0.53 Kpc andwhereTherefore the gravitational potential in the LL approach is given by:

We plot in Fig.6 the variations offorξ=1/8 andξ=?1/8 after settingvhalo=n0=1 for graphical illustration purpose and we plot in Figs.7–10 their 3D variations for different ranges ofξ:

Fig.7(Color online)Variations of for?1/4<ξ<1/4.

Fig.8 (Color online)Contour plot of Fig.7.

Fig.9(Color online)Variations of for?1/4<ξ<1/4.

Fig.10 (Color online)Contour plot of Fig.9.

Forξ=1/8,the global minimum is≈?11367 atr=0 whereas the global maximum is≈11367 atr=0 in unitsvhalo=n0=1.

4 Conclusions

To conclude in this work,we have introduced the notion of LL,which belongs to the class of non-standard Lagrangians.After deriving the corresponding Hamiltonian equations and the modi fied Boltzmann equation we have discussed their implications in stellar dynamics in galaxies.We have discussed two independent classical cases:the weak and the strong gravitational field.For the weak case,the modi fied Jeans equations are more or less similar to the standard ones.More precisely,after introducing a small perturbation around the equilibrium position,it was observed that the dispersion relation is slightly modi fied and depends on the parameterξ.Whenξ=1/4,the medium is governed by the isothermal sound waves.Forξ>1/4,the system is stable and the perturbation varies periodically in time.Stability of the system is achieved as well forξ<1/4 andk?1 whereas it is unstable fork?1.We have deduced as well the Jeans mass and it was observed that a physical solution is obtained only ifξ<1/4.For the strong case,Jeans equations are modified considerably.A new pressure termappears in the theory and amazingly whenthe medium is stable and is governed by isothermal sound waves.Such a relation between the gravitational field and the pressure is motivating and may have interesting consequences in astrophysics and cosmology.In this note,we have tried to prove the importance of NSL in general and LL in particular in astrophysics(the Milky Way’s halo).Starting from simple arguments,we have obtained a number of rich possibilities that deserve further exploration and application.

I would like to thank the anonymous referees for their useful comments and valuable suggestions.

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