~jzck/physics-notes

a9992e549f506dae9f2be4f9ef26af441d92bfe3 — Jack Halford 9 months ago af8a21b
25 done, graph layout changed
M Makefile => Makefile +2 -2
@@ 48,8 48,8 @@ clean:
re: clean all

deploy:
> aws s3 sync --content-type 'text/html;charset=utf-8' www-data s3://www.0x5.be/physics-notes \
> aws s3 sync --content-type 'text/html;charset=utf-8' www-data s3://0x5.be/physics-notes \
	--exclude "*" --include "*.html"
> aws s3 sync www-data s3://www.0x5.be/physics-notes --exclude "*.html"
> aws s3 sync www-data s3://0x5.be/physics-notes --exclude "*.html"

.PHONY: clean all re sync

A formularies/LL1/25.1.tex => formularies/LL1/25.1.tex +1 -0
@@ 0,0 1,1 @@
m\ddot{x}=-kx-\alpha\dot{x}

A formularies/LL1/25.10.tex => formularies/LL1/25.10.tex +1 -0
@@ 0,0 1,1 @@
f_{\text{fr},i}=-\partial F/\partial \dot{x}_i

A formularies/LL1/25.11.tex => formularies/LL1/25.11.tex +1 -0
@@ 0,0 1,1 @@
F=\mfrac{1}{2}\sum_{i,k}\alpha_{ik}\dot{x}_i\dot{x}_k

A formularies/LL1/25.12.tex => formularies/LL1/25.12.tex +3 -0
@@ 0,0 1,3 @@
\frac{\dd{}}{\dd{t}}\frac{\partial L}{\partial \dot{x_i}}=
\frac{\partial L}{\partial x_i}-
\frac{\partial F}{\partial \dot{x}_i}

A formularies/LL1/25.13.tex => formularies/LL1/25.13.tex +1 -0
@@ 0,0 1,1 @@
\dd{E}/\dd{t}=-2F

A formularies/LL1/25.14.tex => formularies/LL1/25.14.tex +1 -0
@@ 0,0 1,1 @@
\sum_k m_{ik}\ddot{x}_k+\sum_k k_{ik}x_k=-\sum_k \alpha_{ik}\dot{x}_k

A formularies/LL1/25.15.tex => formularies/LL1/25.15.tex +1 -0
@@ 0,0 1,1 @@
\sum_k\left( m_{ik}r^2 + \alpha_{ik}r + k_{ik}\right)=0

A formularies/LL1/25.16.tex => formularies/LL1/25.16.tex +1 -0
@@ 0,0 1,1 @@
|m_{ik}r^2 + \alpha_{ik}r + k_{ik}|=0

A formularies/LL1/25.2.tex => formularies/LL1/25.2.tex +2 -0
@@ 0,0 1,2 @@
k/m=\omega_0^2,\qquad
\alpha/m=2\lambda

A formularies/LL1/25.3.tex => formularies/LL1/25.3.tex +1 -0
@@ 0,0 1,1 @@
\ddot{x}+2\lambda\dot{x}+\omega_0^2x=0

A formularies/LL1/25.4.tex => formularies/LL1/25.4.tex +1 -0
@@ 0,0 1,1 @@
x=a e^{-\lambda t}\cos(\omega t+\alpha)

A formularies/LL1/25.6.tex => formularies/LL1/25.6.tex +2 -0
@@ 0,0 1,2 @@
x=c_1 e^{-\left(\lambda-\sqrt{\lambda^2-\omega_0^2}\right)t}
+ c_2 e^{-\left(\lambda+\sqrt{\lambda^2-\omega_0^2}\right)t}

A formularies/LL1/25.7.tex => formularies/LL1/25.7.tex +1 -0
@@ 0,0 1,1 @@
x=(c_1+c_2 t)e^{-\lambda t}

A formularies/LL1/25.8.tex => formularies/LL1/25.8.tex +1 -0
@@ 0,0 1,1 @@
f_{\text{fr},i}=-\sum_k\alpha_{ik}\dot{x}_k

A formularies/LL1/25.9.tex => formularies/LL1/25.9.tex +1 -0
@@ 0,0 1,1 @@
\alpha_{ik}=\alpha_{ki}

A md/damped-oscillations.md => md/damped-oscillations.md +131 -0
@@ 0,0 1,131 @@
---
type: model
title: damped oscillations
---

We consider that the medium exerts some resistance on the moving body. The energy of the moving body will be dissipated into heat.

The full state of the system now needs knowledge of the motion of the medium and the thermal states of the medium and the body, thus this is no longer a mechanical problem in the sense we have seen previously.

<hr>

In the special case of [small oscillations](small-oscillations.md), with frequencies small compared to the dissipative process, we can model the friction by a force which depends only on the velocity of the motion. If the velocity is small, we can expand powers of the force. The zero-order term is zero, since no friction affect a body at rest. So the general form of the friction force on a system executing small oscillations in one dimension if $f_{\text{fr}}=-\alpha \dot{x}$. with a positive $\alpha$, the minus sign because the force resists the velocity.

We add this force on the right hand side of the equation of motion (see `LL1/21.4`)

```eq
LL1/25.1
```

We devide by $m$ and put

```eq
LL1/25.2
```

Thus the equation becomes

```eq
LL1/25.3
```

the roots of the characteristic equation are $r_{1,2}=-\lambda\pm\sqrt{\lambda^2-\omega_0^2}$, thus the solution is of the form

$$ x=c_1 e^{r_1 t}+c_2 e^{r_2 t} $$

We know that these solutions are all stable, since the real parts of $r_{1,2}$ are negative.

