Eulers equations of motion

Consider, as the liquid particle whose motion is to be described, the liquid contained in the element of Fig. 5.2, that is, the liquid contained in an element with edges 6x, by, dz and with one corner located at the point (x, y, z). The liquid particle must obey Newton s laws of motion, so setting up the equations of motion involves determining the net force acting on the liquid particle and equating that to the product of its mass and acceleration. 

The forces acting on the element of Fig. 5.2 are the body forces, with components, say, X, Y, Z per unit mass in the x,y,z directions, respectively, the pressure forces and the viscous forces. For the present it will be assumed that the liquid has zero viscosity (known as an inviscid liquid) and then the pressure forces are perpendicular to the boundary surface. The effect of viscosity will be considered briefly in section 5.12. 

Consider first the motion in the x direction. The net force acting on the element of liquid in the x direction is  

To ensure that all quantities have values appropriate to the point (x, y, z), the limit is taken as 5x, 8y and 6z each tend to zero. Then  

Equations [5.13], [5.14] and [5.15] are known as Euler s equations of motion for a non-viscous liquid. They may be combined into the single vector equation  

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Brownian Motion of a Rigid Body. Many molecules of interest are extended in three dimensions with three principal moments of inertia of comparable size. Gyroscopic forces complicate the discussion of Brownian motion for such bodies and a number of workers have developed convenient formal treatments without much physical novelty emerging. Steele has given a clear treatment of this problem, but obtains tractable expressions only for spherical-top molecules, which have the same moment of inertia I about all axes through the molecular mass centre, so that the Euler equations of motion fall apart in independent variables. 

Another numerical method to supplement the MC method should be the numerical integration of the equations of motion. This kind of calculation for simple molecular systems is called molecular dynamics (MD) mediod where Newton or Newton-Euler equation of motion is solved numerically and some dynamic properties of flie molecule involved can be obtained. 

The Euler equations of motion relate the time derivative of the molecular angular velocity coj, expressed in the body-fixed (principal) axis system, to the torques given above ... 

A constant temperature MD calculation has been carried out for a 3.2 mol% aqueous solution of TBA containing 7 TBA and 209 water molecules at 298 K and 0.9792 g/cc. The computer program used for keeping the temperature of the solution constant is based on the one developed by one of us (Tanaka et al., 1983) [6] as a realization of the proposal of Andersen (1980) [7]. The time step in integrating the Newton-Euler equation of motion for all the molecules is 0.0004 picoseconds and the calculation is extended up to 84,000 time steps (approximately 26 picoseconds). 

Claim 4.1.1 A field t described by the Euler equations of motion of a rigid body fixed in the centre of mass is tangent to the orbits 0 of the adjoint representation in the Lie algebra so(3) and is Hamiltonian on these orbits (which are homeomorphic to the spheres S ). 

Examine a particular case of Theorem 4.2.3 from 2 (Ch. 4). Examine the Euler equations of motion of a multidimensional rigid body realized as Hamiltonian systems on the Lie algebras so(3) and so(4) of small dimension. We have, in fact, considered the case of the algebra so(3) when we demonstrated that the equations integrated by us coincide with the classical Euler equations of motion of a three-dimensional rigid body. The case of the Lie albegra so(4) deserves a more detailed... 

The virtual power form of the Newton-Euler equations of motion for a set of Nh interconnected rigid bodies is... 

In the submodel on the mesoscale simulation, the overspray part of suspension and its influence on the coating layer growth dynamics have been investigated. The droplets have been considered as discrete elements and the Newton—Euler equations of motions have been solved for all droplets. The dynamics of soHd pellets and fluid profile were taken from the previously obtained DPM results. To describe the particle growth the one-dimensional PBM has been used. From the analysis of the obtained results, the following main conclusions can be drawn ... 

By the Lagrange-Euler equation of motion, the required torques 4(k) of the joint actuators of robot i can be calculated [4], The required torque of joint j of robot i must satisfy... 

It has been observed by [27, 24] that the equations of motion of a free rigid body are subject to reduction. (For a detailed discussion of this interesting topic, see [23].) This leads to an unconstrained Lie-Poisson system which is directly solvable by splitting, i.e. the Euler equations in the angular momenta ... 

The Euler Lagrangian approach is very common in the field of dilute dispersed two-phase flow. Already in the mid 1980s, a particle tracking routine was available in the commercial CFD-code FLUENT. In the Euler-Lagrangian approach, the dispersed phase is conceived as a collection of individual particles (solid particles, droplets, bubbles) for which the equations of motion can be solved individually. The particles are conceived as point particles which move... 

For a family of trajectories all starting at the value X(to) and at t=t all arriving at X(t), there is one trajectory that renders the action stationary. The classical mechanical trajectory of a given dynamical system is the one for which 5S=0, i.e. the action becomes stationary. The equation of motion is obtained from this variational principle [59], The corresponding Euler-Lagrange equations are obtained d(3L/3vk)/dt = 9L/dXk. In Cartesian coordinates these equations become Newton s equations of motion for each nucleus of mass Mk ... 

Euler-type equation of motion mpAv/At = — pgradiV + VqU) (4.55)... 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

In the framework of the Euler-Lagrange formalism, we write the equation of motion for the displacements of the atoms as ... 

There has been very much effort devoted to the solution of the diffusion equation of motion for a reactant particle executing Brownian motion. The Euler equation of diffusion... 

Morse and Feshbach [499] have discussed the variation approach to a description of equations of motion for diffusion. Their approach is straightforward and is generalised here to consider the cases where there is an energy of interaction, U, between the pair of particles, separated by a distance r at time t. It is relatively easy to extend this to a many-body situation. The usual Euler form of the equation of motion is the Debye— Smoluchowski equation, which has been discussed in much detail before, viz. 

The equations of motion of the held at the Higgs minimum (the minimum potential energy of the vacuum) are the Euler-Lagrange equations... 

Generalized momenta are defined by pk = and generalized forces are defined by Qk = f - When applied using t as the independent variable, Euler s variational equation for the action integral / takes the form of Lagrange s equations of motion... 

Following the logic of Euler, after integrating by parts to replace the term in Sq by one in Sq, this implies the Lagrangian equations of motion. 

The Euler-Lagrange equation leads to the Car-Parinello equations of motion of the form ... 

Now that we have introduced coordinates and velocities, the next question is how to predict the time evolution of a mechanical system. This is accomplished by solving a set of ordinary differential equations, the equations of motion, which can be derived from the principle of least action. It was discovered by Maupertuis and was further developed by Euler, Lagrange and Hamilton (d Abro (1951)). 

Using this particular form of L in the Euler-Lagrange equations of motion (3.1.6) we get... 

Toward the end of the nineteenth century the science of fluid mechanics began to develop in two branches, namely theoretical hydrodynamics and hydraulics. The first branch evolved from Euler s equations of motion for a frictionless, non-... 

The best way of expressing the equation of motion of the angular velocity u is to refer it to the principal axis system of the real body. This makes the nonlinear term (the Euler term) appear in the second equation of... 

It is regarded as a function of the linear and angular accelerations, (a, p ) are treated like constant parameters. The linear acceleration is denoted by a,-, and here it is assumed to be the rate of change of the peculiar momentum, a, = p,- /m. According to Gauss principle the equations of motion are obtained when C is minimal. It is immediately obvious that when the external field is equal to zero, C is minimal when each term in the sum is equal to zero so that Newton s and Euler s equations are recovered. 

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