~jzck/physics-notes

6971d5669617d128273c86090af9c1700cd7cd0b — Jack Halford 7 months ago 535ed88
big refactor
87 files changed, 795 insertions(+), 124 deletions(-)

M Makefile
M formularies/LL2/1.1.tex
M formularies/LL2/3.1.tex
A formularies/LL2/4.4.tex
A formularies/LL2/4.5.tex
A formularies/LL2/4.6.tex
A formularies/LL2/7.2.tex
A formularies/LL2/7.3.tex
A formularies/LL2/7.4.tex
A formularies/LL2/8.1.tex
A formularies/LL2/8.2.tex
A formularies/LL2/9.1.tex
A formularies/LL2/9.10.tex
A formularies/LL2/9.11.tex
A formularies/LL2/9.12.tex
A formularies/LL2/9.13.tex
A formularies/LL2/9.14.tex
A formularies/LL2/9.15.tex
A formularies/LL2/9.16.tex
A formularies/LL2/9.17.tex
A formularies/LL2/9.18.tex
A formularies/LL2/9.19.tex
A formularies/LL2/9.2.tex
A formularies/LL2/9.20.tex
A formularies/LL2/9.3.tex
A formularies/LL2/9.4.tex
A formularies/LL2/9.5.tex
A formularies/LL2/9.6.tex
A formularies/LL2/9.7.tex
A formularies/LL2/9.8.tex
A formularies/LL2/9.9.tex
M formularies/LL3/15.3.tex
D md/four-velocity.md
M notes.md
R md/lhopital.md => src/18.01x/lhopital.md
A src/18.01x/taylor-series.md
A src/18.02sc/lagrange-multipliers.md
A src/18.02sc/partial-derivative.md
R md/erf.md => src/18.03x/erf.md
R md/integration-factor.md => src/18.03x/integration-factor.md
R md/ammonia-maser.md => src/FLP3/ammonia-maser.md
R img/flp3-9-1.svg => src/FLP3/img/9-1.svg
R md/angular-momentum.md => src/LL1/angular-momentum.md
R md/central-field.md => src/LL1/central-field.md
R md/coplanar-double-pendulum.md => src/LL1/coplanar-double-pendulum.md
R md/damped-oscillations.md => src/LL1/damped-oscillations.md
R md/disintegration.md => src/LL1/disintegration.md
R md/elastic-collisions.md => src/LL1/elastic-collisions.md
R md/energy.md => src/LL1/energy.md
R md/forced-oscillations.md => src/LL1/forced-oscillations.md
R md/galileo-relativity.md => src/LL1/galileo-relativity.md
R md/inertia-tensor.md => src/LL1/inertia-tensor.md
R md/keplers-problem.md => src/LL1/keplers-problem.md
R md/lagrangian.md => src/LL1/lagrangian.md
R md/mechanical-similarity.md => src/LL1/mechanical-similarity.md
R md/momentum.md => src/LL1/momentum.md
R md/natural-reference-frame.md => src/LL1/natural-reference-frame.md
R md/particle.md => src/LL1/particle.md
R md/rigid-body.md => src/LL1/rigid-body.md
R md/scattering.md => src/LL1/scattering.md
R md/small-oscillations.md => src/LL1/small-oscillations.md
R md/solid-angular-momentum.md => src/LL1/solid-angular-momentum.md
R md/two-body-problem.md => src/LL1/two-body-problem.md
R md/einstein-relativity.md => src/LL2/einstein-relativity.md
R md/four-vectors.md => src/LL2/four-vectors.md
A src/LL2/four-velocity.md
R md/intervals.md => src/LL2/intervals.md
R md/lorentz-transformation.md => src/LL2/lorentz-transformation.md
A src/LL2/proper-length.md
R md/proper-time.md => src/LL2/proper-time.md
R problems/LL2/6.1.md => src/LL2/pset6.md
A src/LL2/relativistic-mechanics.md
R md/hamiltonian-operator.md => src/LL3/hamiltonian-operator.md
R md/hermitian-eigenfunctions.md => src/LL3/hermitian-eigenfunctions.md
R md/momentum-operator.md => src/LL3/momentum-operator.md
R md/quantum-operator.md => src/LL3/quantum-operator.md
R md/translation-operator.md => src/LL3/translation-operator.md
R md/wave-function.md => src/LL3/wave-function.md
A src/index.md
A src/index.tpl
M tools/generate-graph.sh
M tools/generate-local-graph.sh
M tools/get
D tools/index.html
M tools/math.tex
M tools/pf-filter.py
M tools/template.html
M Makefile => Makefile +50 -21
@@ 3,20 3,28 @@ SHELL := bash
.SHELLFLAGS := -eu -o pipefail -c
.RECIPEPREFIX = >

MD	=	$(shell ls -1 md/*.md)
GRAPH	=	$(subst md/,graph/,$(MD:.md=))
SVG	=	$(subst md/,www-data/,$(MD:.md=.svg))
TEX	=	$(subst md/,www-data/,$(MD:.md=.tex))
HTML	=	$(subst md/,www-data/,$(MD:.md=.html)) www-data/index.html www-data/style.css
NODES		=	$(shell ls -1 src/*/*.md)
DOMAINS		=	$(shell find src -mindepth 1 -maxdepth 1 -type d)
GRAPH_DATA	=	$(subst src/,graph/,$(NODES:.md=.txt))
GRAPH_DOT	=	$(subst src/,graph/,$(NODES:.md=.dot))
GRAPH_LOCAL	=	$(subst src/,www-data/,$(NODES:.md=.svg))
# GRAPH_LOCAL	=	www-data/LL1/energy.svg
GRAPH_DOMAIN	=	$(subst src/,www-data/,$(addsuffix /index.svg, $(DOMAINS)))
# TEX		=	$(subst md/,www-data/,$(MD:.md=.tex))
HTML		=	$(subst src/,www-data/,$(NODES:.md=.html)) www-data/index.html
DIRS		=	$(subst src/,graph/,$(DOMAINS))
DIRS		+=	$(subst src/,www-data/,$(DOMAINS))
IMG_DIRS	=	$(addsuffix /img,$(subst src/,www-data/,$(DOMAINS)))


ifeq ($(PROD),yes)
PROD_OPT=	"-V prod:yes"
PROD_OPT	=	"-V prod:yes"
endif

all: $(HTML) $(SVG) www-data/graph.svg
all: $(DIRS) $(HTML) $(GRAPH_LOCAL) $(GRAPH_DOMAIN) $(IMG_DIRS) www-data/colors.css www-data/style.css
.PRECIOUS: $(GRAPH_DATA) $(GRAPH_DOT)

PROD = yes
www-data/%.html: md/%.md | www-data/colors.css www-data
www-data/%.html: src/%.md $(WWW_DIRS)
> @printf '$@\n'
> pandoc	\
		-s --filter tools/pf-filter.py \


@@ 32,22 40,43 @@ www-data/%.html: md/%.md | www-data/colors.css www-data
		$< \
		-o $@ &

www-data/index.html: | www-data
> cp tools/index.html $@
www-data/style.css: tools/style.css | www-data
www-data/index.html: src/index.tpl src/index.md
> pandoc \
	--template src/index.tpl \
	src/index.md \
	-o $@

www-data/style.css: tools/style.css
> cp tools/style.css $@
www-data/colors.css: | www-data

www-data/colors.css:
> ./tools/get bg_css >$@
www-data/%.svg: graph/% | www-data
> dot -Tsvg -o $@ <(./tools/generate-local-graph.sh $<) &
www-data/graph.svg: $(GRAPH) | www-data
> dot -Tsvg -o $@ <(./tools/generate-graph.sh)

graph/%: md/%.md | graph
> @tools/get graph $(^F)>$@
# local graph.svg
www-data/%.svg: graph/%.dot
> dot -Tsvg -o $@ $< &

