Narrow Results

In order to simulate rigidly rotating polytropes, we have simulated systems of $N$ point particles, with $N$ up to 1800. Two particles at a distance $r$ interact by an attractive potential $-1∕r$ and a repulsive potential $1∕r^2$. The repulsion simulates the pressure in a polytropic gas of polytropic index $3∕2$. We take the total angular momentum $L$ to be conserved, but not the total energy $E$. The particles are stationary in the rotating coordinate system. The rotational energy is $L^2∕(2I)$ where $I$ is the moment of inertia. Configurations, where the energy $E$ has a local minimum, are stable. In the continuum limit $N\to \infty$, the particles become more and more tightly packed in a finite volume, with the interparticle distances decreasing as $N^{-1∕3}$. We argue that $N^{-1∕3}$ is a good parameter for describing the continuum limit. We argue further that the continuum limit is the polytropic gas of index $3∕2$. For example, the density profile of the nonrotating gas approaches that computed from the Lane–Emden equation describing the nonrotating polytropic gas. In the case of maximum rotation, the instability occurs by the loss of particles from the equator, which becomes a sharp edge, as predicted by Jeans in his study of rotating polytropes. We describe the minimum energy nonrotating configurations for a number of small values of $N$.

The virial theorem and the concept of canonical-statistical distributions represent two fundamental elements of statistical physics. We apply these concepts to hand tremor oscillations recorded from six Parkinson patients. We find that the virial theorem holds for Parkinson tremor oscillations. In contrast, we find that the concept of canonical distributions fails to a certain extent and needs to be replaced by the notion of non-canonical (i.e., canonical-dissipative) distributions. In doing so, our analysis reveals both general statistical aspects and non-equilibrium aspects of Parkinson hand tremor.

We derive the virial theorem for an overdamped system of rotating self-gravitating Brownian particles. We show that, in the two-dimensional case, it takes a closed form that can be used to obtain general results about the dynamics without being required to solve the Smoluchowski–Poisson system explicitly. In particular, we obtain the exact analytical expression of the mean square displacement 〈r²〉(t) of the interacting Brownian particles. We exhibit a critical temperature below which the system collapses, and above which it evaporates, and we determine how this temperature is affected by a solid rotation. We also develop an analogy between self-gravitating systems and two-dimensional point vortices. We derive a virial-like relation for point vortices at statistical equilibrium relating the angular velocity to the angular momentum and the temperature.

The 4-dimensionally covariant approach to multiconstituent Newtonian fluid dynamics presented in the preceding paper of this series is developed by construction of the relevant 4-dimensional stress–energy tensor whose conservation in the non-dissipative variational case is shown to be interpretable as a Noether identity of the Milne spacetime structure. The formalism is illustrated by the application to homogeneously expanding cosmological models, for which appropriately generalized local Bernoulli constants are constructed. Another application is to the Iordanski type generalization of the Joukowski formula for the Magnus force on a vortex. Finally, at a global level, a new (formally simpler but more generally applicable) version of the "virial theorem" is obtained for multiconstituent — neutron or other — fluid star models as a special case within an extensive category of formulae whereby the time evolution of variously weighted mass moment integrals is determined by corresponding space integrals of stress tensor components, with the implication that all such stress integrals must vanish for any stationary equilibrium configuration.

Astrophysical bounds on the cosmological constant are examined for spherically symmetric bodies. Similar limits emerge from the hydrostatical and gravitational equilibrium and the validity of the Newtonian limit. The methods in use seem to be disjoint from the basic principles, however they have the same implication regarding the upper bounds. Therefore we will compare different inequalities and comment on the possible relationship between them. These inequalities are of relevance for the so-called coincidence problem and for the bound of the cosmological constant which comes surprisingly close to the "experimental" value.

Galaxy Clusters are huge physical systems having a generic size of tens of millions of light years. Thus any modification at the center of the cluster will affect the outskirts only tens of millions of years afterwards. Those retardation considerations seem to be neglected in present day analysis used to estimate the total mass of the galaxy cluster, including those estimates which are based on the virial theorem. The significant differences between the velocities predicted by Newtonian action at a distance theory and observations are usually dealt with by either assuming an unobservable type of matter or by modifying gravity. Here we demonstrate that considering general relativistic effects one can explain the apparent excess matter appearing in galaxy clusters virial estimates.

