~jzck/physics-notes

508d894fe48a45d461c3b675e597cedb89692f79 — Jack Halford 1 year, 21 days ago 28a670e
small oscillations
A formularies/LL1/21.1.tex => formularies/LL1/21.1.tex +1 -0
@@ 0,0 1,1 @@
x=q-q_0

A formularies/LL1/21.11.tex => formularies/LL1/21.11.tex +1 -0
@@ 0,0 1,1 @@
x=\re{Ae^{i\omega t}}

A formularies/LL1/21.2.tex => formularies/LL1/21.2.tex +1 -0
@@ 0,0 1,1 @@
U(x)=\mfrac{1}{2}kx^2

A formularies/LL1/21.3.tex => formularies/LL1/21.3.tex +1 -0
@@ 0,0 1,1 @@
L=\mfrac{1}{2}m\dot{x}^2-\mfrac{1}{2}kx^2

A formularies/LL1/21.5.tex => formularies/LL1/21.5.tex +1 -0
@@ 0,0 1,1 @@
\ddot{x}+\omega^2 x=0

A formularies/LL1/21.6.tex => formularies/LL1/21.6.tex +1 -0
@@ 0,0 1,1 @@
w=\sqrt{k/m}

A formularies/LL1/22.1.tex => formularies/LL1/22.1.tex +1 -0
@@ 0,0 1,1 @@
L=\mfrac{1}{2}m\dot{x}^2-\mfrac{1}{2}kx^2+xF(t)

A formularies/LL1/22.10.tex => formularies/LL1/22.10.tex +1 -0
@@ 0,0 1,1 @@
\xi=e^{i\omega t}\left[\int_0^t\mfrac{1}{m}F(t)e^{-i\omega t}+\xi_0 \right]

A formularies/LL1/22.11.tex => formularies/LL1/22.11.tex +1 -0
@@ 0,0 1,1 @@
E=\mfrac{1}{2}m(\dot{x}^2+\omega^2x^2)=\mfrac{1}{2}m|\xi|^2

A formularies/LL1/22.12.tex => formularies/LL1/22.12.tex +1 -0
@@ 0,0 1,1 @@
E=\mfrac{1}{2m}\left|\int_{-\infty}^{\infty}F(t)e^{-i\omega t}\right|^2

A formularies/LL1/22.2.tex => formularies/LL1/22.2.tex +1 -0
@@ 0,0 1,1 @@
\ddot{x}+\omega^2 x=F(t)/m

A formularies/LL1/22.3.tex => formularies/LL1/22.3.tex +1 -0
@@ 0,0 1,1 @@
F(t)=f\cos(\gamma t+\beta)=\re{Fe^{i\gamma t}}

A formularies/LL1/22.4a.tex => formularies/LL1/22.4a.tex +1 -0
@@ 0,0 1,1 @@
x_p=\re{\frac{Fe^{i\gamma t}}{m(\omega^2-\gamma^2)}}=\frac{f\cos(\gamma t+\beta)}{m(\omega^2-\gamma^2)}

A formularies/LL1/22.5a.tex => formularies/LL1/22.5a.tex +1 -0
@@ 0,0 1,1 @@
x_p=t\cdot\re{\frac{Fe^{i\gamma t}}{2im\omega}}=t\cdot\frac{f\sin(\gamma t+\beta)}{2m\omega}

A formularies/LL1/22.6.tex => formularies/LL1/22.6.tex +1 -0
@@ 0,0 1,1 @@
x=\re{Ae^{i\omega t}+Be^{i(\omega+\epsilon)t}}=\re{[A+Be^{i\epsilon t}]e^{i\omega t}}

A formularies/LL1/22.7.tex => formularies/LL1/22.7.tex +1 -0
@@ 0,0 1,1 @@
c^2=a^2+b^2+2ab\cos(\epsilon t+\beta-\alpha)

A formularies/LL1/22.8.tex => formularies/LL1/22.8.tex +1 -0
@@ 0,0 1,1 @@
\dot{\xi}-i\omega\xi=\mfrac{1}{m}F(t)

A formularies/LL1/22.9.tex => formularies/LL1/22.9.tex +1 -0
@@ 0,0 1,1 @@
\xi=\dot{x}+i\omega x

M md/coplanar-double-pendulum.md => md/coplanar-double-pendulum.md +1 -3
@@ 3,6 3,7 @@ title: coplanar double pendulum
type: problem
---

Find the [Lagrangian](lagrangian.md) of a coplanar double pendulum
## Solution

$$ L=\mfrac{1}{2}(m_1+m_2)l_1^2\phi_1^2


@@ 11,6 12,3 @@ $$ L=\mfrac{1}{2}(m_1+m_2)l_1^2\phi_1^2
	+(m_1+m_2)gl_1\cos\phi_1
	+m_2gl_2\cos\phi_2
$$

## Model
 - [Classical mechanics](mechanical-system.md)

M md/erf.md => md/erf.md +2 -2
@@ 8,11 8,11 @@ sources: MIT 18.031x

This trick works for finding a particular solution to $n^{th}$ order ODE with constant factors when the input is of exponential form.

