Euler's rotation theorem

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	TAM674.htm
		From Euler's theorem on Rotation, "Any rotation in space can be represented as a rotation about a fixed axis at a given angle", I derived the rotation matrix R in terms of the Euler parameters lambda_0=cos(u/2) and lambda=sin(u/2)h, where h is the axis of rotation* with unit length and u the angle of rotation.
		As an example of the application of the Lagrange equations I derived the equations of motion for a simple 2D model of an overhead container crane in terms of independent generalized coordinates.
		Start with the constraint equations of motion and add the constraints on the level of the accelerations this gives you the full sparse DAE from which you can solve the accelerations of the cm of the bodies together with the constraint forces.
	www.tam.cornell.edu /~als93/TAM674.htm (6714 words)

	Rotation - Wikipedia, the free encyclopedia
		This is a consequence of Euler's rotation theorem.
		One consequence of the rotation of a planet is the phenomenon of precession.
		Rotation of a planar body is the movement when points of the body travel in circular trajectories around a fixed point called the center of rotation.
	en.wikipedia.org /wiki/Rotation (936 words)

	PlanetPhysics: axis angle of rotation
		The axis angle of rotation is a result of Euler's theorem of rotation and it describes a rotation of a three dimensional rigid body by a unit vector
		This is version 1 of axis angle of rotation, born on 2005-08-28.
		"axis angle of rotation" is owned by bloftin.
	planetphysics.org /encyclopedia/AxisAngleOfRotation.html (56 words)

	THE SYLLABI
		Curves: curvature, torsion, the Serret-Frenet apparatus, the fundamental theorem of curves, rotation index, the four-vertex theorem, Fenchel's theorem, total curvature, the Fray-Milnor theorem.
		Ordinary differential equations: the concepts of existence and uniqueness of solution dependence of solution oninitial conditions and parameters, Sturm seperation and comparison theorems and 2nd order linear requations, systems of equations, calculus of variation, Euler equation, non-linear system of equations, stability theory, Liapounov's method.
		Sets, mappings, divisibility of integers, the fundamental theorem of arithmetic, groups, subgroups, normal subgroups, homomorphisms, rings, ideals, quotient rings, integral domains, fields, fields of quotients, integral domains, Euclidean domains, divisibility in integral domains ring homomorphisms.
	www.kanoonline.com /buk/programs/mathd.htm (56 words)

	Lunar Republic : Craters
		In mechanics, there are Euler angles (to specify the orientation of a rigid body); Euler's theorem (that every rotation has an axis); Euler's equations for motion of fluids; and the Euler-Lagrange equation (that comes from calculus of variations).
		The "Euler's formula" with which most calculus students are familiar defines the exponentials of imaginary numbers in terms of trigonometric functions; however, there is another "Euler's formula" that gives the values of the Riemann zeta function at positive even integers in terms of Bernoulli numbers.
		Georg Christoph ~ (1638-1705), German artist, cartographer and astronomer; created a lunar atlas, the Genuina Corporis Lunaris Facies, with many features misplaced, the outlines of most of the maria in error, and many prominent craters not appearing at all.
	www.lunarrepublic.com /gazetteer/crater_e.shtml (56 words)

	The Crater Company - Moon Crater Catalog - Index Of Named Lunar Craters
		In mechanics, there are Euler angles (to specify the orientation of a rigid body); Euler's theorem (that every rotation has an axis); Euler's equations for motion of fluids; and the Euler-Lagrange equation (that comes from calculus of variations).
		The "Euler's formula" with which most calculus students are familiar defines the exponentials of imaginary numbers in terms of trigonometric functions; however, there is another "Euler's formula" that gives the values of the Riemann zeta function at positive even integers in terms of Bernoulli numbers.
		You don't have to be a star — or even famous — to name a crater.
	www.cratercompany.com /catalog/crater_e.shtml (56 words)

	Guru Gobind Singh Indraprastha University
		Graph Theory : Terminology,isomorphic graphs, Euler’s formula (proof) four color problem (without proof) and the chromatic number of a graph, five color theorem.
		Sets and Combinations : Sets,Subtracts, powersets, binary and unary operations on a set, set operations/set identities, fundamental country principles, principle of inclusion, exclusion and pigeonhole principle, permutation and combination, pascal’s triangles, binominal theorem, representation of discrete structures.
		Relation/function and matrices: Rotation, properties of binary rotation, operation on binary rotation, closures, partial ordering, equivalence relation, Function properties of function, composition of function, inverse, binary and n-ary operations,characteristics for, Permutation function, composition of cycles, Boolean matrices, Boolean matrices multiplication.
	ggsipu.nic.in /students/mca04/mcale04/assignment/shilpakaul/MTechSecondSem.htm (56 words)

	TI-85 BASIC Math Programs - ticalc.org
		Included programs: ram graphs, maclaurin series, volume and rotation of a solid, ram/trapezoid/simpsion, test for convergence, projectile motion, table of domain/range/derivatives/integrals of 12 trig functions and 4 other log/expo functions, exact value/infinite domain of trig functions, Euler's method, Improved Euler's method, Runge-Kutta's method, Newton's method, slope field, and partial sums of a recursive series.
		Displays the formulas for arithmetic operations, exponents/ radicals, factoring special polynomials, binomial theorem, quadratic formula, and inequalities/absolute value separated into categories.
		Finds slope, y-intersept, coin porblems, mixtures, and has a real cool multiplication table that goes up to 99,999.
	www.ticalc.org /pub/85/basic/math (56 words)

	m133
		Background material from solid geometry and vector algebra, planes, incidence geometry of the sphere, the spherical triangle inequality, isometries of the sphere, Euler's formula, spherical triangles, congruence theorems and trigonometry, finite rotation groups and isometry groups of the sphere.
		Cyclic and dihedral groups, conjugate subgroups, Leonardo's theorem, regular polygons and their symmetries and similarities.
		(There are also six appendices on the axiomatic approach, sets and functions, groups, linear algebra, a proof of Theorem 2.2, and the trigonometric and hyperbolic functions.)
	math.ucr.edu /home/UndergradInfo/pages/m133 (263 words)

	ODSAA.html
		The study makes use of infinite dimensional Hamiltonian formulation of the vorticity equation when the rotation of a planet is taken into account [see T. Shepherd, "Hamiltonian Dynamics", Encyclopedia of Atmospheric Sciences, Academic Press, 929-938, 2003]
		Classes of steady and periodic solutions are investigated for the incompressible Euler equation on a shpere.
		Specifically, I will focus on the recent beautiful proof by Simons and Zalinescu of Minty's theorem, which utilizes the Fitzpatrick function and Fenchel duality (cont'd).
	www.uoguelph.ca /~mcojocar/ODSAA.html (502 words)

	The Founders Of Classical Mechanics :: Leonhard Euler
		Just in mechanics, one has Euler angles (to specify the orientation of a rigid body), Euler's theorem (that every rotation has an axis), Euler's equations for motion of fluids, and the Euler-Lagrange equation (that comes from calculus of variations).
		Leonhard Euler (1707-1783) was arguably the greatest mathematician of the eighteenth century (His closest competitor for that title is Lagrange) and one of the most prolific of all time; his publication list of 886 papers and books may be exceeded only by Paul Erdös.
		The Founders Of Classical Mechanics :: Leonhard Euler
	about-physicists.org /euler.html (457 words)

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