# domain index.svg
www-data/%/index.svg: $(GRAPH_DATA)
> dot -Tsvg -o $@ <(./tools/generate-graph.sh $@) &

# graph data
graph/%.txt: src/%.md
> tools/get graph $< >$@

# local graph.dot
graph/%.dot: graph/%.txt 
> ./tools/generate-local-graph.sh $< >$@

# domain graph dot
www-data/%/index.svg: $(GRAPH_DATA)
> dot -Tsvg -o $@ <(./tools/generate-graph.sh $@) &

$(DIRS):
> mkdir -p $@

www-data graph:
> @mkdir -p $@
$(IMG_DIRS):
> cp -rf $(subst www-data/,src/,$@) $@

clean:
> rm -rf www-data graph

M formularies/LL2/1.1.tex => formularies/LL2/1.1.tex +1 -1
@@ 1,1 1,1 @@
c=2.99793\text{ cm/s}
c=299\ 792\ 458\text{ m/s}\qquad\text{(exact value)}

M formularies/LL2/3.1.tex => formularies/LL2/3.1.tex +1 -1
@@ 1,1 1,1 @@
\dd{t'}=\frac{\dd{s}}{c}=\dd{t}\sqrt{1-\frac{v^2}{c^2}}
\dd{s}=c\dd{t'}=c\dd{t}\sqrt{1-\frac{v^2}{c^2}}

A formularies/LL2/4.4.tex => formularies/LL2/4.4.tex +4 -0
@@ 0,0 1,4 @@
x=x'+Vt',\qquad
y=y',\qquad
z=z',\qquad
t=t'+Vx'/c^2

A formularies/LL2/4.5.tex => formularies/LL2/4.5.tex +1 -0
@@ 0,0 1,1 @@
l=l_0\sqrt{1-\frac{V^2}{c^2}}

A formularies/LL2/4.6.tex => formularies/LL2/4.6.tex +1 -0
@@ 0,0 1,1 @@
\mathscr{V}=\mathscr{V}_0\sqrt{1-\frac{V^2}{c^2}}

A formularies/LL2/7.2.tex => formularies/LL2/7.2.tex +4 -0
@@ 0,0 1,4 @@
u^i=\left(
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}},
\frac{\v{v}}{c\sqrt{1-\frac{v^2}{c^2}}}
\right)

A formularies/LL2/7.3.tex => formularies/LL2/7.3.tex +1 -0
@@ 0,0 1,1 @@
u^iu_i=1

A formularies/LL2/7.4.tex => formularies/LL2/7.4.tex +0 -0
A formularies/LL2/8.1.tex => formularies/LL2/8.1.tex +1 -0
@@ 0,0 1,1 @@
S=-mc\int_a^b\dd{s}

A formularies/LL2/8.2.tex => formularies/LL2/8.2.tex +1 -0
@@ 0,0 1,1 @@
L=-mc^2\sqrt{1-\frac{v^2}{c^2}}

A formularies/LL2/9.1.tex => formularies/LL2/9.1.tex +1 -0
@@ 0,0 1,1 @@
\v{p}=\frac{m\v{v}}{\sqrt{1-\frac{v^2}{c^2}}}

A formularies/LL2/9.10.tex => formularies/LL2/9.10.tex +1 -0
@@ 0,0 1,1 @@
\delta S=-mc\left[u_i\delta x^i\right]_a^b+mc\int_a^b\delta x^i\frac{\dd{u}_i}{\dd{s}}\dd{s}

A formularies/LL2/9.11.tex => formularies/LL2/9.11.tex +1 -0
@@ 0,0 1,1 @@
\delta S=-mcu_i\delta x^i

A formularies/LL2/9.12.tex => formularies/LL2/9.12.tex +1 -0
@@ 0,0 1,1 @@
p_i=-\frac{\partial S}{\partial x^i}

A formularies/LL2/9.13.tex => formularies/LL2/9.13.tex +1 -0
@@ 0,0 1,1 @@
p^i=(\mathscr{E}/c,\v{p})

A formularies/LL2/9.14.tex => formularies/LL2/9.14.tex +1 -0
@@ 0,0 1,1 @@
p^i=mcu^i

A formularies/LL2/9.15.tex => formularies/LL2/9.15.tex +4 -0
@@ 0,0 1,4 @@
p_x=\frac{p_x'+\frac{V}{c^2}\mathscr{E'}}{\sqrt{1-\frac{V^2}{c^2}}},\qquad
p_y=p_y',\qquad
p_z=p_z',\qquad
\mathscr{E}=\frac{\mathscr{E}'+Vp_x'}{\sqrt{1-\frac{V^2}{c^2}}},\qquad

A formularies/LL2/9.16.tex => formularies/LL2/9.16.tex +1 -0
@@ 0,0 1,1 @@
p^ip_i=m^2c^2

A formularies/LL2/9.17.tex => formularies/LL2/9.17.tex +1 -0
@@ 0,0 1,1 @@
g^i=\frac{\dd{p}^i}{\dd{s}}=mc\frac{\dd{u}^i}{\dd{s}}

A formularies/LL2/9.18.tex => formularies/LL2/9.18.tex +1 -0
@@ 0,0 1,1 @@
g^i=\left(\frac{\v{f}\cdot\v{v}}{c^2\sqrt{1-\frac{v^2}{c^2}}},\frac{\v{f}}{c\sqrt{1-\frac{v^2}{c^2}}}\right)

A formularies/LL2/9.19.tex => formularies/LL2/9.19.tex +7 -0
@@ 0,0 1,7 @@
\frac{\partial S}{\partial x_i}
\frac{\partial S}{\partial x^i}
\equiv
g^{ik}
\frac{\partial S}{\partial x^k}
\frac{\partial S}{\partial x^i}
=m^2c^2

A formularies/LL2/9.2.tex => formularies/LL2/9.2.tex +1 -0
@@ 0,0 1,1 @@
\frac{\dd{p}}{\dd{t}}=\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}\frac{\dd{\v{v}}}{\dd{t}}

A formularies/LL2/9.20.tex => formularies/LL2/9.20.tex +5 -0
@@ 0,0 1,5 @@
\frac{1}{c^2}\left(\frac{\partial S}{\partial t}\right)^2
-\left(\frac{\partial S}{\partial x}\right)^2
-\left(\frac{\partial S}{\partial y}\right)^2
-\left(\frac{\partial S}{\partial z}\right)^2
=m^2c^2

A formularies/LL2/9.3.tex => formularies/LL2/9.3.tex +1 -0
@@ 0,0 1,1 @@
\frac{\dd{p}}{\dd{t}}=\frac{m}{\left(1-\frac{v^2}{c^2}\right)^{\mfrac{3}{2}}}\frac{\dd{\v{v}}}{\dd{t}}

A formularies/LL2/9.4.tex => formularies/LL2/9.4.tex +1 -0
@@ 0,0 1,1 @@
\mathscr{E}=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}