Consider the following $n^th$ order constant factor ODE.
Consider the following $n^{th}$ order constant factor ODE.

$$ P(D)z=e^{rt} $$

where $P(D)$ is a polynomial of the derivative operator $D$.
where $P(D)$ is a polynomial of the derivative operator $D$, and $r$ is any complex number.

because of the property of exponential we have


A md/forced-oscillations.md => md/forced-oscillations.md +112 -0
@@ 0,0 1,112 @@
---
type: model
title: forced oscillations
---

We consider [small oscillations](small-oscillations.md) on which a variable force $F(t)$ acts, weak enough not to make the oscillations too large.

<hr>

Additionally to the `LL1/21.2` potential, the system now has the potential energy $U_e(x, t)$ resulting from the force. We expand this as a series of the small quantity $x$: $U_e(x,t)\approx U(0,t)+x[\partial U_e/\partial x]_{x=0}$. We can omit the first term because it is a function of time only, and thus is a total derivative of a function of time, which doesn't affect the Lagrangian. The second term is the the external alone is $x[\partial U_e/\partial x]_{x=0}=-xF(t)$. Thus the Lagrangian of the system is

```eq
LL1/22.1
```

According to `LL1/2.6` we have the equation of motion

```eq
LL1/22.2
```

## Periodic force

We consider the special case of a periodic force of frequency $\gamma$

```eq
LL1/22.3
```

We can easily find a particular solution to the equation of motion `LL1/22.2` using the [exponential response function](erf.md), which gives us

```eq
LL1/22.4a
```

### Resonance

in the special case of resonance $\gamma=\omega$, we use a second order [exponential reponse](erf.md)

```eq
LL1/22.5a
```

Thus, the amplitude of oscillations increases linearly with time, until the oscillations are large enough that the current model of small oscillations becomes invalid.

### Beats

Let's consider the case close to resonance $\gamma=\omega+\epsilon$, with $\epsilon\ll\omega$. The solution is a linear combition of the homogenous solitution `LL1/21.11` and the particular solution `LL1/22.4a`

```eq
LL1/22.6
```

We can consider the term $C=[A+Be^{i\epsilon t}]$ as an amplitude varying much slower than the factor $e^{i\omega t}$, because the period is much larger $\mfrac{2\pi}{\epsilon}\gg\mfrac{2\pi}{\omega}$. We find the real amplitude

```eq
LL1/22.7
```

Thus the amplitude varies periodically with frequency $\epsilon$ between the limits $|a-b|\leq c\leq|a+b|$. This phenomenon is called _beats_.

## Arbitrary force

We rewrite `LL1/22.2` as

$$ \frac{\dd{}}{\dd{t}}(\dot{x}+i\omega x)-i\omega(\dot{x}+i\omega x)=\mfrac{1}{m}F(t) $$

or

```eq
LL1/22.8
```

where

```eq
LL1/22.9
```

is a complex equation. We can solve `LL1/22.8` using the [integration factor](integration-factor.md) trick. In our case the integration factor is $\int p\dd{t}=-i\omega t$, such that

```eq
LL1/22.10
```

The function $x(t)$ is given by the imaginary part of `LL1/22.10`, divided by $\omega$.

In forced oscillations, the ernegy is of the system is not conserved, because of the energy gained from the external field. The energy of the system is, from `LL1/6.2`

```eq
LL1/22.11
```

To get the total amount of transferred energy we put the lower bound at $-\infty$ and we integrate until $t=\infty$, we get

$$ |\xi|^2=\mfrac{1}{m^2}\left|\int_{-\infty}^{\infty}F(t)e^{-i\omega t}\right|^2 $$

Subsituting, we obtain

```eq
LL1/22.12
```

which is the squared Fourier component of the force at the intrisic frequency $\omega$ of the system.

## Short acting force

if $F(t)$ act during a short amount of time compared to $1/\omega$, then we can put $e^{-i\omega t}\approx 1$. The transferered energy is simply

$$ E=\mfrac{1}{2m}\left|\int_{-\infty}^{\infty}F(t)\right|^2 $$

This result means that a force of short duration gives the system momentum $\int F\dd{t}$ without bringing a perceptible displacement.

M md/integration-factor.md => md/integration-factor.md +1 -1
@@ 20,7 20,7 @@ $$ (ye^{\int p\dd{x}})'=qe^{\int p\dd{x}} $$

Thus, integration on both sides we have

$$ y=e^{-\int p\dd{x}}\int qe^{\int p\dd{x}}\dd{x}+\text{constant} $$
$$ y=e^{-\int p\dd{x}}\left[\int qe^{\int p\dd{x}}\dd{x}+\text{constant}\right] $$

Which is a general solution for $y$


A md/small-oscillations.md => md/small-oscillations.md +53 -0
@@ 0,0 1,53 @@
---
type: model
title: small oscillations
---

We consider a [mechanical system](mechanical-system.md) near a stable equilibrium.

