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物理代写|理论力学作业代写Theoretical Mechanics代考|The Action and Angle Variable

如果你也在 怎样代写理论力学Theoretical Mechanics这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。理论力学Theoretical Mechanics是一组密切相关的经典力学的替代公式。它是由许多科学家和数学家在18世纪及以后,在牛顿力学之后发展起来的。由于牛顿力学考虑的是运动的矢量,特别是系统中各组成部分的加速度、动量、力,因此由牛顿定律和欧拉定律所支配的力学的另一个名称是矢量力学。

理论力学Theoretical Mechanics使用代表系统整体的运动标量属性–通常是其总动能和势能–而不是牛顿对单个粒子的矢量力。运动方程是由标量通过一些关于标量变化的基本原理推导出来的。分析力学使用代表系统整体的运动标量属性–通常是其总动能和势能–而不是牛顿对单个粒子的矢量力。运动方程是由标量通过一些关于标量变化的基本原理推导出来的。

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物理代写|理论力学作业代写Theoretical Mechanics代考|The Action and Angle Variable

物理代写|理论力学作业代写Theoretical Mechanics代考|Periodic Systems

We now discuss an important modification of the Hamilton-Jacobi method that is especially designed for
periodic systems
for which we are often more interested in the characteristic frequencies of the movement than, for instance, in the actual shape of the trajectory.
What do we understand by ‘periodic’?
In case of only one degree of freedom $(S=1)$ the answer is immediately clear. After a certain time $\tau$, the ‘period’, the system again reaches its initial state. The phase space is the two-dimensional $(p, q)$-plane. In this context one distinguishes two types of periodicities:
(1) Libration
The phase trajectory is a closed curve:
$$
\begin{aligned}
&q(t+\tau)=q(t) \
&p(t+\tau)=p(t)
\end{aligned}
$$
$q$ and $p$ are periodic with the same frequency. That is typical for oscillating systems such as pendulum, spring etc., which move between two states of vanishing kinetic energy .

物理代写|理论力学作业代写Theoretical Mechanics代考|The Action and Angle Variable

The considerations of this section are concerned exclusively with periodic systems.
Let us first summarize the essentials of the Hamilton-Jacobi method:
We look for a canonical transformation
$$
(\mathbf{q}, \mathbf{p}) \longrightarrow(\overline{\mathbf{q}}, \overline{\mathbf{p}})
$$
so that we get:
$$
\begin{aligned}
&\bar{p}{j}=\text { const } \forall j \ &\bar{q}{j}=\left{\begin{array}{l}
\text { const } \forall j \Longleftrightarrow S(\mathbf{q}, \overline{\mathbf{p}}, t) \
\text { cyclic } \forall j \Longleftrightarrow W(\mathbf{q}, \overline{\mathbf{p}})
\end{array}\right.
\end{aligned}
$$
The generating functions $S$ and $W$ are here the solutions of the HJD with the constants of integration $\alpha_{1}, \ldots, \alpha_{S}$, which are identified with the new momenta:
$$
\bar{p}{j}=\alpha{j} \quad \forall j .
$$
One could have just as well equated arbitrary functions of the $\alpha_{j}$ with the $\bar{p}{j}$. The ‘action variables’ $J{j}$
are rather special functions of the $\alpha_{j}$ :
$$
J_{j}=\oint p_{j} d q_{j}, \quad j=1,2, \ldots, S
$$

物理代写|理论力学作业代写THEORETICAL MECHANICS代考|The Kepler Problem

The just discussed example of the harmonic oscillator served only as an illustration. The full usefulness of the method manifests itself more noticeably in connection with the rather sophisticated problems of the planetary and atomic mechanics.
The Kepler problem is defined by the potential
$$
V(\mathbf{r})=-\frac{k}{r} \quad(k>0)
$$
Concrete realizations are for instance:
$$
\begin{aligned}
&k=\gamma m M \Longleftrightarrow \text { gravitation }((2.261), \text { Vol. 1) } \
&k=\frac{q_{1} q_{2}}{4 \pi \varepsilon_{0}} \Longleftrightarrow \text { Coulomb }((2.258), \text { Vol. 1) }
\end{aligned}
$$
Because of (3.125) spherical coordinates are appropriate. The Hamilton function then reads according to $(2.45)$ :
$$
H=\frac{1}{2 m}\left(p_{r}^{2}+\frac{1}{r^{2}} p_{\vartheta}^{2}+\frac{1}{r^{2} \sin ^{2} \vartheta} p_{\varphi}^{2}\right)-\frac{k}{r} .
$$
For the generalized momenta we found already in (2.44):
$$
\begin{aligned}
&p_{r}=m \dot{r}, \
&p_{\vartheta}=m r^{2} \dot{\vartheta}, \
&p_{\varphi}=m r^{2} \sin ^{2} \vartheta \dot{\varphi}=L_{z}=\text { const } .
\end{aligned}
$$
$\varphi$ is cyclic. The $z$-component of the angular momentum $p_{\varphi}=L_{z}$ is therefore a constant of motion. We have as HJD:
$$
\frac{1}{2 m}\left[\left(\frac{\partial W}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial W}{\partial \vartheta}\right)^{2}+\frac{1}{r^{2} \sin ^{2} \vartheta}\left(\frac{\partial W}{\partial \varphi}\right)^{2}\right]-\frac{k}{r}=\alpha_{1}=E .
$$
The problem is separable:
$$
W=W_{r}(r)+W_{\vartheta}(\vartheta)+W_{\varphi}(\varphi) .
$$

