Equations of motion

The equations of motion for fluids are given by the equation for the conservation of matter and Newton s law for the forces acting on the fluid. For an incompressible fluid, the density p of the fluid will be constant  

If the velocity of the fluid is v, then the mass that flows in a unit time across a unit area of surface is the component of p v normal to the surface, so that the divergence of pv is equal to the decrease in density per unit time [1]  

In order to outline the main features of measuring the gravitational field with the help of ballistic gravimeter imagine that a small body falls inside a vacuum cylinder under the action of the gravitational field only. In accordance with Newton s second law in the inertial frame we have 

In essence, we have expanded the field magnitude in a power series and discarded all terms except the first and second ones. It may be proper to point out that we assume that within the interval of measurements the rate of change of the field  

To find a solution of Equation (3.5) we perform its integration that gives ds 

Both constants are determined from two initial conditions  

we suppose that the field varies linearly along the path of movement. Then, the fall of a body is described by Equation (3.4)  

In the harmonic approximation the potential energy of a crystal in which the atoms are vibrating about their equilibrium positions differs from l 0, the potential energy with each atom on its equilibrium site, by 

Because the potential energy is invariant under an arbitrary displacement of the whole crystal, the force constants obey the sum rule 

Equation (11) describes the time dependence of the displacements. In the harmonic approximation the displacements are plane waves 

The 3s linear homogeneous equations (14) have non-trivial solutions if 

Because these equations are homogeneous, the eigenvector components c (/v, q/) are determined only to within a constant factor, which may be chosen to satisfy the orthonormal 

The equations of the Lagrangian incremental description of motion can be derived from the principles of virtual work (i.e., virtual displacements, virtual forces, or mixed virtual displacements and forces). Since our ultimate objective is to develop the finite-element model of the equations governing a body, we will not actually derive the differential equations of motion but utilize the virtual work statements to develop the finite element models. 

The displacement finite element model is based on the principle of virtual displacements. The principle requires that the sum of the external virtual work done on a body and the internal virtual work stored in the body should be equal to zero (see Reddy(46)) 

During the motion of the body, its volume, surface area, density, stresses, and strains change continuously. The stress measure that we shall use is the 2nd Piola-Kirchhofif stress tensor. The components of the 2nd Piola-Kirchhoff stress tensor in Cj will be denoted by To see the meaning of the 2nd Piola-KirchhofiF stress tensor, consider the force dF on surface dS in C2. The Cauchy stress tensor t is defined by 

From the definition it is clear that the 2nd Piola-Kirchhoff stress is measured in C2 but referred to Cj. It can be shown that the components 25 and are related according to 

Similarly, the Green-Lagrange strain tensor E j and the infinitesimal strain tensor e-j are related by 

As any high school student, knows, Newton s second law of motion says that force is equal to mass times acceleration for a system with constant mass M. 

This is the basic relationship that is used in writing the equations of motion for a system. In a slightly more general form, where mass can vary with time, 

Equation (2.33) says that the time rate of change of momentum in the i direction (mass times velocity in the i direction) is equal to the net sum of the forces pushing in the z direction. It can be thought of as a dynamic force balance. Or more eloquently it is called the conservation ofinomentum. 

In the real world there are three directions x, y, and z. Thus, three force balances can be written for any system. Therefore, each system has three equations of motion (plus one total mass balance, one energy equation, and NC — 1 component balances). 

Instead of writing three equations of motion, it is often more convenient (and always more elegant) to write the three equations as one vector equation. We will not use the vector form in this book since all our examples will be simple one-dimensional force balances. The field of fluid mechanics makes extensive use of the conservation of momentum. 

Other potential energy contributions may come from non-bonded interactions among atoms in the molecule, described by some function of interatomic distance, as in equation 2.7, possibly supplemented by coulombic terms over some kind of distribution of charge within the molecule (for example, localized point charges on atoms, as in equation 2.8)  

When potentials are known, obtaining forces is a simple matter let Xi,k be the k-th cartesian component of the position vector of atom i of mass rm the corresponding component of the force at atom i is given by 

Given a reasonable starting configuration (Xi,k)°, integration of the above differential equation gives the trajectories of all atoms in the molecule. Of course the integration cannot be carried out analytically, but numerical integration techniques are cheaply and readily available [2]. 

A single isolated molecule of mass tn would just travel forever on a linear trajectory with constant velocity v and kinetic energy E (kin) = / mv dv = 1/2 mv. When two molecules are close to one another, they interact with a potential such as shown in Fig. 4.4, which will here be denoted simply by (pot, inter). This potential may have 

Pressure can be easily understood for a gas, by the traditional didactic picture of traveling molecules that hit the container walls and transfer a certain amount of momentum to them. But the definition of pressure does not require the presence of a wall and need not invoke collisions between wall and molecules a physical barrier is just convenient for the visualization of an arbitrary plane through the material. Pressure is flow of momentum through any plane, and can thus be defined at any place in any system of any state of aggregation. 