A formularies/LL2/9.5.tex => formularies/LL2/9.5.tex +1 -0
@@ 0,0 1,1 @@
\mathscr{E}=mc^2

A formularies/LL2/9.6.tex => formularies/LL2/9.6.tex +1 -0
@@ 0,0 1,1 @@
\frac{\mathscr{E^2}}{c^2}=p^2+m^2c^2

A formularies/LL2/9.7.tex => formularies/LL2/9.7.tex +1 -0
@@ 0,0 1,1 @@
\mathscr{H}=c\sqrt{p^2+m^2c^2}

A formularies/LL2/9.8.tex => formularies/LL2/9.8.tex +1 -0
@@ 0,0 1,1 @@
\v{p}=\mathscr{E}\frac{\v{v}}{c^2}

A formularies/LL2/9.9.tex => formularies/LL2/9.9.tex +1 -0
@@ 0,0 1,1 @@
p=\frac{\mathscr{E}}{c}

M formularies/LL3/15.3.tex => formularies/LL3/15.3.tex +3 -3
@@ 1,3 1,3 @@
\qop{p}_x\qop{p}y-\qop{p}_y\qop{p}x=0,\qquad
\qop{p}_x\qop{p}z-\qop{p}_z\qop{p}x=0,\qquad
\qop{p}_y\qop{p}z-\qop{p}_z\qop{p}y=0
\qop{p}_x\qop{p}_y-\qop{p}_y\qop{p}_x=0,\qquad
\qop{p}_x\qop{p}_z-\qop{p}_z\qop{p}_x=0,\qquad
\qop{p}_y\qop{p}_z-\qop{p}_z\qop{p}_y=0

D md/four-velocity.md => md/four-velocity.md +0 -12
@@ 1,12 0,0 @@
---
title: four velocity
type: theory
---

From the ordinary three-dimensional velocity vecotr one can form a [four-vector](four-vectors.md). We define the four-velocity as

```eq
LL2/7.1
```

and `LL2/3.1`

M notes.md => notes.md +1 -0
@@ 4,4 4,5 @@ I note the transposed operator ^t instead of a tilde hat

## Mistakes

LL2. expression before 9.10, denominator is ds not sqrt(ds)
LL3. between 4.6 and 4.7, I think that fg-gf is anti-Hermitian iff f and g are Hermitian

R md/lhopital.md => src/18.01x/lhopital.md +0 -0
A src/18.01x/taylor-series.md => src/18.01x/taylor-series.md +12 -0
@@ 0,0 1,12 @@
---
title: Taylor series
type: math
---

For radius of convergence calculations see notes of MIT 18.01x

$\alpha\choose k$ is the <a href=https://en.wikipedia.org/wiki/Binomial_coefficient>binomial coefficient</a>.

$$
(1+x)^\alpha=\sum_{k=0}^\infty {\alpha\choose k}x^k=1+\alpha x+\frac{\alpha(\alpha-1)}{k!}x^2+\text{...}
$$

A src/18.02sc/lagrange-multipliers.md => src/18.02sc/lagrange-multipliers.md +46 -0
@@ 0,0 1,46 @@
---
title: Lagrange multipliers
type: math
source: mit 18.02sc
---

Consider a function of multiple variables $f(x,y,z)$, where the variables are constrained by $g(x,y,z)=c$.

Similarly to how the extremum of a single variable are located at zero derivative points.

At a constrained min/max, the first order change in $f$ in the direction of $g$ is 0.

Therefore we must have, if we note $\v{u}$ any tangeant of g.

$$
\frac{\dd{f}}{\dd{s}}\bigg|_{\hat{u}}=\nabla f\cdot\hat{u}=0
$$

which means

$$
\nabla f\perp\hat{u}
$$

but, because $\v{u}$ is the direction of g, from the definition of the gradient we also have

$$
\nabla g\perp\hat{u}
$$

But because $g$ is a single equation, it's solutions form a hyper-plane (subspace of dimension $n-1$). Therefore the set of vector $\v{u}$ form a hyper-plane. By definition there is only one perpendicular direction to a hyper-plane, therefore

$$
\nabla f\parallel\nabla g
$$

Finally, we find the $n+1$ equations:

$$
\begin{cases}
\nabla f=\lambda\nabla g\\
g=c
\end{cases}
$$

to which the solutions $x_1,x_2,...,x_n,\lambda$ are the locals extremum of $f$.

A src/18.02sc/partial-derivative.md => src/18.02sc/partial-derivative.md +72 -0
@@ 0,0 1,72 @@
---
title: partial derivative
type: math
source: mit 18.02sc
---

When dealing with functions of mutliple variables $f:\mathbb{R}^n\to\mathbb{R}$. We may ask about the rate of change of $f$ in terms of "direction". We will take the example of a function 3 variables $f(x,y,z)$.

## partial derivative

Formally, we define the partial derivative of $f$ with respect to $x$ by

$$
\frac{\partial f}{\partial x}=f_x(x,y,z)=\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y,z)}{\Delta x}
$$

In practive, we differentiate as we usually do with one variable, while keeping all the other variables constant.

## gradient vector

The rate of change of $f$ in a given direction $\v{u}$ is simply the scalar product

$$
\frac{\dd{f}}{\dd{s}}\bigg|_{\v{u}}=\nabla f\cdot \hat{u}
$$

We have the main formula for implicit differentiation

$$
\dd{f}=\nabla f\cdot \dd{\v{r}}=
\frac{\partial f}{\partial x}\dd{x}+
\frac{\partial f}{\partial y}\dd{y}+
\frac{\partial f}{\partial z}\dd{z}
$$

## Constrained partial differentials

When $(x,y,z)$ are constrained (not independant). We may get different values for $\partial f/\partial x$ whether we set $y$ constant of $z$ constant, or a combination with some other function $u(y,z)$ constant.

#### short example

let $f(x,y)=x+y$, we make the change of variable $u=x,\quad v=x+y$, we then have $f(u,v)=2u+v$.

although we have $u=x$, we find $\frac{\partial f}{\partial x}=1\neq 2=\frac{\partial f}{\partial u}$.

#### clearer notation

Evidently $\frac{\partial f}{\partial x}$ _has no meaning_ when $x$ is contrained to the other variables. We must explicit what we keep constant:

$$
\left(\frac{\partial f}{\partial x}\right)_y=\left(\frac{\partial f}{\partial u}\right)_y=1
$$
$$
\left(\frac{\partial f}{\partial x}\right)_v=\left(\frac{\partial f}{\partial u}\right)_v=2
$$

Another explicit way to write the same thing:

$$
\dd{f}=\dd{x}+\dd{y}
$$
$$
\dd{f}=2\dd{u}+\dd{v}
$$

In general, when we write

$$
\dd{f}=f_x\dd{x}+f_y\dd{y}+f_z\dd{z}
$$

we have no ambiguity about $f_x=\frac{\partial f}{\partial x}=\left(\frac{\partial f}{\partial x}\right)_{(y,z)}$

R md/erf.md => src/18.03x/erf.md +0 -0
R md/integration-factor.md => src/18.03x/integration-factor.md +0 -0
R md/ammonia-maser.md => src/FLP3/ammonia-maser.md +2 -3
@@ 1,10 1,9 @@
---
title: the ammonia maser
type: model
sources: FLP3-9
---

![two base states of the ammonia molecule (source: FLP 3, fig 9-1)](../img/flp3-9-1.svg)
![two base states of the ammonia molecule (source: FLP 3, fig 9-1)](img/9-1.svg)

### Experimental data



@@ 14,7 13,7 @@ In order to excite an electron inside an atom, it requires photons in the optica

The geometry of the ammonia molecule $NH3$ is a tetrahedron, in this configuration, the hydrogen triangle can be either side of the nitrogren atom. Because of the tunnel effect, we can model the ammonia molecule as a two state system $\ket{\Psi}=C_L\ket{L}+C_R\ket{R}$.

