Conservation Laws in Mechanics

Because in the framework of Galileo s transforms the distance is invariant, the potential energy is also invariant (remember that kinetic energy is noninvariant in relation to Gahleo s transformations because of the dependence of a body s velocity on the choice of the coordinate system). 

The work of the gravitational forces at a body displacement from some point 1 to a point 2 will generally be expressed as follows  

Conservation laws are the most general, fundamental laws of nature. They have an enormous scientific value. Their importance is defined by the fact that the solution to many kinds of problem can be achieved with their help and without detailed analysis of specific 

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Consider a system of two colliding particles being a closed and a conservative one it is possible to apply to collision both conservation laws the energy and momentum conservation laws in mechanics. The energy conservation law looks as follows ... 

In a number of problems, e.g., the well-known problem of localized explosion solved by Sedov [1], the exponents are found by elementary means from the conservation laws of mechanics (from the conservation of energy in Sedov s example). 

The conservation of momentum and angular momentum is a fundamental law in mechanics. Kepler s second law can be easily justified by the conservation of angular momentum. 

A very remote analogy is conservation laws in classical and quantum physics. Description of quantum mechanics in terms of classical mechanics is not well defined, which happens because of commutativity of classical values and noncommutativity of their quantum analogs. We should regularize it and as a result part of classical symmetries may be realized in such a way that some conservation laws cannot be measured at all (e.g., conservation of the angular momentum as a vector). That example turns our attention to problem of observations. 

We derive the conservation law of mechanical energy, referred to as Stokes power formula, starting with the following equation of motion in an Eulerian framework ... 

Conservation laws at a microscopic level of molecular interactions play an important role. In particular, energy as a conserved variable plays a central role in statistical mechanics. Another important concept for equilibrium systems is the law of detailed balance. Molecular motion can be viewed as a sequence of collisions, each of which is akin to a reaction. Most often it is the momentum, energy and angrilar momentum of each of the constituents that is changed during a collision if the molecular structure is altered, one has a chemical reaction. The law of detailed balance implies that, in equilibrium, the number of each reaction in the forward direction is the same as that in the reverse direction i.e. each microscopic reaction is in equilibrium. This is a consequence of the time reversal syimnetry of mechanics. 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

Takesue [takes87] defines the energy of an ERCA as a conserved quantity that is both additive and propagative. As we have seen above, the additivity requirement merely stipulates that the energy must be written as a sum (over all sites) of identical functions of local variables. The requirement that the energy must also be propagative is introduced to prevent the presence of local conservation laws. If rules with local conservation laws spawn information barriers, a statistical mechanical description of the system clearly cannot be realized in this case. ERCA that are candidate thermodynamic models therefore require the existence of additive conserved quantities with no local conservations laws. A total of seven such ERCA rules qualify. ... 

There are several attractive features of such a mesoscopic description. Because the dynamics is simple, it is both easy and efficient to simulate. The equations of motion are easily written and the techniques of nonequilibriun statistical mechanics can be used to derive macroscopic laws and correlation function expressions for the transport properties. Accurate analytical expressions for the transport coefficient can be derived. The mesoscopic description can be combined with full molecular dynamics in order to describe the properties of solute species, such as polymers or colloids, in solution. Because all of the conservation laws are satisfied, hydrodynamic interactions, which play an important role in the dynamical properties of such systems, are automatically taken into account. 

A control volume is a volume specified in transacting the solution to a problem typically involving the transfer of matter across the volume s surface. In the study of thermodynamics it is often referred to as an open system, and is essential to the solution of problems in fluid mechanics. Since the conservation laws of physics are defined for (fixed mass) systems, we need a way to transform these expressions to the domain of the control volume. A system has a fixed mass whereas the mass within a control volume can change with time. 

Conservation Law for a System Conservation laws (e.g., Newton s second law or the conservation of energy) are most conveniently written for a system, which, by definition, is an identified mass of material. In fluid mechanics, however, since the fluid is free to deform and mix as it moves, a specific system is difficult to follow. The conservation of momentum, leading to the Navier-Stokes equations, is stated generally as... 

A sound understanding of the physical conservation laws is essential to one s ability to specialize them, solve them, and apply the results successfully. Therefore we begin with a derivation of the laws that govern the conservation of mass, momentum, thermal energy, and chemical species. We approach the derivation from a fluid-mechanical point of view, and the reader may find considerable overlap with other books in viscous fluid mechanics. However, we depart from the traditional presentation in two ways. First, because we are principally concerned with chemically reacting flow, we retain many features that may be negligible in fluid flow alone. Second, because we are often concerned with axisymmetric flows, we cast much of the mathematics in cylindrical coordinates rather than cartesian coordinates. While the later choice adds some complexity, it also serves to highlight some important issues that can be overlooked in cartesian coordinates. 

The mathematical formulation of the theory becomes drastically more complicated however, the physical conclusions in the part of the curve relating to the pressure change during the reaction, the selection principle, and the calculation of the detonation velocity and the effect of external losses on the detonation velocity remain practically unchanged. As was to be expected, a theory of pressure and velocity of a detonation wave based on the general conservation laws proves not very sensitive to the mechanism of chemical reaction. 

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