Differentiating and rearranging the terms leads to the following equations of the movement (Paul equation)  

The trajectory of an ion will be stable if the values of x and y never reach r0, thus if it never hits the rods. To obtain the values of either x or y during the time, these equations need to be integrated. The following equation was established in 1866 by the physicist Mathieu in order to describe the propagation of waves in membranes  

In the first term of the Paul equation, replacing f2 by f2 introduces a factor o 2/4. To compensate for this factor, the whole equation must be multiplied by the reverse, 4/co2. In the cosine term, 2 is equal to cot, as needed in the Paul equations. Incorporating these changes and rearranging the terms yields the following expressions  

Stable and unstable trajectories of ions in a quadrupole. Reproduced (modified) from March R.E. and Hughes R.J., Quadrupole Storage Mass Spectrometry, Wiley, New York, 1989, with permission. 

The last terms of both the U and V equations is a constant for a given quadrupole instrument, as they operate at constant . We see that switching from one m/z to another results in a proportional multiplication of au and qu, which means changing the scale of the drawing in U, V coordinates thus the triangular area A will change from one mass to another, like 

Application of the FEM to structural dynamics leads to the discrete equations of motion of an adaptronic structure  

The equations of motion (5.14) can be transformed into the state equations of a state-space system description (5.3, 5.4) if the displacements q and velocities q are chosen as state variables  

A comparison of both systems of equations shows that the state or system matrix is determined by the plant properties K, M and D. 

Alternatively, modal amplitudes and velocities can be chosen as state variables, leading to a desirable decoupling of the state differential equations. 

This method is widely used to simulate motions in liquids and solids and to study rapid diffusions in ionic lattices. Although the application of molecular dynamics (MD) to studying ion transport in polymer systems is still in a seminal state, it is included in this chapter because of its correlation with the expansion of computing power, which will probably result in its prominence as an investigative method in this field. 

The general procedure for applying MD to a solid or molecular system is to define a configuration vector x that comprises the 3N coordinates of the AT constituent atoms of the system. Unlike the configuration vector described in Section 1.4 for static simulation, the present vector x(0 possesses a time dependence, which allows the atoms to explore the configuration space under the forces imposed on them. These forces are defined by the kinetic energies of the particles V2 m and by their mutual interactions, which are derived from potentials like those discussed in Section 1.3 and which are now expressed as a single ftinction 0 

In this way motions of the atoms can be made to emulate those of a real system resulting from thermal energy and mutual interactions. 

The time evolution of the total conformation vector x(r) is implemented by allowing all N atoms to move in accordance with Newtonian dynamics, which couples their positions, masses, and momenta, and the forces acting on them. The general law describing the motion of particle i of mass W/ under the influence of the potential energy V(x) derived from the remaining particles is 

Various algorithms have been proposed to update the elements of the configuration and velocity vectors when is sufficiently small, a commonly used pair is that of Verlet  

As discussed in Section 1.7, for incompressible fluids the continuity equation becomes 

These equations ate given in component form for several coordinate systems in Ihble 1.7.1. 

At solid boundaries, the no-slip, no-penetration conditions generally hold 

At liquid-liquid interfaces, the velocities and stresses of both fluids (a and b) tangent to the intmface must match 

The velocities normal to the liquid-liquid interfaces are again zero, and the normal stress balance must include any interfacial tension r and the surface curvature H 

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Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

The mathematical theory is rather complex because it involves subjecting the basic equations of motion to the special boundary conditions of a surface that may possess viscoelasticity. An element of fluid can generally be held to satisfy two kinds of conservation equations. First, by conservation of mass. 

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

Figure Al.6.1. Gaussian wavepacket in a hannonic oscillator. Note tliat the average position and momentum change according to the classical equations of motion (adapted from [6]).

In the presence of some fomi of relaxation the equations of motion must be supplemented by a temi involving a relaxation superoperator—superoperator because it maps one operator into another operator. The literature on the correct fomi of such a superoperator is large, contradictory and incomplete. In brief, the extant theories can be divided into two kinds, those without memory relaxation (Markovian) Tp and those with memory... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The volume of a Y -space-volume-element does not change in the course of time if each of its points traces out a trajectory in Y space determined by the equations of motion. Equivalently, the Jacobian... 

