# 物理代写|理论力学作业代写Theoretical Mechanics代考|Discussion of the Spinning Top Motion

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Discussion of the Spinning Top Motion

(a) $\vartheta$ : Angle between space-fixed $\hat{z}$ axis and body-fixed $z$ axis. The $\hat{z}$ axis is given according to (4.91) by the direction of the angular momentum $\mathbf{L}$. The $z$ axis is the body axis ( $\zeta$ axis). From that it follows:
The body axis moves with constant aperture angle $\vartheta=\vartheta_0$ and with constant angular velocity $\dot{\varphi}$ around the direction of the angular momentum. The cone described by the body axis is called ‘nutation cone’.
(b) $\dot{\psi}$ : Angular velocity by which the body (more strictly the body-fixed $\eta, \xi$ plane) rotates around the body axis.
(c) $\omega$ : The angular velocity $\omega$ is equal to the vector sum of $\dot{\varphi}$ and $\dot{\psi}$. It always lies in the $\hat{z}, \zeta$ plane, thus rotates together with the body axis around the direction of the angular momentum ( $\hat{z}$ axis) enclosing with the body axis the angle $\gamma(4.89)$. The momentary rotation axis defined by $\omega$ moves therefore on the so-called space cone around the space-fixed angular-momentum direction.

The pole cone rolls off with its outside mantle on the space-fixed space cone and therefore directs the body axis on the nutation cone.

For $A>C$ it is $\dot{\psi} \uparrow \uparrow \mathbf{e}_\zeta$. Then the outside area of the pole cone rolls off the space-cone mantle (Fig. 4.19).

For $A<C$ it is $\dot{\psi} \uparrow \downarrow \mathbf{e}_\zeta$. The pole cone rolls off with its inside area on the space-fixed space cone where again the body axis is directed on the nutation cone (Fig. 4.20).

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Elements of Differential Calculus

One defines the following types of numbers:
$$\begin{array}{ll} \mathbb{N}={1,2,3, \ldots} & \text { natural numbers } \ \mathbb{Z}={\ldots,-2,-1,0,1,2,3, \ldots} & \text { integer numbers } \ \mathbb{Q}=\left{x ; x=\frac{p}{q} ; p \in \mathbb{Z}, q \in \mathbb{N}\right} & \text { rational numbers } \ \mathbb{R}={x ; \text { continuous number line }} & \text { real numbers } . \end{array}$$
Therefore
$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} .$$

The body of complex numbers $\mathbb{C}$ will be introduced and discussed later in Sect. 2.3.5. For the above-mentioned set of numbers the basic operations addition and multiplication are defined in the well-known manner. We will remind here only shortly to the process of raising to a power.
For an arbitrary real number $a$ the $n$-th power is defined as:
$$a^n=\underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}{n \text {-fold }} \quad n \in \mathbb{N} .$$ There are the following rules: 1. $$(a \cdot b)^n=\underbrace{(a \cdot b) \cdot(a \cdot b) \cdot \ldots \cdot(a \cdot b)}{n \text {-fold }}=a^n \cdot b^n$$
2.
$$a^k \cdot a^n=\underbrace{a \cdot a \cdot \ldots \cdot a}{k \text {-fold }} \cdot \underbrace{a \cdot a \cdot \ldots \cdot a}{n \text {-fold }}=a^{k+n}$$
3.
$$\left(a^n\right)^k=\underbrace{a^n \cdot a^n \cdot \ldots \cdot a^n}_{k \text {-fold }}=a^{n-k} .$$

# 理论力学代写

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Discussion of the Spinning Top Motion

(A) $\vartheta$ : 空间固定之间的角度 $\hat{z}$ 轴体固定 $z$ 轴。这 $\hat{z}$ axis 根据 (4.91) 由角动量的方向给出 L. 这 $z$ 轴是身体轴 ( $\zeta$ 轴)。由此得出:

(二) $\dot{\psi}$ : 身体的角速度 (更严格的身体固定 $\eta, \xi$ 平面) 绕体轴旋转。
(C) $\omega$ : 角速度 $\omega$ 等于向量和 $\varphi$ 和 $\psi$. 它总是位于 $\hat{z}, \zeta$ 平面，因此与体轴一起围绕角动量 的方向旋转 ( $\hat{z} a x i s$ ) 与 body axis 包围的角度 $\gamma(4.89)$. 瞬时旋转轴定义为 $\omega$ 因此围绕空间 固定角动量方向在所谓的空间雉上运动。

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Elements of Differential Calculus

$\backslash$ begin ${$ array $}{|} \backslash$ |mathbb ${\mathrm{N}}={1,2,3, \backslash$ dots $} \& \backslash$ text ${$ 自然数 $} \backslash \backslash \operatorname{mathbb}{Z}={\backslash$ dots, $, 2,-1,0,1,2,3, \backslash$ dd

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$

$$a^n=\underbrace{a \cdot a \cdot a \cdot \ldots \cdot a} n \text {-fold } \quad n \in \mathbb{N}$$

$$(a \cdot b)^n=\underbrace{(a \cdot b) \cdot(a \cdot b) \cdot \ldots \cdot(a \cdot b)} n \text {-fold }=a^n \cdot b^n$$
2.
$$a^k \cdot a^n=\underbrace{a \cdot a \cdot \ldots \cdot a} k \text {-fold } \cdot \underbrace{a \cdot a \cdot \ldots \cdot a} n \text {-fold }=a^{k+n}$$
3.
$$\left(a^n\right)^k=\underbrace{a^n \cdot a^n \cdot \ldots \cdot a^n}_{k \text {-fold }}=a^{n-k}$$

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