Momentum method

J. C. Simo and N. Tarnow. The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. ZAMP, 43 757-793, 1992. 

S. B. Smith, Y. Cui, and C. Bustamante, An optical-trap force transducer that operates by direct measurement of light momentum. Methods Enzymol. 361, 134-162 (2003). 

Various methods have been developed for dealing with the anomalous commutation relationships in molecular quantum mechanics, chief among them being Van Vleck s reversed angular momentum method [10]. Most of these methods are rather complicated and require the introduction of an array of new symbols. Brown and Howard [15], however, have pointed out that it is quite possible to handle these difficulties within the standard framework of spherical tensor algebra. If matrix elements are evaluated directly in laboratory-fixed coordinates and components are referred to axes mounted on the molecule only when necessary, it is possible to avoid the anomalous commutation relationships completely. Only the standard equations given earlier in this chapter are used to derive the required results it is just necessary to keep a cool head in the process ... 

The retention time is determined from the peak maximum in the case of symmetrical peak shapes. For well-packed columns symmetrical peaks should be achieved as long as the amount injected into the column is in the linear concentration range of the adsorption isotherm. If increased amounts of substances in the nonlinear concentration range of the adsorption isotherm are injected the peak is often heavily distorted and asymmetric. In that case the retention time has to be calculated from the centroid following the momentum method (Eq. 2.27). 

All three equations yield correct results only if the peak has a Gaussian shape. This is hardly ever the case with real-life chromatograms. Correct values for asymmetric peaks are obtained by the momentum method. Approximately correct values are obtained by the equation ... 

In section 4 we indicate how the probit function (lognormal distribution), the normal and WeibuU distribution function parameters are obtained from experimental data. We use the graphical method, point estimator and momentum method within a resambling approach. In section 5 we summarize and conclude. 

We compared several methods (point estimator determined by the hkelihood principle or the momentum method and the graphical method) that determine the parameters of the lognormal, the normal and the Weibull distribution. We foimd that in aU cases resampling must be applied because of the few data that is available. 

Figure 3.16. Measurement of /cli a using the momentum method (Dang et al., 1977 Nikolaev et al., 1976) assuming gas and electrode effects known from the response curve of the electrode, respectively the aeration system, and assuming a step function signal.

Eq. (8) is the starting-point for further manipulations using angular momentum methods. For our present purposes, however, we now transform from the standard spherical tensor components to closely-related real quantities, since this gives an expression which is easier to comprehend and more suited to computation. Following Griffith [4], we define the quantities and for /c > 0, by... 

Momentum Methods. Density can be measured by several momentum methods. The density of the fluid is obtained by measuring fluid flow rate and fluid flow momentum. For example, the rotating paddle method is based on the principle that an aerodynamic foil, if rotated through a fluid, will experience a measurable drag, which can be related to density. The rotating paddle serves both as a density measurement device and as a stirrer for the mixture. The equipment required is relatively simple. Disadvantages include low frequency response and sensitivity, and the presence of moving parts in the fluid. 

Density of a fluid can also be measured by the angular momentum method, which measures the angular momentum of a rotating fluid and relates this momentum to density. As in the rotating paddle method, the angular momentum method measures the density and stirs the mixture. The equipment required could be relatively simple. Disadvantages of this method include the presence of bulky, moving parts in the fluid and the lack of any... 

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