In this paper, the geometric approach to the virial theorem (VT) developed in [J. F. Cariñena, F. Falceto and M. F. Rañada, A geometric approach to a generalized virial theorem, J. Phys. A: Math. Theor. 45 (2012) 395210, 19 pp.] is written in terms of quasi-velocities (see [J. F. Cariñena, J. Nunes da Costa and P. Santos, Quasi-coordinates from the point of view of Lie algebroid structures, J. Phys. A: Math. Theor. 40 (2007) 10031–10048]). A generalization of the VT for mechanical systems on Lie algebroids is also given, using the geometric tools of Lagrangian and Hamiltonian mechanics on the prolongation of the Lie algebroid.

The generalization of the virial theorem, introduced by Clausius in statistical mechanics, has recently been carried out in the framework of geometric approaches to Hamiltonian and Lagrangian theories and it has been formulated in an intrinsic way. It is here revisited not only in its more general situation of a generic vector field but mainly from the perspective of contact and cosymplectic geometry. The previous generalizations allowing virial like relations from one-parameter groups of non-strictly canonical transformations and the rôle of Killing and conformal Killing vector fields for Lagrangians of a mechanical type are here completed with the theory for contact Hamiltonian, as well as gradient and evolution, vector fields. The corresponding theories in the framework of cosymplectic geometry and the particular case of time-dependent vector fields are also developed.

A generalization of virial theorems and virial stresses to micropolar continuum mechanics is explored. The linear momentum balance in dyadic product with translation leads to (i) the first virial theorem of micropolar continuum mechanics involving the infinite-time limit of the kinetic translational energy and (ii) a classical formula for computing the virial force-stress known in molecular dynamics. The angular momentum balance in dyadic product with rotation leads to (i) the second virial theorem of micropolar continuum mechanics involving the infinite-time limit of the kinetic rotational energy and (ii) a virial couple-stress along with a formula for its computation. The latter stress is also uncovered in the dyadic product of linear momentum balance with rotation. The virial force-stress and virial couple-stress contain, respectively, the Reynolds force-stress and turbulent couple-stress.

The rotation of the galactic objects has been seen by asymmetric Doppler shift in the CMB data. Molecular hydrogen clouds at virial temperature may contribute to the galactic halo dark matter and they might be the reason for the observed rotational asymmetry in the galactic halos. We present a method to constrain the parameters of these virial clouds given that they are composed of a single fluid. The method is such that it should be possible to extend it to more than one fluid.

Recently the role of huge magnetic fields in white dwarfs (WDs) has been explore. It was proposed the existence of WDs with a magnetic field of 10¹⁸ G with a critical mass M_max ≈ 2.58M_⊙ much larger than the Chandrasekhar limit ∼ 1.4M_⊙. These Ultra-magnetized super-Chandrasekhar white dwarfs were obtained not considering some physical aspects as virial theorem, breaking of spherical symmetry, inverse β-decay, and pycnonuclear fusion reactions, making them unstable. Taking in account these points, ultra-magnetized, supermassive and stable white dwarfs, with magnetic fields in their interior at maximum of the order of 10¹³ − 10¹⁴G, even difficult to be formed, are possible to exist at least theoretically.

The following sections are included:

• Introduction
• Formulating Classical Mechanics
• Numerical Methods for Differential Equations
  • The Størmer-Verlet Centered-Difference Algorithm
  • Runge-Kutta Integration
  • Compiler and Graphics Software
• Thermomechanical Equilibrium from Gibbs’ Entropy
• Development of Newtonian Molecular Dynamics
• “Corresponding States” At and Not At Equilibrium
  • van der Waals’ Equation and Corresponding States
• Gauss’ and Nosé-Hoover Thermal Mechanics
  • Example Problem in Gaussian Isokinetic Mechanics
  • “Stiff” Equations and the Nosé-Hoover Oscillator
  • Adaptive Integration Solves Nosé’s Equations
  • Is the Restriction $\mathcal{H}$ = 0 Signi?cant?
• Lattice Dynamics of One-Dimensional Chains
• A Classical ManyBody Example Problem
• Simulating Hydrodynamic Manybody Properties
  • The Virial Theorem for the Pressure Tensor P
  • The Pressure Tensor is Necessarily Symmetric
  • The Heat Theorem for the Heat-Flux Vector Q
• Summary of Lecture 1
• References