Because of the symmetry of the system, we can write the [Hamiltonian](hamiltonian-operator.md) in the $L,R$ basis
Because of the symmetry of the system, we can write the [Hamiltonian](LL3/hamiltonian-operator) in the $L,R$ basis

$$ \qop{H}=\begin{pmatrix} E& -A\\ -A& E \end{pmatrix} $$


R img/flp3-9-1.svg => src/FLP3/img/9-1.svg +0 -0
R md/angular-momentum.md => src/LL1/angular-momentum.md +1 -1
@@ 26,7 26,7 @@ LL1/9.2
LL1/5
```

We now write the condition that the [Lagrangian](lagrangian.md) is unchanged by rotiation
We now write the condition that the [Lagrangian](LL1/lagrangian.md) is unchanged by rotiation

$$
\delta L=\sum_a \left(

R md/central-field.md => src/LL1/central-field.md +1 -1
@@ 11,7 11,7 @@ This potential generates a radial force $\v{F}=-\frac{\partial U(r)}{\partial\v{

## Conservation of $M_z$

We have shown that the [angular-momentum](angular-momentum.md) $\v{M}=\v{r}\times\v{p}$ of a system in a central field is conserved, because of the symmetry of the field around the center.
We have shown that the [angular-momentum](LL1/angular-momentum.md) $\v{M}=\v{r}\times\v{p}$ of a system in a central field is conserved, because of the symmetry of the field around the center.

Since $\v{M}$ is perpendicular to $\v{r}$, the constancy of $\v{M}$ show that the radius vector $\v{r}$ must remain must remain in the plane perpendicular to $\v{M}$. (like planets in the solar system, rotate each on a single plane)


R md/coplanar-double-pendulum.md => src/LL1/coplanar-double-pendulum.md +1 -1
@@ 3,7 3,7 @@ title: coplanar double pendulum
type: problem
---

Find the [Lagrangian](lagrangian.md) of a coplanar double pendulum
Find the [Lagrangian](LL1/lagrangian.md) of a coplanar double pendulum

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

R md/damped-oscillations.md => src/LL1/damped-oscillations.md +1 -1
@@ 9,7 9,7 @@ The full state of the system now needs knowledge of the motion of the medium and

<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.
In the special case of [small oscillations](LL1/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`)


R md/disintegration.md => src/LL1/disintegration.md +1 -1
@@ 14,4 14,4 @@ type: model
\begin{equation} T_{10,\text{max}}=(M-m_1)\epsilon/M \end{equation}

## Assumptions
 - [Two body problem](two-body-problem.md)
 - [Two body problem](LL1/two-body-problem.md)

R md/elastic-collisions.md => src/LL1/elastic-collisions.md +1 -1
@@ 27,4 27,4 @@ v_2'&=\frac{2m_1v}{m_1+m_2}\sin\mfrac{1}{2}\chi
\begin{equation} v_1'=v\cos\mfrac{1}{2}\chi,\qquad v_2'=v\sin\mfrac{1}{2}\chi \end{equation}

## System
 - [Two body problem](two-body-problem.md)
 - [Two body problem](LL1/two-body-problem.md)

R md/energy.md => src/LL1/energy.md +1 -1
@@ 7,7 7,7 @@ Here we find that the conservation of energy arises from the _homogeneity of tim

<hr>

Let us suppose a closed system, we can then write the total time derivative of the [Lagrangian](lagrangian.md) as
Let us suppose a closed system, we can then write the total time derivative of the [Lagrangian](LL1/lagrangian.md.md) as

$$
\frac{\dd{L}}{\dd{t}}=\sum_i\frac{\partial L}{\partial q_i}\dot{q}_i+\sum_i\frac{\partial L}{\partial \dot{q}_i}\ddot{q}_i

R md/forced-oscillations.md => src/LL1/forced-oscillations.md +4 -4
@@ 3,7 3,7 @@ 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.
We consider [small oscillations](LL1/small-oscillations) on which a variable force $F(t.md)$ acts, weak enough not to make the oscillations too large.

<hr>



@@ 27,7 27,7 @@ We consider the special case of a periodic force of frequency $\gamma$
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
We can easily find a particular solution to the equation of motion `LL1/22.2` using the [exponential response function](18.03x/erf.md), which gives us

```eq
LL1/22.4a


@@ 35,7 35,7 @@ LL1/22.4a

### Resonance

in the special case of resonance $\gamma=\omega$, we use a second order [exponential reponse](erf.md)
in the special case of resonance $\gamma=\omega$, we use a second order [exponential reponse](18.03x/erf.md)

```eq
LL1/22.5a


@@ 77,7 77,7 @@ where
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
is a complex equation. We can solve `LL1/22.8` using the [integration factor](18.03x/integration-factor.md) trick. In our case the integration factor is $\int p\dd{t}=-i\omega t$, such that

```eq
LL1/22.10

R md/galileo-relativity.md => src/LL1/galileo-relativity.md +0 -0
R md/inertia-tensor.md => src/LL1/inertia-tensor.md +1 -1
@@ 48,7 48,7 @@ LL1/32.11
```

## Model
 - [Rigid body](rigid-body.md)
 - [Rigid body](LL1/rigid-body.md)

## Properties
 - rank 2 tensor

R md/keplers-problem.md => src/LL1/keplers-problem.md +1 -1
@@ 3,7 3,7 @@ title: Kepler's problem
type: model
---

Kepler's problem is a special case of the [two body problem](two-body-problem.md) with a $k=-1$ potential (e.g. Newtonian or Coulombian)
Kepler's problem is a special case of the [two body problem](LL1/two-body-problem) with a $k=-1$ potential (e.g. Newtonian or Coulombian.md)

<hr>


R md/lagrangian.md => src/LL1/lagrangian.md +3 -3
@@ 61,9 61,9 @@ $$

i.e. they only differ by a quantity which is constant, which means $\delta{S}=\delta{S'}$, the equations of motion will be unchanged for these two systems $L$ and $L'$, i.e. these two systems are one and the same. Thus the Lagrangian is defined to within an additive total time derivative of any function of co-ordinates and time $\dd{f(q,t)}/\dd{t}$.

## For a free [particle](particle.md)
## For a free [particle](LL1/particle.md)

[Galileo's relativity](galileo-relativity.md) tells us that the Lagrangian of a particle only depends on the square of it's velocity `LL1/3.1`. Let's consider two inertial frames of reference $K$ and $K'$, moving with a velocity difference $\mathbf{\epsilon}$, i.e. $\mathbf{v'}=\mathbf{v}+\mathbf{\epsilon}$. Since the two frames are inertial, we expect the laws of physics to be the unchanged, i.e. we expect to Lagrangian $L$ and $L'$ to be equivalent.
[Galileo's relativity](LL1/galileo-relativity.md) tells us that the Lagrangian of a particle only depends on the square of it's velocity `LL1/3.1`. Let's consider two inertial frames of reference $K$ and $K'$, moving with a velocity difference $\mathbf{\epsilon}$, i.e. $\mathbf{v'}=\mathbf{v}+\mathbf{\epsilon}$. Since the two frames are inertial, we expect the laws of physics to be the unchanged, i.e. we expect to Lagrangian $L$ and $L'$ to be equivalent.

We have $L'=L(v'^2)=L(v^2+2\mathbf{v}\cdot\epsilon+\epsilon^2)$. We expand this expression is powers of $\epsilon$ and find



@@ 135,7 135,7 @@ LL1/5.1
```
We call the first term $T$ the _kinetic energy_, and the second term $U$ is the _potential energy_.