### Periodic damping

In the case $\lambda\lt\omega_0$, we have $r_{1,2}$ are complex. The solutions take the form

```eq
LL1/25.4
```
with $\omega=\sqrt{\omega_0^2-\lambda^2}$ and $a$,$\alpha$ real constants.

### Aperiodic damping

In the case $\lambda\gt\omega_0$, we have $r_{1,2}\lt0$
```eq
LL1/25.6
```
where x decreases monotonously towards 0.

### Critical damping

In the case $\lambda=\omega_0$, we have the double root $r_{1,2}=-\lambda$. The solutions of the differential equation are

TODO: math note
```eq
LL1/25.7
```

# Dissipative function

For a system with more than one degree of freedom, the generalised frictional forces corresponding to the co-ordinates $x_i$ are of the form

```eq
LL1/25.8
```

From mechanical arguments we can deduce no symmetry properties for the coefficients $\alpha_{ik}$. But from statistical physics (TODO: note to book 5), we can demonstrate that

```eq
LL1/25.9
```

Henre `LL1/25.8` can be written as the derivatives

```eq
LL1/25.10
```

of the quadratic form

```eq
LL1/25.11
```

Which we call the _dissipative function_.

The forces `LL1/25.10` must be added to the right-hand side of Lagranges equation `LL1/2.6`

```eq
LL1/25.12
```

The dissipative function's physical significance is the rate of dissipation of energy in the system. We have from `LL1/6.1`

\begin{aligned}
\frac{\dd{E}}{\dd{t}} &=
\frac{\dd{}}{\dd{t}}\left(\sum_i\dot{x}_i\frac{\partial L}{\partial\dot{x}_i}-L\right) \\
&= \sum_i \dot{x}_i\left(\frac{\dd{}}{\dd{t}}\left[\frac{\partial L}{\partial\dot{x}_i}\right]-\frac{\partial L}{\partial x_i}\right) \\
&= -\sum_i \dot{x}_i\frac{\partial F}{\partial\dot{x}_i}
\end{aligned}

Since F is a quadratic function of velocities, Euler's theorem on homogeneous functions (TODO: math note) shows that the sum on the right-hand side is $2F$, Thus

```eq
LL1/25.13
```

Since dissipative processes lead to loss of energy, it follows that $F\gt0$, i.e, the quadratic form `LL1/25.11` is positive definite.

The equations of small oscillations are the `LL1/23.5` plus the friction forces `LL1/25.8`

```eq
LL1/25.14
```

This is a set of $i$ homogeneous ODEs.

TODO: nxn ODEs math course

```eq
LL1/25.15
```

```eq
LL1/25.16
```

D md/density-matrix.md => md/density-matrix.md +0 -6
@@ 1,6 0,0 @@
---
title: density matrix
type: theory
---

paragraph 14

M md/energy.md => md/energy.md +1 -1
@@ 33,7 33,7 @@ LL1/6.1

This property doesn't hold for a system with a potential dependant on time.

As we know, the Lagrangian has the form $L=T(q,\dot{q})-U(q)$, where $T$ is a quadratic function of velocities. Using Euler theorem on homogenous functions
As we know, the Lagrangian has the form $L=T(q,\dot{q})-U(q)$, where $T$ is a quadratic function of velocities. Using Euler theorem on homogenous functions (TODO: math note)

$$
\sum_i\dot{q_i}\frac{\partial L}{\partial \dot{q_i}}=

D md/integrals-of-motion.md => md/integrals-of-motion.md +0 -10
@@ 1,10 0,0 @@
---
title: integrals of motion
type: theory
---

During the motion of a system, we can find functions of the co-ordinates which are constant, we call them _integrals of the motion_.

The number of integrals of motion for a system wit $s$ degrees of freedom is $2s-1$.

TODO p6

D md/matrix-mechanics.md => md/matrix-mechanics.md +0 -6
@@ 1,6 0,0 @@
---
title: matrix mechanics
type: theory
---

paragraph 11

M md/natural-reference-frame.md => md/natural-reference-frame.md +0 -1
@@ 59,4 59,3 @@ with the following proof
```eq
LL1/8.5
```


M tools/generate-graph.sh => tools/generate-graph.sh +2 -2
@@ 21,8 21,8 @@ generate_one_node() {
echo 'digraph {'
echo '    bgcolor="#ffffff00"'
echo '    node  [style="rounded,filled", shape=box]'
echo '    rankdir=RL'
echo '    layout=fdp'

for node in md/*.md; do generate_one_node $node& done; wait
for node in md/*.md; do generate_one_node $node& done | sort; wait 

echo "}"

M tools/get => tools/get +1 -1
@@ 6,7 6,7 @@ colors[theory]="#c5edf4"	#blue
colors[problem]="#f4fde4"	#yellow
colors[model]="#dbfae0"		#greeen
colors[math]="#f5d6e5"		#pink
colors[DEFAULT]="#fbe7da"	#pink
colors[DEFAULT]="#fff8e7"	#cosmic latte

_or_default() { grep ^ || echo DEFAULT; }
slug() { echo $1; }

M tools/index.html => tools/index.html +1 -1
@@ 1,7 1,7 @@
<script async defer data-website-id="b9b10aa0-177f-42d5-8493-f3a8062db28e" src="https://umami.pourtan.eu/umami.js"></script>
<link rel="stylesheet" type="text/css" href="colors.css">
<link rel="stylesheet" type="text/css" href="style.css">
<body class='_experiment'>
<body class='_DEFAULT'>
<h1>Main Graph</h1>
<div style="margin:auto"><object data=graph.svg type=image/svg+xml></object></div>
</body>