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Angular velocity in Euler angles: $ω_{3} = ψ^{'} + φ^{'} \cos θ$ and $ω_{1}^{2} + ω_{2}^{2} = θ^{' 2} + φ^{' 2} \sin^{2} θ$

Statement

Let $J \subseteq R$ be an interval with at least two points, let $φ, θ, ψ : J \to R$ be differentiable with $θ (J) \subseteq (0, π)$ , and put $R (t) : = Θ (φ (t), θ (t), ψ (t)) = R_{z} (φ (t)) R_{y} (θ (t)) R_{z} (ψ (t)),$ the Euler-angle parametrization of The Euler-angle parametrization and the heavy symmetric top system. Then $R$ is a differentiable $S O (3)$ -valued curve, and its body angular velocity $ω : J \to R^{3}$ in the sense of Rigid placements, body and spatial angular velocity, and material velocities (the unique vector with $R (t)^{T} R^{'} (t) = [ω (t)]_{\times}$ ) is, in the standard basis, $ω = (- φ^{'} \sin θ \cos ψ + θ^{'} \sin ψ) e_{1} + (φ^{'} \sin θ \sin ψ + θ^{'} \cos ψ) e_{2} + (ψ^{'} + φ^{'} \cos θ) e_{3},$ all functions evaluated at $t$ . Consequently $ω_{3} = ψ^{'} + φ^{'} \cos θ, ω_{1}^{2} + ω_{2}^{2} = θ^{' 2} + φ^{' 2} \sin^{2} θ, ∥ ω ∥_{2}^{2} = θ^{' 2} + φ^{' 2} \sin^{2} θ + (ψ^{'} + φ^{'} \cos θ)^{2} .$ Moreover, if $A, C > 0$ and $I : R^{3} \to R^{3}$ is the linear map with $I e_{1} = A e_{1}$ , $I e_{2} = A e_{2}$ , $I e_{3} = C e_{3}$ , then $ω^{T} I ω = A (θ^{' 2} + φ^{' 2} \sin^{2} θ) + C (ψ^{'} + φ^{'} \cos θ)^{2}$ .

Facts & Assumptions

Given: Differentiable $φ, θ, ψ : J \to R$ with $θ \in (0, π)$ , the curve $R = R_{z} (φ) R_{y} (θ) R_{z} (ψ)$ , and the matrices $R_{z}, R_{y}$ of The coordinate rotation matrices $R_{z} (a)$ and $R_{y} (a)$ : values in $S O (3)$ , composition law, and derivatives.

[L1]

$R_{z} (a), R_{y} (a) \in S O (3)$ for all $a$ , $R_{z} (a) R_{z} (b) = R_{z} (a + b)$ , $R_{y} (a) R_{y} (b) = R_{y} (a + b)$ , and for differentiable $a$ one has $(R_{z} \circ a)^{'} = a^{'} R_{z} (a) [e_{3}]_{\times} = a^{'} [e_{3}]_{\times} R_{z} (a)$ and $(R_{y} \circ a)^{'} = a^{'} R_{y} (a) [e_{2}]_{\times} = a^{'} [e_{2}]_{\times} R_{y} (a)$ The coordinate rotation matrices $R_{z} (a)$ and $R_{y} (a)$ : values in $S O (3)$ , composition law, and derivatives.

[L2]

The product rule for matrix-valued curves, the skewness of $R^{T} R^{'}$ for an $S O (3)$ -valued curve, and the equivalence $R^{T} R^{'} = [ω]_{\times} \Leftrightarrow R^{'} = R [ω]_{\times}$ hold; moreover $[ω]_{\times} z = ω \times z$ and $R^{T} [u]_{\times} R = [R^{T} u]_{\times}$ Derivative rules for differentiable matrix-valued curves, and skewness of $R^{T} R^{'}$ for a curve in $S O (3)$ , The skew-matrix/cross-product correspondence $[ω]_{\times} z = ω \times z$ and its conjugation law $R [ω]_{\times} R^{T} = [R ω]_{\times}$ , Rigid placements, body and spatial angular velocity, and material velocities.

[L3]

For $R \in S O (3)$ , $R^{T} R = I_{3}$ ; the matrices $e_{1} e_{1}^{T} + e_{2} e_{2}^{T} + e_{3} e_{3}^{T} = I_{3}$ , $[e_{3}]_{\times} e_{3} = 0$ and $[e_{2}]_{\times} e_{2} = 0$ The skew-matrix/cross-product correspondence $[ω]_{\times} z = ω \times z$ and its conjugation law $R [ω]_{\times} R^{T} = [R ω]_{\times}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes.

[L4]

$(\sin \circ a)^{'} = \cos (a) a^{'}$ and $(\cos \circ a)^{'} = - \sin (a) a^{'}$ ; the algebra of derivatives holds The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Sine and cosine defined by their real power series. For the identity used below, [L1] gives $R_{z} (ψ) \in S O (3)$ , hence $R_{z} (ψ)^{T} R_{z} (ψ) = I_{3}$ by [L3]. Its first column is $(\cos ψ, \sin ψ, 0)$ , so the first diagonal entry of this product is $\cos^{2} ψ + \sin^{2} ψ = 1$ by the matrix-product formula The coordinate rotation matrices $R_{z} (a)$ and $R_{y} (a)$ : values in $S O (3)$ , composition law, and derivatives, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes.

