Final answer:
The components of angular velocity along space axes in terms of Euler angles represent the relationship between the rates of change of these angles and their trigonometric functions.
Step-by-step explanation:
To show that the components of the angular velocity along the space set of axes are given in terms of the Euler angles, we utilize the definition that angular velocity is the change in angle per unit time. Specifically, for a rotating body, the Euler angles—typically denoted as φ (phi), θ (theta), and ψ (psi)—describe the orientation of the body in three-dimensional space. They are used to express the angular position of a frame of reference.
According to the question, the components of the angular velocity along the x, y, and z axes can be expressed as:
- ωx = ˙θ cosφ + ˙ψ sinθ sinφ
- ωy = ˙θ sinφ - ˙ψ sinθ cosφ
- ωz = ˙ψ cosφ
This illustrates that ωx, ωy, and ωz are directly related to the rates of change of the Euler angles and their sine and cosine components. These expressions allow for the determination of the angular velocity vector's components in a fixed coordinate system given the body's rotation rates around its own moving axes.