Statistics-lab™可以为您提供cornell.edu PHYS3318 Analytical Mechanics分析力学课程的代写代考和辅导服务！

PHYS3318 Analytical Mechanics课程简介

Forbidden Overlap: Students may not receive credit for both PHYS 3314 and PHYS 3318.
Prerequisite: strong performance in PHYS 2218 or permission of instructor; AEP 4210 or appropriate course(s) in mathematics. Intended for students with exceptional background in physics/math. PHYS 3314 covers similar material, while assuming less prior knowledge. Makes use of Fourier analysis, linear differential equations, linear algebra, and vector analysis.

PREREQUISITES 

Covers Newtonian mechanics of particles and systems of particles, including rigid bodies, oscillating systems, gravitation and planetary motion, moving coordinate systems, Euler’s equations, Lagrange and Hamilton formulations, normal modes and small vibrations, and perturbation theory. At the level of Classical Mechanics by Goldstein, Mechanics by Landau and Lifshitz, and Analytical Mechanics by Hand and Finch.

PHYS3318 Analytical Mechanics HELP（EXAM HELP， ONLINE TUTOR）

Proof The action $\bar{S}$ associated with $\bar{L}$ is
$$
\bar{S}=\int_{t_1}^{t_2} \bar{L}(q, \dot{q}, t) d t=\int_{t_1}^{t_2} L(q, \dot{q}, t) d t+\int_{t_1}^{t_2} \frac{d f}{d t} d t=S+f\left(q\left(t_2\right), t_2\right)-f\left(q\left(t_1\right), t_1\right)
$$
Since the variation of the action leaves the endpoints $q\left(t_1\right)$ and $q\left(t_2\right)$ fixed, $\delta \bar{S}=\delta S$. Therefore, the conditions $\delta \bar{S}=0$ and $\delta S=0$ are identical, showing that $\bar{L}$ and $L$ engender exactly the same equations of motion.

Proof We follow Gallavotti $(1983)$. Let $\left[t_1, t_2\right]$ be a time interval so short that a single physical path goes through the fixed endpoints $x\left(t_1\right)$ and $x\left(t_2\right)$. Let $\eta:\left[t_1, t_2\right] \rightarrow \mathbb{R}$ be an infinitely differentiable function such that $\eta\left(t_1\right)=\eta\left(t_2\right)=0$. The physical path shall be denoted simply by $x(t)$. The action for the varied path $\bar{x}=x+\eta$ is
$$
S[\bar{x}]=\int_{t_1}^{t_2}\left[\frac{m}{2}(\dot{x}+\dot{\eta})^2-V(x+\eta)\right] d t
$$
By Taylor’s theorem with Lagrange’s form of the remainder (Spivak, 1994) we can write
$$
V(x+\eta)=V(x)+V^{\prime}(x) \eta+\frac{1}{2} V^{\prime \prime}(x+\xi) \eta^2, \quad|\xi| \leq|\eta| .
$$
Since we are only interested in establishing that the action is a local minimum for the physical path, let us choose $\eta$ such that $|\eta(t)| \leq 1$ for all $t \in\left[t_1, t_2\right]$, which implies $|\xi| \leq 1$. Substituting (2.57) into $(2.56)$, we are led to
$$
\begin{aligned}
S[\bar{x}] & =\int_{t_1}^{t_2}\left[\frac{m}{2} \dot{x}^2+m \dot{x} \dot{\eta}+\frac{m}{2} \dot{\eta}^2-V(x)-V^{\prime}(x) \eta-\frac{1}{2} V^{\prime \prime}(x+\xi) \eta^2\right] d t \
& =S[x]+\int_{t_1}^{t_2}\left[m \dot{x} \dot{\eta}-V^{\prime}(x) \eta\right] d t+\int_{t_1}^{t_2}\left[\frac{m}{2} \dot{\eta}^2-\frac{1}{2} V^{\prime \prime}(x+\xi) \eta^2\right] d t .
\end{aligned}
$$
Integrating by parts the term containing $\dot{\eta}$ and taking into account that the physical path obeys the equation of motion $m \ddot{x}+V^{\prime}(x)=0$, we obtain
$$
S[\bar{x}]-S[x]=\int_{t_1}^{t_2}\left[\frac{m}{2} \dot{\eta}^2-\frac{1}{2} V^{\prime \prime}(x+\xi) \eta^2\right] d t
$$

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供cornell.edu PHYS3318 Analytical Mechanics分析力学课程的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。