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El-Nabulsi Rami Ahmad. Gravitational Field as a Pressure Force from Logarithmic Lagrangians and Non-Standard Hamiltonians: The Case of Stellar Halo of Milky Way. Communications in Theoretical Physics, 2018, 69(3): 233  
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Gravitational Field as a Pressure Force from Logarithmic Lagrangians and Non-Standard Hamiltonians: The Case of Stellar Halo of Milky Way
El-Nabulsi Rami Ahmad *
Athens Institute for Education and Research, Mathematics and Physics Divisions, 8 Valaoritou Street, Kolonaki, 10671 Athens, Greece

 

† Corresponding author. E-mail: nabulsiahmadrami@yahoo.fr

Abstract
Abstract

Recently, the notion of non-standard Lagrangians was discussed widely in literature in an attempt to explore the inverse variational problem of nonlinear differential equations. Different forms of non-standard Lagrangians were introduced in literature and have revealed nice mathematical and physical properties. One interesting form related to the inverse variational problem is the logarithmic Lagrangian, which has a number of motivating features related to the Liénard-type and Emden nonlinear differential equations. Such types of Lagrangians lead to nonlinear dynamics based on non-standard Hamiltonians. In this communication, we show that some new dynamical properties are obtained in stellar dynamics if standard Lagrangians are replaced by Logarithmic Lagrangians and their corresponding non-standard Hamiltonians. One interesting consequence concerns the emergence of an extra pressure term, which is related to the gravitational field suggesting that gravitation may act as a pressure in a strong gravitational field. The case of the stellar halo of the Milky Way is considered.

PACS: ;02.30.Xx;;98.35.Df;
Keyword:logarithmic Lagrangian;non-standard Hamiltonians;modified Boltzmann equation;stellar dynamics;Milky way
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 Liénard-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 fluids. 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 modified as well. We are curious to know the implications of the modified 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 time be the base manifold, M the n-dimensional configuration space on t with coordinates qk(t), (k = 1, 2, …, n), TM the tangent bundle of M with coordinates . The LL of the system is denoted by with assumed to be a C2 function with respect to all its arguments. The action functional along a curve q(t) in M with endpoints a and b is defined by:

where ξ is a free parameter available for the theory. One can easily check that if q(t) is a local minimizer to the action (1), then it satisfies the following modified Euler-Lagrange equation:

Here .

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 Lagrangian and take a Legendre transformation to get the Hamiltonian in the family and the family of the action functionals is expressed as:

It is noteworthy that in our arguments, we have from Eq. (2) and . In fact, since we have:

then:

Using the fact that:

which gives

we obtain from Eqs. (5) and (6) the following canonical equations:

It is easy to check that for time-independent LL, the energy is a constant of motion. To prove this, we find the derivative of the non-standard Hamiltonian with respect to time, which gives:

It is interesting to obtain conservation of energy in the LL approach and standard forms of the canonical equations.

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 in N-particle phase space where and are 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 flow of collisionless stars by:[36]

For the case of a star of mass m moving in a gravitational field Φ, the Hamiltonian is . Usually, the gravitational field obeys the Poisson equation where G is the gravitational constant and is the mass density. Then from Eq. (11), we find using :

Using the fact that p = ∂ ln L/∂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.

In fact, the first three integrals are standard and their derivations are found in many hydrodynamics books, e.g. Refs. [38–39]. The last integral gives:

where

Now, we have:

and accordingly we find:

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

Finally, the 2nd of the modified Jeans equations is

We can now introduce in Eq. (21) the tensor velocity dispersion defined by .[39] Since we can rewrite Eq. (22) as:

We simplify this equation by multiplying Eq. (15) by ⟨vj⟩ and after simple arrangements,[33] we find:

and after substituting into Eq. (23), we find the 3rd of the modified Jeans equations:

We can simplify this equation to:

where and represents the tensor . Surprisingly for ξ = 1/4, the gravitational part disappears and Eq. (26) will be reduced to:

A comparison with the Euler equation in fluid dynamics shows that the term has a similar effect as a pressure P. 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 ≡ ρ is the density of the gas assumed to be at rest with initial density ρ0. 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 + n1, P0 + P1, v1, Φ0 + Φ1) and that the gas is isothermal, i.e. where is the isothermal sound speed, k, μ and mu are 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 that , k is the wave vector and ω is the corresponding frequency. Therefore, we obtain after simple algebra the following dispersion relation:

It is obvious that when ξ = 1/4, we fall into the isothermal sound waves. If ξ > 1/4 then the system is stable and the perturbation varies periodically in time. If ξ < 1/4 and k ≫ 1 the system is stable whereas for k ≪ 1, the system is unstable. Notice that the critical wave-number now is and the critical wavelength is . We can deduce now the modified Jeans mass, which is . Hence we argue that a physical solution is obtained only if ξ < 1/4. Any gas of cloud with mass larger that the modified Jeans mass is unstable and hence will collapse due to its own gravitational field. It is notable at the end that after setting (n, P, v, Φ) = (n0 + n1, P0 + P1, v1, Φ0 + Φ1) it is easy to check that the perturbed part of the density obeys the sound wave equation where .

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 modified. To check this, we can write:

and then the modified continuity equation takes the form:

Multiplying Eq. (35) by ⟨vk⟩ and substituting the resulting equation into Eq. (33), we find:

where . Defining as an extra-pressure, we can simplify Eq. (34) to:

In Eq. (37) the pressure is coupled to the gravitational field and accordingly rending the system of equation coupled altogether and the Fourier decomposition can not be applied. Obviously when , Eq. (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 relation is 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-pressure and 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 potential where v0 is a constant (assuming the velocity components are isotropic). For the Milky Way’s halo, observations show that the velocity of gas on circular orbits is v0 ≈ 220 · s−1 (rotation is negligible) and that n ∝ r−3.5.[46] The extra-pressure is then given by . This pressure is tiny for large r but significant for small r and vanishes at r = 1 (in units v0 = 1). We first plot in Fig. 1 the variations of for ξ = 1/8 and ξ = − 1/8 after setting v0 = 1 for graphical illustration purpose:

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

For ξ = 1/8 the global minimum (≈ −8/7e) occurs at r = e2/7 and for ξ = − 1/8 the global maximum (≈ 8/7e) occurs as well at r = e2/7. In Figs. 2–5 we plot in 3D the variations of for different ranges of ξ and their corresponding contour plot:

Fig. 2 (Color online) Variations of 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.

More generally, the dark halo potential of the Milky Way is characterized by the logarithmic potential with d = 12 Kpc and vhalo = 131.5 km · s−1. The density distribution is given by the Plummer profile where r0 = 0.53 Kpc and where M = 107M⊙.[47] Therefore the gravitational potential in the LL approach is given by:

We plot in Fig. 6 the variations of for ξ = 1/8 and ξ = − 1/8 after setting vhalo = n0 = 1 for graphical illustration purpose and we plot in Figs. 7–10 their 3D variations for different ranges of ξ:

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

For ξ = 1/8, the global minimum is ≈ − 11367 at r = 0 whereas the global maximum is ≈ 11367 at r = 0 in units vhalo = 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 modified 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 modified 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 modified and depends on the parameter ξ. When ξ = 1/4, the medium is governed by the isothermal sound waves. For ξ > 1/4, the system is stable and the perturbation varies periodically in time. Stability of the system is achieved as well for ξ < 1/4 and k ≫ 1 whereas it is unstable for k ≪ 1. We have deduced as well the Jeans mass and it was observed that a physical solution is obtained only if ξ < 1/4. For the strong case, Jeans equations are modified considerably. A new pressure term appears in the theory and amazingly when the 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.

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