[L5]

Evaluation of $R_{z} (a) e_{3} = e_{3}$ , $R_{z} (a) e_{1} = \cos a e_{1} + \sin a e_{2}$ , $R_{z} (a) e_{2} = - \sin a e_{1} + \cos a e_{2}$ and $R_{y} (a) e_{1} = \cos a e_{1} - \sin a e_{3}$ , $R_{y} (a) e_{3} = \sin a e_{1} + \cos a e_{3}$ is the column reading of the displayed matrices The coordinate rotation matrices $R_{z} (a)$ and $R_{y} (a)$ : values in $S O (3)$ , composition law, and derivatives.

Proof

technique · direct

1.1givenL1L2

By [L1] each of the three factors is a differentiable $S O (3)$ -valued curve; by the product rule [L2] and closure of $S O (3)$ under multiplication, $R$ is differentiable with values in $S O (3)$ . Hence $R^{T} R^{'}$ is skew and there is a unique $ω$ with $R^{T} R^{'} = [ω]_{\times}$ by [L2].

1.2givenL2L5algebra

By the conjugation identity $S^{T} [u]_{\times} S = [S^{T} u]_{\times}$ of [L2] with $S = R_{y} (θ)$ and $S = R_{z} (ψ)$ , and the column readings of [L5], $R_{y} (θ)^{T} [e_{3}]_{\times} R_{y} (θ) = [R_{y} (θ)^{T} e_{3}]_{\times} = [- \sin θ e_{1} + \cos θ e_{3}]_{\times}, R_{z} (ψ)^{T} [e_{2}]_{\times} R_{z} (ψ) = [\sin ψ e_{1} + \cos ψ e_{2}]_{\times} .$

2.1step 1.1givenL1L2algebra

Differentiating the product and applying [L1], $\dot{R} = φ^{'} R_{z} (φ) [e_{3}]_{\times} R_{y} (θ) R_{z} (ψ) + θ^{'} R_{z} (φ) R_{y} (θ) [e_{2}]_{\times} R_{z} (ψ) + ψ^{'} R_{z} (φ) R_{y} (θ) R_{z} (ψ) [e_{3}]_{\times} .$

3.1step 2.1givenL2L3algebra

Multiplying on the left by $R^{T} = R_{z} (ψ)^{T} R_{y} (θ)^{T} R_{z} (φ)^{T}$ and using $R_{z} (φ)^{T} R_{z} (φ) = I_{3}$ [L3] term by term, $R^{T} \dot{R} = φ^{'} R_{z} (ψ)^{T} R_{y} (θ)^{T} [e_{3}]_{\times} R_{y} (θ) R_{z} (ψ) + θ^{'} R_{z} (ψ)^{T} [e_{2}]_{\times} R_{z} (ψ) + ψ^{'} [e_{3}]_{\times} .$

4.1step 3.11.2givenL2L5algebra

Conjugating the first vector of step 1.2 by $R_{z} (ψ)^{T}$ and using [L5], $R_{z} (ψ)^{T} (- \sin θ e_{1} + \cos θ e_{3}) = - \sin θ \cos ψ e_{1} + \sin θ \sin ψ e_{2} + \cos θ e_{3}$ ; substituting into step 3.1 gives $R^{T} \dot{R} = [ω]_{\times}$ with $ω$ exactly as displayed in the statement.

5.1step 4.1givenL4algebra

Reading off the components gives $ω_{3} = ψ^{'} + φ^{'} \cos θ$ ; expanding the squares of the first two components and using $\sin^{2} ψ + \cos^{2} ψ = 1$ from [L4] cancels the mixed terms and gives $ω_{1}^{2} + ω_{2}^{2} = φ^{' 2} \sin^{2} θ + θ^{' 2}$ , whence $∥ ω ∥_{2}^{2} = θ^{' 2} + φ^{' 2} \sin^{2} θ + (ψ^{'} + φ^{'} \cos θ)^{2}$ .

6.1step 4.15.1givenL3algebra∎

For the diagonal map $I$ with $I e_{1} = A e_{1}$ , $I e_{2} = A e_{2}$ , $I e_{3} = C e_{3}$ , the quadratic form is $ω^{T} I ω = A (ω_{1}^{2} + ω_{2}^{2}) + C ω_{3}^{2}$ , and substituting step 5.1 gives the displayed identity.

Remarks

Relation to the kinetic energy. Combined with clause 2 of Kinetic energy and angular momentum of a rigid placement: $K = \frac{1}{2} m ∥ c^{'} ∥_{2}^{2} + \frac{1}{2} ω^{T} I ω$ and $L_{c} = R I ω$ , the last identity says that a rigid placement with principal moments $(A, A, C)$ has rotational kinetic energy $\frac{1}{2} A (θ^{' 2} + φ^{' 2} \sin^{2} θ) + \frac{1}{2} C (ψ^{'} + φ^{'} \cos θ)^{2}$ , which is the kinetic part of the heavy-top function of The Euler-angle parametrization and the heavy symmetric top system. No statement about a physical top is made by that comparison; it is the consuming physical items that identify the parameters.

Angular velocity in Euler angles: $ω_{3} = ψ^{'} + φ^{'} \cos θ$ and $ω_{1}^{2} + ω_{2}^{2} = θ^{' 2} + φ^{' 2} \sin^{2} θ$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Angular velocity in Euler angles: ω3=ψ′+φ′cos⁡θω3​=ψ′+φ′cosθ and ω12+ω22=θ′2+φ′2sin⁡2θω12​+ω22​=θ′2+φ′2sin2θ

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Angular velocity in Euler angles: $ω_{3} = ψ^{'} + φ^{'} \cos θ$ and $ω_{1}^{2} + ω_{2}^{2} = θ^{' 2} + φ^{' 2} \sin^{2} θ$