Classical Mechanics is the theory governing the motion of particles. The theory Arnold refers to this observation as the principle of Newtonian determinacy. Newton formulated what is now called 'classical mechanics'. Since his time the theory has been reformulated and generalized in various ways. Newtonian Mechanics. Introduction. The world we live in is a complex place, and we must expect any theory that describes it accurately to share that.

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Both quantum mechanics and relativity were invented in the twentieth century; the laws of classical mechanics were stated by Sir. Isaac Newton in [New02] . Newtonian Mechanics – Single Particle. (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 2). Since our course on the. Chapter 2. Newtonian Mechanics – Single Particle. In this Chapter we will review what Newton's laws of mechanics tell us about the motion of a single particle.

Skip to main content. Log In Sign Up. Newtonian mechanics of a many-particle assembly coupled to an external body potential. Newtonian mechanics of a many-particle assembly coupled to an external body potential Physical Review A , Robert E Salvino. Salvino Sverdrup Technology, Inc. Lewis Research Center Group Mail Stop Cleveland, Ohio Abstract The Newtonian mechanics of a many-particle system evolving in time under the influence of an external body potential, that is, an external potential that couples to the center of mass only, is exam- ined. The lack of any other external fields allows the complete sep- aration of the center-of-mass or external-field-dependent equations from the internal or external-field-independent dynamics. The com- plete solution of the center-of-mass motion then allows an analytical evaluation of the external body potential contribution to the thermo- physical properties of the system. The phenomena of field-induced heating, Taylor-Aris hydrodynamic form for the diffusion tensor, and an analogous hydrodynamic form for the viscosity tensor are derived from microscopic principles. A brief and model-dependent description of equilibrium phenomena is also presented.

Equation 4. For long times, the external field contributions to 4.

The significance of this long-time divergence will be discussed below within the context of thermophysical parameters. All of the gravitational effects are contained in the center-of-mass terms given by the kinetic-energy contribution of the center of mass in Eq. The internal component of 4. Finally, a position-velocity outer product [7] can also be constructed: Again, since the internal center of mass is at zero and the internal center-of-mass drift velocity is also zero [see Eq. The position-velocity outer product 4.

All of the above dynamical functions have separated into a contribution that is independent of the external gravitational field and a contribution from the external gravitational field through the center-of-mass motion only. This contribution depends on gross properties of the system only and not on the details of the internal dynamics in any way.

However, not all ther- mophysical properties will have such a simple separation. In particular, the pressure tensor time correlation function will not separate in this way [see Eq. V Some Macroscopic State Variables: Nonequilibrium Case The connection of the microscopic theory with the macroscopic descrip- tion of the many-particle system is made through the thermophysical prop- erties of the system.

These are related to the time average or time integral of various microscopic variables. With the discussion at the end of Sec. The focus here will be on the external field or gravitational contribution to the thermophysical properties. As noted in Sec. IV, there are some quantities that are independent of the external field. These consist of any dynamical functions that depend on particle coordinate differences only and, consequently, any thermophysical property which depends exclusively on such dynamical functions will also be independent of the external field.

Two examples are the potential energy of interaction between particles and the interparticle force contribution to the pressure and pressure tensor. Another class of thermophysical properties are simple time averages of the corresponding microscopic dynamical function that depend in some way on the center-of-mass coordinates and velocity.

The first velocity-dependent term in 5. As can be seen from 5. Unless a mechanism is provided e.

This contribution lends to lower the total energy of the system of particles. For a given position, a particle can always find a new position with lower energy in a system unbounded from below. A third class of thermophysical properties are those related to the time integral of a time correlation function of dynamical variables. A time- integrated correlation function which has the character of 5. In an Earth gravitational environment, the gravitational term will be of the same order of magnitude as the internal contribution to the diffusion tensor for a Lennard-Jones system [8] for cutoff times on the order of or, in real time, on the order of 0.

The elements of the shear viscosity tensor can be obtained from 5. On the other hand, the second term, the pure center-of-mass contri- bution, can be evaluated exactly. For simplicity, the center of mass will be considered to be initially located at the origin and to be initially at rest.

However, the third term in 5. Unlike the pure center-of-mass contribution to the viscosity 5. It is clear that the pure center-of-mass contribution 5. Again, the divergent behavior is to be attributed to the constant external force driving the system and supplying an unlimited amount of energy to the system.

It will be shown below how, for each thermophysical quantity examined in Sec. The contribution to the thermophysical properties from the external field, for the most part, is due to the initial period of the system, the initial approach to equilibrium.

However, if the system is allowed enough time to equilibrate, that is, if the time origin for equilibrium is chosen correctly so that it coincides with or is later than the wall time, then only equilibrium states will be sampled and this model then shows there will be no external field contribution to equilibrium properties due to the initial transient period.

For instance, the external field effect on the temperature defined in Sec. IV is changed from 5. If the time origin is chosen to be equal to or greater than the wall time, the g terms are eliminated completely. With the initial velocity of the center of mass set to zero, use of Eq. It may be possible to use such a signature to indicate if unwanted transients were permitted to enter the equilibrium thermophysics.

A similar effect can be calculated for the gravitational contribution to the energy. The result 5. That is, the term that is explicitly linear in g [there is a g dependence hidden in zcm tw ] may he considered to be the equilibrium contribution and the quadratic term is the transient contribution.

A similar structure shows up in the pressure and pressure tensor [see Eq. Hence, the inertial frame of reference only makes sense in the relative near movements. In the same way the acceleration of a body depends of the relative movements.

We are going to see that the inertial mass coincides with the gravitational mass only in an inertial frame. Consider a galaxy or a galaxy system of gravitational mass M gc and inertial mass M ic.

The galaxy or the galaxy system in its inertial frame feels the constant acceleration a0 induced by the rest of the matter of the universe i. In this inertial frame the experimental body is not the one who feels the constant acceleration a0, but it is the rest of the universe who feels this constant acceleration a0.

If we now consider, applying the relative movement, that is our planet who feels the constant acceleration a0 and hence we are not in an inertial frame. Moreover, Jacob D. Concluding remarks We have seen that the inertia theory of Assis as well as the inertia theory from General Relativity take us to the same conclusions in a simple case. However, this is not an extra assumption because in fact the inertial frames of reference do not exist, as we have seen in the previous section.

Moreover, in the framework of the inertia theory of Assis this assumption it is already incorporated. Another important open problem is to understand the origin of gravity. Moreover, in [17] the origin of gravity is considered in connection with electromagnetism. In: Edwards Matthew R, editor. Montreal: Apeiron; Found Phys Lett ;— Relational mechanics. Relativistic gravitation theory for the MOND paradigm.

Phys Rev D ; D 70, ]. An alternative to the dark matter paradigm: relativistic MOND gravitation. In: JHEP conference proceedings of science; Does the missing mass problem signal the breakdown of Newtonian gravity?

Astrophys J ;— Phys Lett A ;—6. When treating large degrees of freedom at the macroscopic level, statistical mechanics becomes useful. Statistical mechanics describes the behavior of large but countable numbers of particles and their interactions as a whole at the macroscopic level. Statistical mechanics is mainly used in thermodynamics for systems that lie outside the bounds of the assumptions of classical thermodynamics.

In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. In case that objects become extremely heavy i. In that case, General relativity GR becomes applicable.