## In one dimension

A stable equilibrium at a position $q_0$ is where the potential $U(q)$ is a local minimum at $U(q_0)$. A movement away from this position leads to setting up a force $-\dd{U}/\dd{q}$ which tends to return the system to equilibrium. We choose a co-ordinate system where the equilibrium corresponds to $x=0$

```eq
LL1/21.1
```

We put $U(q_0)=0$ as a base energy  
We put $U'(q_0)=0$ because we don't consider asymetrical potentials  
We note $U''(q_0)=k\neq 0$  because we don't consider potential of higher order.  
Then consider the series expansion of $U(q-q_0)$, and keeping the lowest terms for small deviations of equilibrium we have

```eq
LL1/21.2
```

The kinetic energy, with one degree of freedom is of the form $\mfrac{1}{2}a(q)\dot{q}^2=\mfrac{1}{2}a(q)\dot{x}^2$. In the same approximation as above, $a(q)=a(q_0)$.

We note $a(q_0)=m$, this is the mass only if $x$ is the Cartesian co-ordinate.  

```eq
LL1/21.3
```

Using `LL1/2.6` we derive the equation of motion, which is called the harmonic oscillator

```eq
LL1/21.5
```

where 

```eq
LL1/21.6
```

The general solution the harmonic oscillator is

```eq
LL1/21.11
```

where $A=ae^{i\alpha}$ is the complex amplitude, composed of the real amplitude $a$ and the phase $\alpha$ which depend on the initial condition of the system $x(0)$ and $\dot{x}(0)$.

we note that the frequency $\omega$ doesn't depend on the inital condition, but only on the parameters of the system $k$ and $m$.

M tools/generate-graph.sh => tools/generate-graph.sh +6 -2
@@ 5,10 5,15 @@ GET=tools/get
generate_one_node() {
    node=$(basename ${1%.md})
    
    # [ "$($GET type $node)" == "problem" ] && return
    type=$(./tools/get type $node)
    [ "$type" == "math" ] && return
    [ "$type" == "problem" ] && return
    $GET graph_node $node
    title=$($GET title $node)
    [ -f graph/$node ] && for line in $(cat graph/$node); do
	type=$(./tools/get type $line)
	[ "$type" == "math" ] && return
	[ "$type" == "problem" ] && return
	printf  '    "%s" -> "%s"\n' "$($GET title $line)" "$title"
    done
}


@@ 18,7 23,6 @@ echo '    bgcolor="#ffffff00"'
echo '    node  [style="rounded,filled", shape=box]'
echo '    rankdir=RL'

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

echo "}"

M tools/index.html => tools/index.html +0 -2
@@ 1,9 1,7 @@
<link rel="stylesheet" type="text/css" href="colors.css">
<link rel="stylesheet" type="text/css" href="style.css">

<!-- <div id="container"> -->
<h1>Main Graph</h1>
<!-- </div"> -->

<body>
<div style="margin:auto"><object data=graph.svg type=image/svg+xml></object></div>

M tools/math.tex => tools/math.tex +2 -1
@@ 6,9 6,10 @@

\newcommand{\dd}[2][]{\mathrm{d}^{#1}#2}
\newcommand{\v}[1]{\mathbf{#1}}
\newcommand{\re}[1]{\text{re}\left[#1\right]}
\newcommand{\qop}[2][]{\hat{#2}\vphantom{#2}^{#1}}
\newcommand{\mfrac}[2]{\textstyle\frac{#1}{#2}\displaystyle}
\newcommand{\ptag}[2][]{\label{#1/#2}\tag*{(#2)$_{\text{#1}}$}}
\newcommand{\ptag}[2][]{\label{#1/#2}\tag*{(#2)}}

\newenvironment{nalign}{
\begin{equation}

M tools/style.css => tools/style.css +1 -2
@@ 21,12 21,11 @@ hr { display: block; height: 1px;

.tooltip .tooltiptext {
  visibility: hidden;
  width: 120px;
  background-color: #555;
  color: #fff;
  text-align: center;
  border-radius: 6px;
  padding: 5px 0;
  padding: 10px 10px;
  position: absolute;
  z-index: 1;
  bottom: 125%;

M tools/template.html => tools/template.html +2 -0
@@ 60,6 60,7 @@ $endif$
$body$

$if(sources)$
<hr>
<h3 class="sources">Sources</h3>
<ul>
$for(sources)$


@@ 67,6 68,7 @@ $for(sources)$
$endfor$
</ul>
$endif$
<hr>
<h3>See also</h3>
<center><object data=$slug$.svg type=image/svg+xml></object></center>
</body>