Angles of Euler

See also: Euler

The movement of a solid compared to a reference frame utilizes 6 coordinates, which are, for example, the three coordinates describing the position of its center of mass (or of an unspecified point of the solid) and three angles, named the angles of Euler , cf the diagram opposite. The angles of Euler can be also used to represent the Orientation of a solid compared to a reference mark; one can see them like a generalization of the spherical coordinated .

Mechanics of the solid

One is interested only here in the description of the movement of the solid in unspecified rotation around the point O, which can be a fixed point of the solid in the reference frame of reference Oxyz or it center of mass.

The angles of Euler are selected in order to allow a simple memorizing of the construction of the instantaneous vector rotation of the solid, necessary to the study of the Cinématique of the solid.

One passes from the reference frame fixes Oxyz at the reference frame related to the solid Ox' y' z' by three successive rotations.

• the Precession $\backslash \; psi$, around the axis OZ , makes pass from Oxyz to the reference frame Ouvz .

• the Nutation $\backslash \; theta$, around the axis Or , makes pass from Ouvz to Ouwz' .

• the clean Rotation $\backslash \; varphi$, around the axis Oz' , makes pass from Ouwz' to the reference frame related to the solid Ox' y' Z' .

NB. The axis Or is carried by the intersection of the plans Oxy and Ox' y'

Under these conditions the instantaneous vector rotation of the solid is given by the simple amount:

$\backslash \; vec\; \{\backslash \; Omega\}\; =\; \backslash \; dowry\; \{\backslash \; psi\}\; \backslash ,\; \backslash \; vec\; \{Z\}\; +\; \backslash \; dowry\; \{\backslash \; theta\}\; \backslash ,\; \backslash \; vec\; \{U\}\; +\; \backslash \; dowry\; \{\backslash \; varphi\}\; \backslash ,\; \backslash \; vec\; \{z\text{'}\}$,

where the vectors appearing in the member of right-hand side are the unit vectors of the corresponding axes. It will be noticed that the preceding simple expression uses a nonorthogonal base.

The use of these angles is very general in mechanics and astronomy, for example to describe the movement of the Gyroscope.

Crystalline orientation

In Science of the materials, the angles of Euler are used to describe the crystalline Orientation (orientation of a Cristallite compared to the axes of the sample), in particular in the field of the Texture (preferential orientation). The angles are then in general noted (φ₁, Φ, φ₂) with:

• φ₁ = ψ
• Φ = θ
• φ₂ = φ

One uses also sometimes another alternative in which the second rotation (nutation) is done according to the axis OV instead of Or ; the angles are then noted (Ψ, Θ, Φ) without that having a relationship with the notations of the mechanics, which is not without risk of confusion.

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