The fact that the potential energy $U$ depends only on the positions of the particles means the particles interact instantaneously, this is related to the fact that we have placed ourselves in [Galilean relativity](galileo-relativity.md), where time is the same in all reference frames.
The fact that the potential energy $U$ depends only on the positions of the particles means the particles interact instantaneously, this is related to the fact that we have placed ourselves in [Galilean relativity](LL1/galileo-relativity.md), where time is the same in all reference frames.

The form `LL1/5.1` shows, that time is also _isotropic_, i.e. $L(q,\dot{q},t)=L(q,\dot{q},-t)$, which means that the reverse motion is equally as possible as the forward motion. Classical mechanical systems are reversible.


R md/mechanical-similarity.md => src/LL1/mechanical-similarity.md +1 -1
@@ 3,7 3,7 @@ title: mechanical similarity
type: theory
---

We have found that the equations of motion are unchanged by multiplication of the [Lagrangian](lagrangian.md) by any constant. This allows us to determine some preperties of motion without necessarily solving the equations.
We have found that the equations of motion are unchanged by multiplication of the [Lagrangian](LL1/lagrangian.md) by any constant. This allows us to determine some preperties of motion without necessarily solving the equations.

Let us consider a homogeneous potential $U(q)$


R md/momentum.md => src/LL1/momentum.md +1 -1
@@ 8,7 8,7 @@ Here we find that the conservation of momentum arises from the _homogeneity of s

The homogeneity of space tells us that if we displace all the particles in a closed system by the same amount $\epsilon$, the mechanical properties should be conserved. Note that this displacement will change the Lagrangian in the case of an open system because the external field will be relatively displaced.

We can write this change in [Lagrangian](lagrangian.md) as
We can write this change in [Lagrangian](LL1/lagrangian.md) as

$$
\delta L=\sum_a \frac{\partial L}{\partial \mathbf{r}_a}\delta\mathbf{r}_a

R md/natural-reference-frame.md => src/LL1/natural-reference-frame.md +2 -2
@@ 10,7 10,7 @@ We generalise concepts of positon, velocity and energy to systems with multiple 

## Rest frame

The [momentum](momentum.md) of a system takes different values in different frames of reference.
The [momentum](LL1/momentum.md) of a system takes different values in different frames of reference.

Let's consider a frame $K'$ that moves with velocity $\mathbf{V}$ relative to a frame $K$, then we have the velocity of a particle $\mathbf{v}_a=\mathbf{v}'_a+\mathbf{V}$. We can then relate the momenta $\mathbf{P}$ and $\mathbf{P}'$.



@@ 48,7 48,7 @@ $\mathbf{R}$ is a natural definition for the "position of the system".

### Energy of the system (rest energy)

We call the [energy](energy.md) in the rest frame the _internal energy_ $E_i$ of thesystem. The energy of a system moving as a whole can be written
We call the [energy](LL1/energy.md) in the rest frame the _internal energy_ $E_i$ of thesystem. The energy of a system moving as a whole can be written

```eq
LL1/8.4

R md/particle.md => src/LL1/particle.md +0 -0
R md/rigid-body.md => src/LL1/rigid-body.md +1 -1
@@ 3,7 3,7 @@ title: rigid body
type: model
---

We consider a group of [particles](particle.md) which are firmly attached, with no deformation. Movements occurs across the whole solid without delay.
We consider a group of [particles](LL1/particle.md) which are firmly attached, with no deformation. Movements occurs across the whole solid without delay.

```eq
LL1/31.1

R md/scattering.md => src/LL1/scattering.md +1 -1
@@ 14,4 14,4 @@ type: model
\begin{equation} \dd{\sigma}=\frac{\rho}{\sin\chi}\left|\frac{\dd{\rho}}{\dd{\chi}}\right|\dd{o} \end{equation}

## Assumptions
 - [Two body problem](two-body-problem.md)
 - [Two body problem](LL1/two-body-problem.md)

R md/small-oscillations.md => src/LL1/small-oscillations.md +1 -1
@@ 3,7 3,7 @@ type: model
title: small oscillations
---

We consider a [mechanical system](particle.md) near a stable equilibrium.
We consider a [mechanical system](LL1/particle.md) near a stable equilibrium.

## In one dimension


R md/solid-angular-momentum.md => src/LL1/solid-angular-momentum.md +1 -1
@@ 4,4 4,4 @@ type: theory
---

## Model
 - [Rigid bodies](rigid-body.md)
 - [Rigid bodies](LL1/rigid-body.md)

R md/two-body-problem.md => src/LL1/two-body-problem.md +1 -1
@@ 3,7 3,7 @@ title: two body problem
type: model
---

We show that a two body problem can be formally reduced to that of a one body plus a [central field](central-field.md)
We show that a two body problem can be formally reduced to that of a one body plus a [central field](LL1/central-field.md)

<hr>


R md/einstein-relativity.md => src/LL2/einstein-relativity.md +6 -9
@@ 3,7 3,7 @@ title: Einstein's relativity
type: model
---

We can deduce a whole new theory of mechanics by changing one assumption: interactions between particles doesn't happen instantaneously.
We can deduce a whole new theory of mechanics by changing one assumption: interactions between particles don't happen instantaneously.

<hr>



@@ 23,18 23,15 @@ From the principle of relativity, it follows that the velocity of propagation of
LL2/1.1
```

The large value of this velocity shows that Galileo's is sufficiently accurate in most cases. The formal passage to Galileo's relativity, in which interaction is instantaneous is done by passing to the limit $c\to\infty$.

## Consequences


The large value of this velocity shows that Galileo's is sufficiently accurate in most cases. The formal passage to Galileo's relativity, in which interaction is instantaneous is done by passing to the limit $c\to\infty$.

## Summary of differences with Galilean relativity
<!-- ## Summary of differences with Galilean relativity -->

$K$ and $K'$ are two inertial frames of reference  
<!-- $K$ and $K'$ are two inertial frames of reference -->  
  
_relative_: is not necessarily the same in $K$ as in $K'$.  
_absolute_: is the same in $K$ and in $K'$.
<!-- _relative_: is not necessarily the same in $K$ as in $K'$. -->  
<!-- _absolute_: is the same in $K$ and in $K'$. -->

<!-- --- | Classical mechanics | Relativistic mechanics -->
<!-- --- | --- | --- -->

R md/four-vectors.md => src/LL2/four-vectors.md +2 -2
@@ 1,6 1,6 @@
---
title: four vectors
type: theory
type: math
---

The coordinates of an event $(ct,x,y,z)$ can be considered as the components of a four-dimensional radius vector. We denote its components by $x^i$.


@@ 12,7 12,7 @@ x^2=y,\qquad
x^3=z
$$

We generalize the concept to any four quantities $A^0,A^1,A^2,A^3$ which transform according to the [Lorentz transform](lorentz-transformation.md)
We generalize the concept to any four quantities $A^0,A^1,A^2,A^3$ which transform according to the [Lorentz transform](LL2/lorentz-transformation.md)

