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Three angular parameters that specify the orientation of a body with respect to reference axes. They are used for describing rotating systems such as gyroscopes, tops, molecules, and nonspherical nuclei. They are not symmetrical in the three angles but are simpler to use than other rotational parameters.
Unfortunately, different definitions of Euler's angles are used, and therefore it is confusing to compare equations in different references. The definition given here is the majority convention according to H. Margenau and G. Murphy.
Let OXYZ be a right-handed cartesian (right-angled) set of fixed coordinate axes and Oxyz a set attached to the rotating body (see illustration).
The orientation of Oxyz can be produced by three successive rotations about the fixed axes starting with Oxyz parallel to OXYZ. Rotate through (1) the angle ψ counterclockwise about OZ, (2) the angle Θ counterclockwise about OX, and (3) the angle &phgr; counterclockwise about OZ again. The line of intersection OK of the xy and XY planes is called the line of nodes.
Denote a rotation about OZ, for example, by Z (angle). Then the complete rotation is, symbolically, given by the equation below where the rightmost operation is done
(or Eulerian angles), the angles φ, θ, and ψ that define the position of a rectangular Cartesian coordinate system OXYZ with respect to another rectangular Cartesian coordinate system Oxyz of the same orientation (see Figure 1).
Let OK be an axis, called the line of nodes, that coincides with the line of intersection of the coordinate plane Oxy of the first system with the coordinate plane OXY of the second system and is directed so that the axes Oz, OZ, and OK form a trihedral of the same orientation. The Euler angle φ, or angle of spin, is the angle between the axes Ox and OK, which is measured in the plane Oxy from Ox in the direction of the shortest rotation from Ox to Oy. The Euler angle θ, or angle of nutation, is the angle between Oz and OZ; it does not exceed π. The Euler angle ψ, or angle of precession, is the angle between the axes OK and OX, which is measured in the plane OXY from Ok in the direction of the shortest rotation from OX to OY. When θ = 0 or π, the Euler angles are not defined.
The Euler angles were introduced by L. Euler in 1748 and are used extensively in the dynamics of solids—for example, in the theory of the gyroscope—and in celestial mechanics.