Lagrangian plane

Definition 1.2.2 We say that a plane in a symplectic space R is isotropic if it is skew- orthogonal to itself, that is to say, a skew-scalar product of any two vectors of the plane is equal to zero. If A is equal to n (that is, to half the dimension of R then an isotropic plane will be called a Lagrangian plane,... 

Lagrangian planes are isotropic planes of maximal possible dimension. Since, for the sake of convenience, we model a symplectic structure on R, this permits setting a skew-symmetric product in a distinct form. If... 

Lemma 1.2.4. A symplectic transformation maps any isotropic plane again into an isotropic plane. In particular, an image of a Lagrangian plane is a Lagrangian plane. 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

Figure 7.11 Qualitative predicted particle trajectories in the transformed frame (Lagrangian) a) down-channel flows induced by the drag motion of the moving barrel, and b) recirculation flow in thex-y plane...

U(l), whose group space is a circle. This result is another internal inconsistency, because the group space of a gauge theory is a circle, there can be no physical quantity in free space perpendicular to that plane. It is necessary but not sufficient, in this view, that the Lagrangian in U(l) field theory be invariant [6] under U(l) gauge transformation. 

The fields E, and % are orthogonal components in the complex phase plane for the oscillations due to the small displacement of the scalar field, which is thereby characterized completely. The scalar field Lagrangian becomes... 

These equations are now in convenient form for a finite-difference scheme along lines similar to that used above. Ail alternative approach developed in some detail employs the Lagrangian interpolation formula to follow the motion of the boundary in the (x/a, r) plane. This is a means of developing finite-difference approximations to derivatives based on functional values and not necessarily equally spaced in the argument. Crank points out that the application of Lagrangian interpolation formulas involves a relatively large number of steps in time, whereas the fixedboundary procedures require iterative solutions at each time interval, which are, however, far fewer in number. 

Since the fictitious particle moves in a central force field described by a spherically symmetric potential function U(r), its angular momentum is conserved. Therefore, the motion of the fictitious particle will be in a plane defined by the velocity and the radius vectors. The Lagrangian may then be conveniently expressed in polar coordinates as... 

The simplest examples of such systems in quantum physics are the interaction of the charged quantum particle with the Lagrangian L = mr2/2 + ejcAr and the solenoidal magnetic field A — e x r/r 2 (the Aharonov-Bohm effect) or the interaction of two anions in 2 + 1-field theory [14]. In both cases, the configurational space is the plane with one point removed. 

Now we see that the slope of the jump condition on the P-u plane is also a function of shock velocity. The interesting part to note is that U in this equation is the shock velocity in laboratory or Eulerian coordinates. The quantity (U -Uq) is the shock velocity in Lagrangian coordinates, or relative to the material. So we see that for the jump condition on the P-u plane... 

Since the slope of the line joining two states on the P-u Hugoniot is poU (Lagrangian, relative to the particles, or material), equating this to dPIdu at the pressure of interest will give us the rarefaction-wave velocity relative to the material into which it is moving. The equation on the Hugoniot of the P-u plane is... 

Remember that the Uq in Eqs. (19.2) and (19.3) is not the u of the material into which this rarefaction is moving but merely a constant equal to the value of the M-axis intercept for the Hugoniot. The u calculated for the front of the rarefaction is Lagrangian it is the velocity of the rarefaction relative to the material into which the rarefaction is moving. Therefore, when plotting this velocity on the x-t plane, it must be corrected for the additional particle velocity of the material ahead, into which it is moving. The x-t plane, again, shows slopes of Eulerian velocity reciprocals. 

Car and Parrinello in their celebrated 1985 paper [2] proposed an alternative route for molecular simulations of electrons and nuclei altogether, in the framework of density functional theory. Their idea was to reintroduce the expansion coefficients Cj(G) of the Kohn-Sham orbitals in the plane wave basis set, with respect to which the Kohn-Sham energy functional should be minimized, as degrees of freedom of the system. They then proposed an extended Car-Parrinello Lagrangian for the system, which has dependance on the fictitious degrees of freedom Cj(G) and their time derivative Cj (G) ... 

For the Lagrangian element, the source of energy is the work done on the boundaries of the system as they move—with power a k -ek on the pair of faces normal to z and —a k ek on the pair normal to x (where energy is expended rather than absorbed) but what is the energy source for the thermodynamic system with stationary boundaries The boundaries being stationary, no work is done on this system the system is maintained in a state that is steady through time by fluxes of mass, inward across boundary planes normal to z and outward across bounding planes normal to x. Let... 

In terms of the new coordinates, the Lagrangian function L and the equations of motion have the same form as previously, because the first choice of axis direction was quite arbitrary. However, since the coordinates have been chosen so that the plane of the motion is the x y plane, the angle t is always equal to a constant, w/2. Inserting this value of in Equation 1-33 and writing it in terms of x instead of tp, we obtain... 

In terms of variables r and x in the plane of motion, the Lagrangian equations of motion are... 

The advection problem is thus described by a periodically driven non-autonomous Hamiltonian dynamical system. In such case, besides the two spatial dimensions an additional variable is needed to complete the phase space description, which is conveniently taken to be the cyclic temporal coordinate, r = t mod T, representing the phase of the periodic time-dependence of the flow. In time-dependent flows ip is not conserved along the trajectories, hence trajectories are no longer restricted to the streamlines. The structure of the trajectories in the phase space can be visualized on a Poincare section that contains the intersection points of the trajectories with a plane corresponding to a specified fixed phase of the flow, tq. On this stroboscopic section the advection dynamics can be defined by the stroboscopic Lagrangian map... 

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