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## 物理代写|统计力学代写STATISTICAL MECHANICS代写|PHYS112 Classical Mechanics

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## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Newton’s Laws

It is obvious from the definition given here that we are dealing with an idealization, which is nevertheless approximately realized in many situations: the fact that the Earth rotates around the Sun and around itself makes a frame of reference attached to the Earth, strictly speaking, non inertial; but it can nevertheless be considered inertial for most experiments performed in laboratories.

A basic principle of mechanics is the equivalence of all inertial frames of reference, also called Galilean invariance: the laws of motion take the same form in all inertial frames of reference and the transformations between such frames consist of (constant in time) rotations and translations on a straight line at constant velocity of the origin of coordinates. This invariance implies conservation laws for total momentum, total angular momentum and energy (checking the first conservation laws will be left as exercises). ${ }^{1}$

Here we will always work in a fixed inertial frame, so we will not be concerned with Galilean invariance. Moreover, we will not discuss conservations laws apart form the conservation of energy. Newton’s first law, says, in modern terminology, that there exist inertial reference frames; since we decided to work in one such frame, we will not discuss it further.

Consider $N$ particles in $\mathbb{R}^{3}$ of masses $m_{1}, m_{2}, \ldots, m_{N}$. The position of the $i$ th particle is represented by a vector $\vec{q}{i} \in \mathbb{R}^{3}$ and the positions of all the particles of the system by a vector $\mathbf{q}=\left(\vec{q}{1}, \vec{q}{2}, \ldots, \vec{q}{N}\right) \in \mathbb{R}^{3 N}$.
Newton’s second law states that:
$$m_{i} \frac{d^{2} \vec{q}{i}}{d t^{2}}=\sum{j=1, j \neq i}^{N} \vec{F}{i j}\left(\vec{q}{i}, \vec{q}{j}\right)+\sum{i=1}^{N} \vec{F}{i}\left(\vec{q}{i}\right)$$
where $\vec{F}{i j}\left(\vec{q}{i}, \vec{q}{j}\right)$ is the force exerted on the particle of index $i$ by the one of index $j$ and $F{i}\left(\vec{q}{i}\right)$ represents the force exerted on the system by bodies located outside of it. ${ }^{2}$ We will assume that the forces are “conservative” or “derive from a potential”, namely that, for each pair $i, j$, there are smooth functions $V{i j}: \mathbb{R}^{6} \rightarrow \mathbb{R}, V_{i}: \mathbb{R}^{3} \rightarrow \mathbb{R}$, such that:
$$\vec{F}{i j}\left(\vec{q}{i}, \vec{q}{j}\right)=-\nabla{\vec{q}{i}} V{i j}\left(\vec{q}{i}, \vec{q}{j}\right)$$
$$\vec{F}{i}\left(\vec{q}{i}\right)=-\nabla_{\vec{q}{i}} V{i}\left(\vec{q}{i}\right)$$ where $\nabla{\vec{q}{i}}$ is the gradient with respect to $\vec{q}{i}$.

## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Hamilton’s Equations

It is often convenient to rewrite Newton’s equations (3.2.6) in Lagrangian form or in Hamiltonian form. We will only use the latter one. ${ }^{4}$ To do so, we will introduce the phase space $\mathbb{R}^{6 N}$, and write a vector $\mathbf{x} \in \mathbb{R}^{\mathbf{6} \mathbf{N}}$ as a pair $\mathbf{x}=(\mathbf{q}, \mathbf{p})$, with $\mathbf{q}=$ $\left(\vec{q}{1}, \vec{q}{2}, \ldots, \vec{q}{N}\right) \in \mathbb{R}^{3 N}, \mathbf{p}=\left(\vec{p}{1}, \vec{p}{2}, \ldots, \vec{p}{N}\right) \in \mathbb{R}^{3 N}$.
The Hamiltonian is a function $H: \mathbb{R}^{6 N} \rightarrow \mathbb{R}$ :
$$H(\mathbf{q}, \mathbf{p})=K(\mathbf{p})+V(\mathbf{q})$$
with a kinetic energy
$$K(\mathbf{p})=\sum_{i=1}^{N} \frac{\left|\vec{p}{i}\right|^{2}}{2 m{i}}$$
and a potential energy $V(\mathbf{q})$ given by (3.2.5).
Then Hamilton’s equations are given by the following pair:
$$\frac{d \vec{q}{i}(t)}{d t}=\nabla{\vec{p}{i}} H(\mathbf{q}(t), \mathbf{p}(t)),$$ and $$\frac{d \vec{p}{i}(t)}{d t}=-\nabla_{\vec{q}{i}} H(\mathbf{q}(t), \mathbf{p}(t)),$$ for $i=1, \ldots, N$. With $H$ defined by (3.3.1), (3.3.2), these equations are: $$\frac{d \vec{q}{i}(t)}{d t}=\frac{\vec{p}{i}(t)}{m{i}}$$
and
$$\frac{d \vec{p}{i}(t)}{d t}=-\nabla{\vec{q}_{i}} V(\mathbf{q}(t)) .$$

## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Newton’s Laws

$$m_{i} \frac{d^{2} \vec{q} i}{d t^{2}}=\sum j=1, j \neq i^{N} \vec{F} i j(\vec{q} i, \vec{q} j)+\sum i=1^{N} \vec{F} i(\vec{q} i)$$

$$\vec{F}{i j}(\vec{q} i, \vec{q} j)=-\nabla \vec{q} i V i j(\vec{q} i, \vec{q} j)$$ $$\vec{F} i(\vec{q} i)=-\nabla{\vec{q} i} V i(\vec{q} i)$$

## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Hamilton’s Equations

$$H(\mathbf{q}, \mathbf{p})=K(\mathbf{p})+V(\mathbf{q})$$

$$K(\mathbf{p})=\sum_{i=1}^{N} \frac{|\vec{p} i|^{2}}{2 m i}$$

$$\frac{d \vec{q} i(t)}{d t}=\nabla \vec{p} i H(\mathbf{q}(t), \mathbf{p}(t)),$$

$$\frac{d \vec{p} i(t)}{d t}=-\nabla_{\vec{q} i} H(\mathbf{q}(t), \mathbf{p}(t)),$$

$$\frac{d \vec{q} i(t)}{d t}=\frac{\vec{p} i(t)}{m i}$$

$$\frac{d \vec{p} i(t)}{d t}=-\nabla \vec{q}_{i} V(\mathbf{q}(t)) .$$

## MATLAB代写

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