```eq
LL2/6.1

A src/LL2/four-velocity.md => src/LL2/four-velocity.md +52 -0
@@ 0,0 1,52 @@
---
title: four velocity
type: theory
---

From the ordinary three-dimensional velocity vecotr one can form a [four-vector](LL2/four-vectors.md). We define the four-velocity as

```eq
LL2/7.1
```

from `LL2/3.1`, we have

$$
\dd{s}=c\dd{t}\sqrt{1-\frac{v^2}{c^2}}
$$

where $v$ is the ordinary three-dimension velocity of the particle. Thus

$$
u^1=\frac{\dd{x}^1}{\dd{s}}=\frac{\dd{x}}{c\dd{t}\sqrt{1-\frac{v^2}{c^2}}}=\frac{v_x}{c\sqrt{1-\frac{v^2}{c^2}}}
$$

etc. Thus

```eq
LL2/7.2
```

We note that the four-velocity is a dimensionless quantity.

Furthermore, the components of the four-velocity are no independant. Noting that \dd{x}_i\dd{x}^i=\dd{s}^2, we have

```eq
LL2/7.3
```

Geometrically, $u^i$ is a unit four-vector tangent to the world line of the particle.

### four-acceleration

Similarly to the definition of the four-velocity, the second derivative

$$
w^i=\frac{\dd[2]{x^i}}{\dd{s}^2}=\frac{\dd{u^i}}{\dd{s}}
$$

may be called the four-acceleration. Differentiating `LL2/7.3` we find

```eq
LL2/7.4
```

R md/intervals.md => src/LL2/intervals.md +1 -1
@@ 11,7 11,7 @@ Let us evaluate the interval between two events in two different reference frame
LL2/2.1
```

In the second frame $K'$, the velocity of light is the same $c$, this is a postulate of [relativistic mechanics](einstein-relativity.md). We have similarly
In the second frame $K'$, the velocity of light is the same $c$, this is a postulate of [relativistic mechanics](LL2/einstein-relativity.md). We have similarly

```eq
LL2/2.2

R md/lorentz-transformation.md => src/LL2/lorentz-transformation.md +29 -5
@@ 5,13 5,13 @@ type: theory

Let us consider a frame $K'$ moving with constant velocity $V$ relative to $K$.

The question arises of how do we transform from coordinates of one frame of reference $K$ to another frame of reference $K'$ and vice-versa. In classical mechanics, the times being equal we have
The question arises of how do we transform from coordinates of one frame of reference $K'$ to another frame of reference $K$ and vice-versa. In classical mechanics, the times being equal we have

```eq
LL2/4.1
```

In the context of special relativity, we obtain the transformation between two frames of references by using the fact that the [interval](intervals.md) $s$ between two events must be the same in every inertial frame.
In the context of special relativity, we obtain the transformation between two frames of references by using the fact that the [interval](LL2/intervals.md) $s$ between two events must be the same in every inertial frame.

The interval can be regarded as a distance of two points in a four dimensional system of coordinates.



@@ 19,7 19,7 @@ We neglect parallel displacements because these operations leave the distances u

We are left with rotations in the four-dimensional space $(x,y,z,ct)$. We can resolve these into 6 planar rotations $xy,zy,xz,tx,ty,tz$. The first three corresponds to spatial rotation that we already know.

Let us consider the $tx$ rotation, in this rotation $y,z$ don't change. This rotation must leave $(ct)^2-x^2$ unchanged, the square of the "distance" of the point (ct,x) from the origin. The general form of the rotation is
Let us consider the $tx$ rotation, in this rotation $y,z$ don't change. This rotation must leave $(ct)^2-x^2$ unchajged, the square of the "distance" of the point (ct,x) from the origin. The general form of the rotation is


```eq


@@ 28,7 28,7 @@ LL2/4.2

where $\psi$ is the hyperbolic angle of rotation. A simple check shows that $(ct)^2-x^2=(ct')^2-x'^2$.

Let's consider a frame $K$ moving with velocity $V$ relative to $K'$ along the axis $x$. The angle $\psi$ may only depends on the relative velocity $V$. We consider the motion of the origin of $K'$, in which $x'=0$, then `LL2/4.2` takes the form
Let's consider a frame $K'$ moving with velocity $V$ relative to $K$ along the axis $x$. The angle $\psi$ may only depends on the relative velocity $V$. We consider the motion of the origin of $K'$, in which $x'=0$, then `LL2/4.2` takes the form

$$
x=ct'\sh\psi,\qquad


@@ 59,11 59,35 @@ Substituting in `LL2/4.2`, we find
```eq
LL2/4.3
```
If we put in $\beta=V/c$ and $\gamma=(1-\beta^2)^{-\mfrac{1}{2}}$, we can write `LL2/4.3` as

$$
\begin{pmatrix}
ct\\ x\\ y\\ z
\end{pmatrix}
=
\begin{pmatrix}
\gamma & \beta\gamma & 0 & 0 \\
\gamma\beta & \gamma & 0 & 0 \\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{pmatrix}
\cdot
\begin{pmatrix}
ct'\\ x'\\ y'\\ z'
\end{pmatrix}
$$

The inverse formulas are obtained by changing $V$ to $-V$, since the $K$ system moves with velocity $-V$ relative to $K'$.

We see that the Lorentz transform `LL2/4.3` becomes Galileo's transform `LL2/4.1` when we take the limit $c\to\infty$

# Proper length
### first order approximation

For velocities small compared to the speed of light, we can use the first order expansion of `LL2/4.3`

```eq
LL2/4.4
```

# Transformation of velocities

A src/LL2/proper-length.md => src/LL2/proper-length.md +31 -0
@@ 0,0 1,31 @@
---
title: proper length
type: theory
---

We consider a rod, in the $K$ system, with _proper length_ $\Delta x=x_2-x_1$. In the $K'$ system we use the [Lorentz transform](LL2/lorentz-transformation.md) to convert $x_1\to x_1'$ and $x_2\to x_2'$.

$$
x_1=\frac{x_1'+Vt'}{\sqrt{1-V^2/c^2}},\qquad
x_2=\frac{x_2'+Vt'}{\sqrt{1-V^2/c^2}}
$$

The length of the rod is then

$$
\Delta x=\frac{\Delta x'}{\sqrt{1-\frac{V^2}{c^2}}}
$$

We see that the length $\Delta x'$ from the point of vue of $K'$ is always smaller than the proper length $\Delta x$. This phenomenon is called _Lorentz contraction_. We have in general

```eq
LL2/4.5
```

### Volume contraction

Since the transverse dimensions do not chnge because of its motion, the volume $\mathscr{V}$ of a bodt decreases according to the similar formula

