# 物理代写|分析力学代写Analytical Mechanics代考|Euler Angles

## 物理代写|分析力学代写Analytical Mechanics代考|Euler Angles

In a Lagrangian formulation of rigid body dynamics, the nine direction cosines $a_{i j}$ are not the most convenient coordinates for describing the instantaneous orientation of a rigid body because they are not mutually independent. The nine equations (3.8) impose only six conditions on the direction cosines, so they can be expressed in terms of three independent parameters. Indeed, there are three distinct conditions corresponding to the diagonal part of Eqs. (3.8), but the six conditions corresponding to the off-diagonal part $(i \neq j)$ are pairwise identical. For example, Eq. (3.8) for $j=1, k=2$ is the same as the one for $j=2, k=1$.
From the practical point of view, a convenient way to parameterise the rotation matrix is by means of the Euler angles. The transformation from the Cartesian system $\Sigma(x, y, z)$ to the $\Sigma^{\prime}\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ system is accomplished in three successive stages, each serving to define one of the Euler angles (Fig. 3.4).
(a) Rotation of axes $(x, y, z)$ about the $z$-axis by angle $\phi:(x, y, z) \stackrel{\mathcal{D}}{\longrightarrow}(\xi, \eta, \zeta)$.
The transformation equations are the same as equations (3.20) with $x_1^{\prime}=\xi$ and $x_2^{\prime}=\eta$ supplemented by equation $x_3^{\prime}=\zeta=z$. Therefore, the rotation matrix $\mathcal{D}$ is written
$$\mathcal{D}=\left(\begin{array}{ccc} \cos \phi & \sin \phi & 0 \ -\sin \phi & \cos \phi & 0 \ 0 & 0 & 1 \end{array}\right)$$
(b) Rotation of axes $(\xi, \eta, \zeta)$ about the $\xi$-axis by angle $\theta:(\xi, \eta, \zeta) \stackrel{\mathcal{C}}{\longrightarrow}\left(\xi^{\prime}, \eta^{\prime}, \zeta^{\prime}\right)$.

## 物理代写|分析力学代写Analytical Mechanics代考|Commutativity of Infinitesimal Rotations

Let $g$ be an arbitrary vector submitted to a counterclockwise infinitesimal rotation by an angle $d \Phi$ about an axis defined by the unit vector $\hat{\boldsymbol{n}}$ (active point of view). According to $(2.136)$ we have $$g^{\prime}=g+d \Omega \times g,$$
where
$$d \Omega=d \Phi \hat{\boldsymbol{n}}$$
The symbols $d \Phi$ and $d \Omega$ are to be understood as mere names for infinitesimal quantities, not as differentials of a scalar $\Phi$ or of a vector $\boldsymbol{\Omega}$. In fact, generally speaking, there is no vector $\boldsymbol{\Omega}$ whose differential is equal to $\boldsymbol{d} \boldsymbol{\Omega}$ (Problem 3.6).
For successive rotations, with associated vectors $d \boldsymbol{\Omega}1$ and $d \Omega_2$, we write \begin{aligned} g^{\prime} & =g+d \boldsymbol{\Omega}_1 \times g, \ g^{\prime \prime} & =g^{\prime}+d \Omega_2 \times g^{\prime}, \end{aligned} whence, neglecting second-order infinitesimals, $$g^{\prime \prime}=g+d \Omega{12} \times g$$
with
$$d \boldsymbol{\Omega}{12}=d \boldsymbol{\Omega}_1+d \boldsymbol{\Omega}_2$$ This last result shows that successive infinitesimal rotations commute $\left(\boldsymbol{d} \boldsymbol{\Omega}{12}=\boldsymbol{d} \boldsymbol{\Omega}_{21}\right)$ as a consequence of the commutativity of vector addition. Furthermore, the vector associated with successive infinitesimal rotations is the sum of the vectors associated with the individual infinitesimal rotations, a property which will be of great value for the forthcoming developments.

It is rewarding to describe infinitesimal rotations in matrix language. Adopting the active point of view, Eq. (3.53) can be written in the form
$$g_{\Sigma}^{\prime}=\mathcal{A} g_{\Sigma}=(I+\varepsilon) g_{\Sigma}$$
where $\varepsilon$ is an infinitesimal matrix. The commutativity of infinitesimal rotations is easily checked, since neglecting second-order infinitesimals and taking into account that matrix sum is commutative,
$$\mathcal{A}_1 \mathcal{A}_2=\left(\boldsymbol{I}+\boldsymbol{\varepsilon}_1\right)\left(\boldsymbol{I}+\boldsymbol{\varepsilon}_2\right)=\boldsymbol{I}+\boldsymbol{\varepsilon}_1+\boldsymbol{\varepsilon}_2=\boldsymbol{I}+\boldsymbol{\varepsilon}_2+\boldsymbol{\varepsilon}_1=\mathcal{A}_2 \mathcal{A}_1$$

# 分析力学代考

## 物理代写|分析力学代写Analytical Mechanics代考|Euler Angles

(3.8) 对于 $j=1, k=2$ 和那个一样 $j=2, k=1$.

(a) 轴的旋转 $(x, y, z)$ 有关 $z$-按角度轴 $\phi:(x, y, z) \stackrel{\mathcal{D}}{\longrightarrow}(\xi, \eta, \zeta)$.

$$\mathcal{D}=\left(\begin{array}{llllllll} \cos \phi & \sin \phi & 0-\sin \phi & \cos \phi & 0 & 0 & 0 & 1 \end{array}\right)$$
(b) 轴的旋转 $(\xi, \eta, \zeta)$ 有关 $\xi$-按角度轴 $\theta:(\xi, \eta, \zeta) \stackrel{\mathcal{C}}{\longrightarrow}\left(\xi^{\prime}, \eta^{\prime}, \zeta^{\prime}\right)$.

## 物理代写|分析力学代写Analytical Mechanics代考|Commutativity of Infinitesimal Rotations

$$g^{\prime}=g+d \Omega \times g$$

$$d \Omega=d \Phi \hat{\boldsymbol{n}}$$

$$g^{\prime}=g+d \boldsymbol{\Omega}1 \times g, g^{\prime \prime}=g^{\prime}+d \Omega_2 \times g^{\prime},$$ 从那里，忽略二阶无穷小， $$g^{\prime \prime}=g+d \Omega 12 \times g$$ 和 $$d \boldsymbol{\Omega} 12=d \boldsymbol{\Omega}_1+d \boldsymbol{\Omega}_2$$最后的结果表明连续的无穷小旋转通勤 $\left(\boldsymbol{d} \Omega 12=\boldsymbol{d} \boldsymbol{\Omega}{21}\right)$ 作为矢量加法的交换性的结 果。此外，与连续无穷小旋转相关的矢量是与各个无穷小旋转相关的矢量的总和，这一 特性对于即将到来的发展具有重要价值。

$$g_{\Sigma}^{\prime}=\mathcal{A} g_{\Sigma}=(I+\varepsilon) g_{\Sigma}$$

$$\mathcal{A}_1 \mathcal{A}_2=\left(\boldsymbol{I}+\boldsymbol{\varepsilon}_1\right)\left(\boldsymbol{I}+\boldsymbol{\varepsilon}_2\right)=\boldsymbol{I}+\boldsymbol{\varepsilon}_1+\boldsymbol{\varepsilon}_2=\boldsymbol{I}+\boldsymbol{\varepsilon}_2+\boldsymbol{\varepsilon}_1=\mathcal{A}_2 \mathcal{A}_1$$

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