Euler

This was first derived by Isaac Newton in 1666. Remarkably, the binomial formula is also valid for negative, fractional, and even complex values of n, which was proved by Niels Henrik Abel in 1826. (It is joked that Newton did not prove the binomial theorem for noninteger n because he was not Abel.) Here are a few interesting binomial expansions that you can work out for yourself  

Imagine there is a bank in your town that offers you 100% annual interest on your deposit (we will let pass the possibility that the bank might be engaged in questionable loan-sharking activities). This means that if you deposit 1 on January 1, you will get back 2 1 year later, Another bank across town wants to get in on the action and offers 100% annual interest compounded semiannually. This means that you get 50% interest credited after half a year, so that your account is worth 1.50 on July 1. But this total amount then grows by another 50% in the second half of the year. This gets you, after 1 year, 

A third bank picks up on the idea and offers to compound your money quarterly. Your 1 there would grow after a year to 

Competition continues to drive banks to offer better and better compounding options, until the Eulergenossenschaftsbank apparently blows away all the competition by offering to compound your interest continuously—every second of every day Let us calculate what your dollar would be worth there after 1 year. Generalization from Eq. (3.97) suggests that compounding n times a year produces (1 + 1/n) . Here are some numerical values for increasing n  

This number was designated e by the great Swiss mathematician Leonhard Euler(possibly after himself). Euler (pronounced approximately like oiler ) also first introduced the symbols i, ir, and /(x). After n itself, e is probably the most famous transcendental number, also with a never-ending decimal expansion. The tantalizing repetition of 1828 is just coincidental. 

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Figure Al.4.4. The definition of the Euler angles (0, ( ), x) that relate the orientation of the molecule fixed (x, y, z) axes to the (X, Y, Z) axes. The origin of both axis systems is at the nuclear centre of mass O, and the node line ON is directed so that a right handed screw is driven along ON in its positive direction by twisting it from Z to z through 9 where 0 < 9 < n. ( ) and x have the ranges 0 to In. x is measured from the node line.

We consider rotations of the molecule about space-fixed axes in the active picture. Such a rotation causes the (x, y, z) axis system to rotate so that the Euler angles change... 

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

Mathematically equation (A2.1.25) is the direct result of the statement that U is homogeneous and of first degree in the extensive properties S, V and n.. It follows, from a theorem of Euler, that... 

The time dependence of the displacement coordinate for a mode undergoing hannonic oscillation is given by V = V j cos2tiv /, where is the amplitude of vibration and is the vibrational frequency. Substitution into equation (Bl.2.9) witii use of Euler s half-angle fomuila yields... 

Here/(9,(p, i ) is the probability distribution of finding a molecule oriented at (0,cp, li) within an element dQ of solid angle with the molecular orientation defined in tenus of the usual Euler angles (figure B 1.5.10). 

FigureBl.5.10 Euler angles and reference frames for the discussion of molecular orientation laboratory frame (v, y, z) and molecular frame (x y, z).

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

S is the path length between the points a and b. The Euler equation to this variation problem yields the condition for the reaction path, equation (B3.5.14). A similar method has been proposed by Stacho and Ban [92]. 

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

It is now convenient to introduce hyperspherical coordinates (p, 0, and <])), which specify the size and shape of the ABC molecular triangle and the Euler... 

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

Figure 1. The space-fixed (ATZ) and body-fixed (xyz) frames. Any rotation of the coordinate system XYZ) to (xyz) may be performed by three successive rotations, denoted by the Euler angles (a, 3, y), about the coordinate axes as follows a) rotation about the Z axis through an angle a(0 < a < 2n), (b) rotation about the new yi axis through an angle P(0 < P < 7i), (c) rotation about the new zi axis through an angle y(0 Y < 2n). The relative orientations of the initial and final coordinate axes are shown in panel (d).

Next, Euler s angles are employed for deriving the outcome of a general rotation of a system of coordinates [86]. It can be shown that R(k, 0) is accordingly presented as... 

In Section V.B, we discussed to some extent the 3x3 adiabatic-to-diabatic transformation matrix A(= for a tri-state system. This matrix was expressed in terms of three (Euler-type) angles Y,y,r = 1,2,3 [see Eq. (81)], which fulfill a set of three coupled, first-order, differential equations [see Eq. (82)]. 

The implicit-Euler (IE) scheme, for example, discretizes system (1) as ... 

C. S. Peskin and T. Schlick. Molecular dynamics by the backwsird Euler s method. Comm. Pure App. Math., 42 1001-1031, 1989. 

T. Schlick, S. Figueroa, and M. Mezei. A molecular dynamics simulation of a water droplet by the implicit-Euler/Langevin scheme. J. Chem. P%s., 94 2118-2129, 1991. 

G. Zhang and T. Schlick. The Langevin/implicit-Euler/Normal-Mode scheme (LIN) for molecular dynamics at large time steps. J. Chem. Phys., 101 4995-5012, 1994. 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

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