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

We consider the motion of a large particle in a fluid composed of lighter, smaller particles. We also suppose that the mean free path of the particles in the fluid, X, is much smaller than a characteristic size, R, of the large particle. The analysis of the motion of the large particle is based upon a method due to Langevin. Consider the equation of motion of the large particle. We write it in the fonn... 

In the general case, (A3.2.23) caimot hold because it leads to (A3.2.24) which requires GE = (GE ) which is m general not true. Indeed, the simple example of the Brownian motion of a hannonic oscillator suffices to make the point [7,14,18]. In this case the equations of motion are [3, 7]... 

Using the Heisenberg equation of motion, (AS,2,40). the connnutator in the last expression may be replaced by the time-derivative operator... 

In this section we discuss the frequency spectrum of excitations on a liquid surface. Wliile we used linearized equations of hydrodynamics in tire last section to obtain the density fluctuation spectrum in the bulk of a homogeneous fluid, here we use linear fluctuating hydrodynamics to derive an equation of motion for the instantaneous position of the interface. We tlien use this equation to analyse the fluctuations in such an inliomogeneous system, around equilibrium and around a NESS characterized by a small temperature gradient. More details can be found in [9, 10]. 

If the surface tension is a fiinction of position, then there is an additional temi, da/dx, to the right-hand side in the last equation. From the above description it can be shown drat the equation of motion for the Fourier component of the broken synnnetry variable is... 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

To generalize what we have just done to reactive and inelastic scattering, one needs to calculate numerically integrated trajectories for motions in many degrees of freedom. This is most convenient to develop in space-fixed Cartesian coordinates. In this case, the classical equations of motion (Hamilton s equations) are given... 

Quantum mechanically, the time dependence of the initially prepared state of A is given by its wavefimc /("f), which may be detennined from the equation of motion... 

Vibrational motion is thus an important primary step in a general reaction mechanism and detailed investigation of this motion is of utmost relevance for our understanding of the dynamics of chemical reactions. In classical mechanics, vibrational motion is described by the time evolution and l t) of general internal position and momentum coordinates. These time dependent fiinctions are solutions of the classical equations of motion, e.g. Newton s equations for given initial conditions and I Iq) = Pq. 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anliannonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when tire displacement of the electron becomes significant under strong drivmg fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, v, of the electron from equilibrium as... 

The equations of motion within such a field are given by ... 

The concept of relaxation time was introduced to the vocabulary of NMR in 1946 by Bloch in his famous equations of motion for nuclear magnetization vector M [1] ... 

The basic equation [8] is tlie equation of motion for the density matrix, p, given in equation (B2.4.25), in which H is the Hamiltonian. 

This Liouville-space equation of motion is exactly the time-domain Bloch equations approach used in equation (B2.4.13). The magnetizations are arrayed in a vector, and anything that happens to them is represented by a matrix. In frequency units (1i/2ti = 1), the fomial solution to equation (B2.4.26) is given by equation (B2.4.27) (compare equation (B2.4.14H. 

Relaxation or chemical exchange can be easily added in Liouville space, by including a Redfield matrix, R, for relaxation, or a kinetic matrix, K, to describe exchange. The equation of motion for a general spin system becomes equation (B2.4.28). 

Hamiltonian in the second-quantization fomi, only one //appears in this fmal so-called equation of motion (EOM) f//, <7/]+ = AJr 7 p(i e. in the second-quantized fomi, // and //are one and the same). 

Rowe D J 1968 Equation-of-motion method and the extended shell model Rev. Mod. Phys. 40 153-66 I applied these ideas to excitation energies in atoms and molecules in 1971 see equation (2.1)-(2.6) in ... 

Molecular dynamics consists of the brute-force solution of Newton s equations of motion. It is necessary to encode in the program the potential energy and force law of interaction between molecules the equations of motion are solved numerically, by finite difference techniques. The system evolution corresponds closely to what happens in real life and allows us to calculate dynamical properties, as well as thennodynamic and structural fiinctions. For a range of molecular models, packaged routines are available, either connnercially or tlirough the academic conmuinity. 

As an alternative to sampling the canonical distribution, it is possible to devise equations of motion for which the iiiechanicaT temperature is constrained to a constant value [84, 85, 86]. The equations of motion are... 

NVT, and in die course of the simulation the volume V of the simulation box is allowed to vary, according to the new equations of motion. A usefid variant allows the simulation box to change shape as well as size [89, 90], It is also possible to extend the Liouville operator-splitting approach to generate algoritlnns for MD in these ensembles examples of explicit, reversible, integrators are given by Martyna et al [91],... 

Ryckaert J-P, Ciccotti G and Berendsen H J C 1977 Numerical integration of the Cartesian equations of motion of a system with constraints molecular dynamics of n-alkanes J. Comput. Phys. 23 327-41... 

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