物理代写|理论力学作业代写Theoretical Mechanics代考|The Action and Angle Variable

理论力学代写

物理代写|理论力学作业代写THEORETICAL MECHANICS代考|PERIODIC SYSTEMS

我们现在讨论 Hamilton-Jacobi 方法的一个重要修改,它是专门为
周期性系统设计
的,我们通常对运动的特征频率比对轨迹的实际形状更感兴趣。
我们对“周期性”的理解是什么?
在只有一个自由度的情况下(小号=1)答案马上就清楚了。一定时间后τ,“期间”,系统再次达到其初始状态。相空间是二维的(p,q)-飞机。在这种情况下,人们区分了两种类型的周期性:
1Libration
相位轨迹是一条闭合曲线:
q(吨+τ)=q(吨) p(吨+τ)=p(吨)
q和p是周期性的,频率相同。这是典型的振荡系统,如钟摆、弹簧等,它们在两种消失的动能状态之间移动。

物理代写|理论力学作业代写THEORETICAL MECHANICS代考|THE ACTION AND ANGLE VARIABLE

本节的考虑只涉及周期系统。
让我们首先总结一下 Hamilton-Jacobi 方法的要点:
我们寻找一个典型的变换
$$
(\mathbf{q}, \mathbf{p}) \longrightarrow(\overline{\mathbf{q}}, \overline{\mathbf{p}})
$$
所以我们得到:
$$
\begin{aligned}
&\bar{p}{j}=\text { const } \forall j \ &\bar{q}{j}=\left{\begin{array}{l}
\text { const } \forall j \Longleftrightarrow S(\mathbf{q}, \overline{\mathbf{p}}, t) \
\text { cyclic } \forall j \Longleftrightarrow W(\mathbf{q}, \overline{\mathbf{p}})
\end{array}\right.
\end{aligned}
$$
The generating functions $S$ and $W$ are here the solutions of the HJD with the constants of integration $\alpha_{1}, \ldots, \alpha_{S}$, which are identified with the new momenta:
$$
\bar{p}{j}=\alpha{j} \quad \forall j .
$$
One could have just as well equated arbitrary functions of the $\alpha_{j}$ with the $\bar{p}{j}$. The ‘action variables’ $J{j}$
are rather special functions of the $\alpha_{j}$ :
$$
J_{j}=\oint p_{j} d q_{j}, \quad j=1,2, \ldots, S
$$

物理代写|理论力学作业代写THEORETICAL MECHANICS代考|THE KEPLER PROBLEM

刚才讨论的谐振子的例子只是一个例子。该方法的全部实用性在与行星和原子力学的相当复杂的问题相关联时表现得更加明显。
开普勒问题由势能定义
五(r)=−到r(到>0)
具体实现例如:
到=C米米⟺ 引力 ((2.261), 卷。1)  到=q1q24圆周率e0⟺ 库仑 ((2.258), 卷。1) 
因为3.125球坐标是合适的。汉密尔顿函数然后根据(2.45):
H=12米(pr2+1r2pϑ2+1r2没有2⁡ϑp披2)−到r.
对于我们已经在2.44:
pr=米r˙, pϑ=米r2ϑ˙, p披=米r2没有2⁡ϑ披˙=大号和= 常量 .
披是循环的。这和- 角动量的分量p披=大号和因此是一个运动常数。我们有 HJD:
12米[(∂在∂r)2+1r2(∂在∂ϑ)2+1r2没有2⁡ϑ(∂在∂披)2]−到r=一种1=和.
问题是可分离的:
在=在r(r)+在ϑ(ϑ)+在披(披).

物理代写|理论力学作业代写Theoretical Mechanics代考

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