```eq
LL2/4.6
```

R md/proper-time.md => src/LL2/proper-time.md +7 -3
@@ 3,7 3,7 @@ title: proper time
type: theory
---

Let us observe clock which are moving relative to us arbitrarily. In the course of time $\dd{t}$, we can attach an inertial reference frame to these clocks, and they will have gone a distance $\sqrt{\dd{x}^2+\dd{y}^2+\dd{z}^2}$. In the coordinate system of the moving clock, the clock is at rest $\dd{x'}=\dd{y'}=\dd{z'}=0$. Because of the invariance of [intervals](intervals.md)
Let us observe clock which are moving relative to us arbitrarily. In the course of time $\dd{t}$, we can attach an inertial reference frame to these clocks, and they will have gone a distance $\sqrt{\dd{x}^2+\dd{y}^2+\dd{z}^2}$. In the coordinate system of the moving clock, the clock is at rest $\dd{x'}=\dd{y'}=\dd{z'}=0$. Because of the invariance of [intervals](LL2/intervals.md)

$$
\dd{s}^2=c^2\dd{t}^2-\dd{x}^2-\dd{y}^2-\dd{z}^2=c^2\dd{t'}^2


@@ 12,7 12,7 @@ $$
from which

$$
\dd{t'}^2=\dd{t}\sqrt{1-\frac{\dd{x}^2+\dd{y}^2+\dd{z}^2}{c^2\dd{t}^2}}
\dd{t'}=\dd{t}\sqrt{1-\frac{\dd{x}^2+\dd{y}^2+\dd{z}^2}{c^2\dd{t}^2}}
$$

But


@@ 63,7 63,11 @@ $$
\frac{1}{c}\int_a^b\dd{s}
$$

If the clock is at rest, then its world line is clearly parallel to the $t$ axis. If the clock carries an arbitrary motion and returns to the starting point, then its world line will be a curve passing through the two points $a$ and $b$. We saw that the clock at rest always indicates a greater time interval than the moving one. Thus we arrive at the result that the integral
taken along the world line of the clock.

If the clock is at rest, then its world line is clearly parallel to the $t$ axis. If the clock carries an arbitrary motion and returns to the starting point, then its world line will be a curve passing through the two points $a$ and $b$.

We saw that the clock at rest always indicates a greater time interval than the moving one. Thus we arrive at the result that the integral

$$
\int_a^b\dd{s}

R problems/LL2/6.1.md => src/LL2/pset6.md +6 -0
@@ 1,3 1,9 @@
---
title: problem set 6
type: problem
---
## problem 1

Find the law of transformation of the components of a symmetric four-tensor $A^{ik}$ under Lorentz transformations `LL2/6.1`

:::solution

A src/LL2/relativistic-mechanics.md => src/LL2/relativistic-mechanics.md +268 -0
@@ 0,0 1,268 @@
---
title: relativistic mechanics
type: theory
---

We start studying the motin of particles the same as we did in classical emchanics, from the principle of least action. The _principle of least action_ states that for each mechanical system there exists an integral $S$ called the _action_, which has a minimum value for the actual motion, so that its variation $\delta S$ is zero.

## Lagrangian of a free particle

Let us consider a free particle (LL2/a particle not under the influence of any external force). From the principle of relativity, the integral $S$ must not depend on our choice of reference system, that is, it must be invariant under Lorentz transform. But, the only scalar of this kind we can construct for a free particle is the [interval](intervals.md) $\alpha\dd{s}$, where $\alpha$ is some constant. The aciton of a free particle must have the form.

$$
S=-\alpha\int_a^b\dd{s}
$$

We saw that $\int_a^b\dd{s}$ has its maximum value along a straight line. We also saw from `LL2/3.2` that we can make the integral it arbitrarily small by choosing a path carefully. Therefore, for the straight line to a minimum of $S$, $\alpha$ must be positive.

With the aid of `LL2/3.1` we find

$$
S=-\alpha c\int_{t_1}^{t_2}\dd{t}\sqrt{1-\frac{v^2}{c^2}}
$$

From the definition of the action `LL1/2.1` we recognize the Lagrangian

$$
L=-\alpha c\sqrt{1-v^2/c^2}
$$

Let us find a known meaning for $\alpha$.  The passage from relativistic mechanics to classical mechanics is done by $c\to\infty$, in the case of our expression this is formally equivalent to saying $v\to 0$. Therefore we may say

$$
L_{classical} = \lim_{v\to 0}L_{relativistic}=-\alpha c+\frac{\alpha v^2}{2c}+\o(v^4)
$$

We recognize from the classical Lagrangian of a free particle `LL1/4.1`, that we must have $\alpha=mc$.

Thus we have the action for a free particle

```eq
LL2/8.1
```

and the Lagrangian is

```eq
LL2/8.2
```

## momentum

By _momentum_ we can mean the vector $\v{p}=\partial L/\partial \v{v}$, using `LL2/8.2` we find

```eq
LL2/9.1
```

For small velocities ($v\ll c$), the expression goes over into the classical $\v{p}=m\v{v}$. For $v=c$ the momentum becomes infinite.

The time derivative of momentum is the force acting on the particle.

Suppose the velocity of the particle changes only in direction ($\v{F}\bot\v{v}$), then

```eq
LL2/9.2
```

Now suppose the velocity only changes in magnitude ($\v{F}\parallel\v{v}$)

```eq
LL2/9.3
```

We see that the ratio of force to acceleration is different in the two cases.

## energy

The _energy_  $\mathscr{E}$ of a particle is defined as the quantity (see `LL1/6.1`)

$$
\mathscr{E}=\v{p}\cdot\v{v}-L
$$

substituting `LL2/8.2` and `LL2/9.1` we find

```eq
LL2/9.4
```

Contrary to classical mechanics, in relativistic mechanics the energy of a free particle does not go to zero for $v=0$, but rather takes on a finite value

```eq
LL2/9.5
```

For small velocities ($v/c\ll 1$), we have, expanding `LL2/9.4` in series in powers of $v/c$,

$$
\mathscr{E}=mc^2+\frac{mv^2}{2}
$$

## non conservation of mass

The formula `LL2/9.5` is valid for any mechanical system at rest, in particular for a composite body. The energy of a body at rest contains the rest ernegies of its constituent particles, the kinetic energy of each constituent particle and the interactions between the particles. In other words, $mc^2$ is not equal to $\sum m_ac^2$, and so $m$ is not equal to $\sum m_a$. Thus in relativistic mechanics the conservation of mass does not hold: the mass of a composite body is not equal to the mass of its parts.

## Hamiltonian

Squaring `LL2/9.1` and `LL2/9.4`, we get the relation

```eq
LL2/9.6
```

The energy expressed in terms of the momentum is called the Hamiltonian function $\mathscr{H}$

```eq
LL2/9.7
```

For low velocities $p\ll mc$ we have

$$
\mathscr{H}=mc^2+\frac{p^2}{2m}+\o{(p^4)}
$$

which is, except for the rest energy, the classical expression for the Hamiltonian.

### light-speed movement

From `LL2/9.1` and `LL2/9.4` we have the relation

```eq
LL2/9.8
```

For the limiting case $v=c$, the momentum and the energy of the particle would become infinite. The only case wherethe momentum and energy don't diverge is for zero mass particles, such particles would then have momentum

```eq
LL2/9.9
```

The same formula also holds approximately for particles with nonzero mass in the so-called _ultrarelativistic_ case, when the particle energy $\mathscr{E}$ is large compared to its rest energy $mc^2$

# Four dimensional form

According to the principle of least action

$$
\delta S=-mc\delta\int_a^b\dd{s}=0
$$

we notice that $\dd{s}\equiv\sqrt{\dd{x}_i\dd{x}^i}$, therefore

$$
\delta S=-mc\int_a^b\frac{\dd{x}_i\delta\dd{x}^i}{\dd{s}}=-mc\int_a^bu_i\dd{\delta x^i}
$$

Integrating by parts we obtain

