---
tags: Mechanics, ss, ncu
author: N0-Ball
title: Introduction
GA: UA-208228992-1
---

[ToC]

# Research Fields of Mechanics

## Kinematics
:::info
Motions of particals
Can be shadows or wave
:::
$$
\frac{d \vec x}{dt} = \vec v \\[1em]
\frac{d \vec v}{dt} = \vec a
$$

## Statics
:::info
A study of equilibrium states or solitary waves with $\frac{\partial} {\partial t} = 0$
:::

:::danger
動能＋位能 只有在不隨時間變化的位能場中，才會守恆
:::
-> Solution steps:
1. None Conservation
2. Conservation
3. Answer
4. None Conservation

## Dynamics
:::info
a self-consisten study of macroscopic fluid motion
:::

- A motion of a group of particals (large view)
- Applies $\vec F = \frac{d \vec p}{dt}$
- Examples
- Hydrodynamics
- Atmospheric Dynamics
- Fluid Dynamics
- Magnetohydrodynamics
- Relativistics Dynamics

:::danger
A system that can't define force can't be Dynamics
:::

## Kinetics
:::info
a self-coinsistent study of microscopic collective motion & changes on heat and entropy
:::

- A motion of a group of particals (micro view)
- Only heat and entropy
- example
- Kinetics plasma physics

## Mechanics
:::info
Can't be define simply by only Force
:::

- Always associate with waves
- example
- Quantum Mechanics
- Wave Mechanics
- Statistical Mechanics
- Relativistic Mechanics
- Fluid Mechanics

# Dimensional Analysis

:::info
A dimension is a measure of a physical variable (without numerical values), while a unit is
a way to assign a number or measurement to that dimension.
For example, length is a dimension, but it is measured in units of feet (ft) or meters (m).
:::

- Only same dimesional system can they add or distract

## Normalization

:::info
Change the equation to dimensionless
:::

## SI Unit

- L : **Length**
- T : **Time**
- M : **Mass**
- Q : **Electric charge**

## Caussian Unit

:::info
The charge is included in the dimensions of the electric
field and the magnetic field in the Gaussian unit
:::

- L : **Length**
- T : **Time**
- M : **Mass**

# Vector calculations


## Spherical Coordinate system


## Chain rule

:::info
Let $f(x) = (3x + 5)^2 + (3x + 5) + 2x + 5$
Find $f'(x) = \frac{df(x)}{dx}$
:::

## Solution 1. (stupid method)

$$
\begin{aligned}
f(x) &= (3x + 5)^2 + (3x + 5) + 2x + 5 \\[1em]
&= 27x^3 + 144x^2 + 257x + 155
\end{aligned}
$$

Thus,

$$
f'(x) = 81x^2 + 288x + 257
$$

## Solution 2. (cool method)

$$
Let\ g(x) = 3x + 5\\[1em]
\Rightarrow f(x) = F[g(x), x] = g^3 + g^2 + g - x
$$

Thus,

$$
\begin{aligned}
f'(x) = \frac{d F[g(x), x]}{dx} &= \frac{\partial F}{\partial g} \frac{\partial g}{\partial x} + \frac{\partial F}{\partial x} = (3g^2 + 2g + 1) * (3) - 1 \\[1em]
&= 81x^2 + 288x + 257
\end{aligned}
$$

## Result

Sol 1. and Sol 2. have the same answer

## Different Coordinate system

for cylindrical coordinate system

$$
\frac{d \hat r[\theta(t)]}{dt} = \frac{d \hat r}{d\theta} \frac{d\theta}{dt}
$$

Likewise for spherical coordinate system

$$
\frac{d\hat r[\theta(t), \phi(t)]}{dt} = \frac{\partial \hat r}{\partial \theta } \frac{d \theta}{dt} + \frac{\partial r}{\partial \phi}\frac{d \phi}{dt}
$$