Conservation of linear momentum

We could stop here in the discussion of the translational group. However, for the purpose of understanding the relation between translational symmetry and the conservation of linear momentum, we now show how the... 

Here (3A — Nc) is the number of degrees of freedom, equal to three times the number of particles minus the number of constraints, which typically will be 3 (corresponding to conservation of linear momentum). In a standard MC simulation the temperature is fixed NVT conditions), while it is a derived quantity in a standard MD simulation NVE conditions). 

Viscometric flow theories describe how to extract material properties from macroscopic measurements, which are integrated quantities such as the torque or volume flow rate. For example, in pipe flow, the standard measurements are the volume flow rate and the pressure drop. The fundamental difference with spatially resolved measurements is that the local characteristics of the flows are exploited. Here we focus on one such example, steady, pressure driven flow through a tube of circular cross section. The standard assumptions are made, namely, that the flow is uni-directional and axisymmetric, with the axial component of velocity depending on the radius only. The conservation of mass is satisfied exactly and the z component of the conservation of linear momentum reduces to... 

Equation (5) expresses the conservation of linear momentum that defines the position of the center of mass of the molecule, while Eq. (6) is an approximate statement of the conservation of angular momentum of the system These conditions, which are usually attributed to Eckart lead to the relation... 

For a fixed mass, the conservation of linear momentum is equivalent to Newton s second law ... 

An object moves in a 3-dimensional space where its potential energy is the same at every point. The expression describing the potential does not explicitly contain the coordinates x, y, or i. That is. the system is invariant with respeet to translation of the origin of the coordinate system in any direction. This symmetry is associated with conservation of linear momentum the momentum in all three dimensions is a constant. 

Consider a planar collision of a sphere of mass m and radius a against an initially stationary sphere of mass m2 and radius a2. Select the Cartesian coordinates so that the x- and y-axes are normal and tangential to the contact surfaces, respectively, as shown in Fig. 2.2. The conservation of linear momentum yields... 

Newton s Second Law is a statement of conservation of linear momentum for a system ... 

Conservation of linear momentum is due to spatial homogeneity. Consider i particles of mass m at coordinates xj, as before. Assume an interaction potential that only depends on the separation Xi — Xj between particle pairs. This potential is assumed because it is independent of the location of the system relative to the coordinate origin and the laws governing the evolution of the system are therefore spatial-displacement symmetric. The time rate of change of the total momentum... 

In a similar manner the homogeneity in space leads to the law of conservation of linear momentum [52] [43]. In this case L does not depend explicitly on qi, i.e., the coordinate qi is said to be cyclic. It can then be seen exploring the Lagrange s equations (2.14) that the quantity dLjdqi is constant in time. By use of the Lagrangian definition (2.6), the relationship can be written in terms of more familiar quantities ... 

This result implies, when integrated in time, that the conservation of linear momentum yields ... 

Before proceeding further, let us return briefly to the derivation based upon a fixed control volume and conservation of linear momentum. In this alternative approach, momentum is transported through the surface of the control volume by convection at a rate pu(u n) at each point, and this is treated as an additional contribution to the rate at which linear momentum is accumulated or lost from the control volume. Of course, there is no term in (2-23) that corresponds to a flux of momentum across the surface of the material (control) volume. Because all points within Vm(t) and on its surface Am( ) are material points, they move precisely with the local continuum velocity u and there is no flux of mass or momentum across the surface that is due to convection. 

It is, perhaps, well to pause for a moment to take stock of our developments to this point. We have successfully derived DEs that must be satisfied by any velocity field that is consistent with conservation of mass and Newton s second law of mechanics (or conservation of linear momentum). However, a closer look at the results, (2-5) or (2-20) and (2 32), reveals the fact that we have far more unknowns than we have relationships between them. Let us consider the simplest situation in which the fluid is isothermal and approximated as incompressible. In this case, the density is a constant property of the material, which we may assume to be known, and the continuity equation, (2-20), provides one relationship among the three unknown scalar components of the velocity u. When Newton s second law is added, we do generate three additional equations involving the components of u, but only at the cost of nine additional unknowns at each point the nine independent components of T. It is clear that more equations are needed. 

To evaluate the term, p v — p4v, we apply the principle of the conservation of linear momentum to the section of flow between stations 3 and 4 . The net force in the direction of flow is equal to the rate of momentum leaving minus the rate of momentum entering the section. Assuming pressure equalization at p3 along the whole flow front at station 3, we may write ... 

Conservation of linear momentum requires that, if two photons are emitted, their directions be opposite. A consequence of this is, as we shall show, that states with J == 1 cannot decay by double quantum emission. For, the angular momentum carried by the photons is 2S or zero the former is unacceptable if, before annihilation, the total angular momentum was % (J = 1) the latter is also unacceptable on grounds of symmetry, since states with J =1, Mr = 0 change sign under a rotation of 180° about the a -axis, but this is not true of the two-quantum state with M = 0, i.e., that which has angular momentum zero. 

We can describe the conservation of linear momentum by noting the analogy between the time-dependent Schrodinger equation, (equation A1 4.108). and (equation A 1.4.99). For an isolated molecule, //does not depend explicitly on t and we can repeat the arguments expressed in (equation Al.4.981. (equation Al.4.991. (equation Al.4.1001. (equation A 1.4.101) and (equation Al.4.102) with X replaced by t and P replaced by -// to show that... 

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