```eq
LL2/9.10
```

To get the equations of motion we compare different trajectories for two fixed points, i.e. at the limits $(\delta x^i)_a=(\delta x^i)_b=0$. Then the actual trajectory is determined from the condition $\delta S=0$. From `LL2/9.10` we obtain the equation $\dd{u_i}/\dd{s}=0$, that is, as we expect, a constant velocity for the free particle in four-dimensional form.

To get an expression of the action as a function of coordinates, we consider a fixed point $a$, so that $(\delta x^i)_a=0$. The second point is the variable coordinate that we write $(\delta x^i)_b=\delta x$.

We only consider actual motions, i.e. those for which the integral is zero, we are left with

```eq
LL2/9.11
```

(TODO: mathlet: point b is sliding on the actual trajectory, $\delta S$ is zero when wigling the trajectory but is not zero along the path.)

### four momentum

The four-vector

```eq
LL2/9.12
```

is called the momentum four-vector. We know from classical mechanics that

$$
\partial S/\partial x=p_x,\qquad
...,\qquad
\partial S/\partial t=-\mathscr{E}
$$

Thus the contravariant components of the four momentum are

```eq
LL2/9.13
```


From `LL2/9.11`, the four momentum for free particle is

```eq
LL2/9.14
```

Thus, in relativistic mechanics, momentum and energy are the components of a single four-vector. Substituing `LL2/9.13` into the general transformation formula `LL2/6.1` we find

```eq
LL2/9.15
```

From the definition of the four momentum `LL2/9.14` and the identity $u^iu_i=1$, we have the square of the four momentum for a free particle

```eq
LL2/9.16
```

Substituting `LL2/9.13` we get back `LL2/9.6`

### four force

By analogy with the usual definition of the force, the four force is defined as the derivative

```eq
LL2/9.17
```

Its components satisfy the identity $g_iu^i=0$. In terms of the three dimensional force its components are

```eq
LL2/9.18
```

the tim component is related to the work done by the force.

### the Hamilton-Jacobi equation

We obtain the equation by susbstituting `LL2/9.12` in `LL2/9.16`

```eq
LL2/9.19
```

writing the sum explicitly we have

```eq
LL2/9.20
```

To make the transition to the classical Hamilton-Jacobi, we must use another action that accounts for the relativistic rest energy $\mathscr{E}_r-\mathscr{E}_c=mc^2$. Because we have $\mathscr{E}=-(\partial S/\partial t)$, we must offset the relativistic action $S_r$ by:

$$
S_r-S_c=-mc^2t
$$

substituting into `LL2/9.20` we find

$$
\frac{1}{2m}\left[
\left(\frac{\partial S_c}{\partial x}\right)^2
+\left(\frac{\partial S_c}{\partial y}\right)^2
+\left(\frac{\partial S_c}{\partial z}\right)^2
\right]
-\frac{1}{2mc^2}\left(\frac{\partial S_c}{\partial t}\right)^2
+\frac{\partial S_c}{\partial t}
=0
$$

In the limit $c\to\infty$, this equation goes over into the classical Hamilton-Jacobi equation.

R md/hamiltonian-operator.md => src/LL3/hamiltonian-operator.md +2 -2
@@ 3,7 3,7 @@ title: the Hamiltonian operator
type: theory
---

The [wave function](wave-function.md) completely determines the system, and also the future states of the system. This means that the derivative $\partial\Psi/\partial t$ must be determined by the function itself at an instant, and, by the principle of superpostion, the relationship must be linear
The [wave function](LL3/wave-function.md) completely determines the system, and also the future states of the system. This means that the derivative $\partial\Psi/\partial t$ must be determined by the function itself at an instant, and, by the principle of superpostion, the relationship must be linear

$$ i\partial\Psi/\partial t=\qop{L}\Psi $$



@@ 29,7 29,7 @@ the slowly varying amplitude $a$ need not be differentiated.

Comparing this to the definition $\partial\Psi/\partial t=-i\qop{L}\Psi$ we have $\qop{L}=-(1/\hbar)\partial S/\partial t$.

As we know from classical mechanics, the derivative $-\partial S/\partial t$ is just Hamilton's function $H$ for a mechanical system. In quantum mechanics we call this the Hamiltonian [operator](quantum-operator.md), or the Hamiltonian of a system.
As we know from classical mechanics, the derivative $-\partial S/\partial t$ is just Hamilton's function $H$ for a mechanical system. In quantum mechanics we call this the Hamiltonian [operator](LL3/quantum-operator.md), or the Hamiltonian of a system.

```eq
LL3/8.1

R md/hermitian-eigenfunctions.md => src/LL3/hermitian-eigenfunctions.md +1 -1
@@ 3,7 3,7 @@ type: problem
title: hermitian eigenfunctions
---

We have seen that real physical quantities are represented by hermitian linear integral [operators](quantum-operator.md) $(3.15)$. Show that the eingenfunctions of these operators are orthogonal.
We have seen that real physical quantities are represented by hermitian linear integral [operators](LL3/quantum-operator) `LL3/3.15`. Show that the eingenfunctions of these operators are orthogonal.

:::solution
Let $f_n$ and $f_m$ be two different eigenvalues of the quantity $f$, and $\Psi_n$, $\Psi_m$ the corresponding eigenfunctions

R md/momentum-operator.md => src/LL3/momentum-operator.md +1 -1
@@ 7,7 7,7 @@ Considering a system of particles not in an external field. The system should be

$$ \psi(r_1+\delta r,r_2+\delta r,...)=\psi(r_1,r_2,...)+\delta r\sum_a \nabla_a\psi $$

The expresion $\qop{O}=1+\delta r\sum_a\nabla_a$ can be regarded as the "infinitely small displacement" operator. This displacement should not change the [Hamiltonian operator](hamiltonian-operator.md), by this we mean that the order of application doesn't matter
The expresion $\qop{O}=1+\delta r\sum_a\nabla_a$ can be regarded as the "infinitely small displacement" operator. This displacement should not change the [Hamiltonian operator](LL3/hamiltonian-operator.md), by this we mean that the order of application doesn't matter

$$ \qop{H}\qop{O}-\qop{O}\qop{H}=0 $$


R md/quantum-operator.md => src/LL3/quantum-operator.md +2 -2
@@ 3,7 3,7 @@ type: theory
title: quantum operators
---

Real physical quantities $f$ are contained in the [wave function](wave-function.md)  $\Psi$, measurement is done by the application of an operator $(\qop{f}\Psi)$. The values that are taken by a physical quantities are eigenvalues $f_n$ of it's operator. The set of eigenvalues form a "spectrum". The spectrum can be discrete (e.g. energy) or continous (e.g. position) or a mix of both (see anharmonic oscillator). To each eingenvalue is associate a eigen-wavefunction $\Psi_n$ which is also normalized.
Real physical quantities $f$ are contained in the [wave function](LL3/wave-function)  $\Psi$, measurement is done by the application of an operator $(\qop{f}\Psi)$. The values that are taken by a physical quantities are eigenvalues $f_n$ of it's operator. The set of eigenvalues form a "spectrum". The spectrum can be discrete (e.g. energy) or continous (e.g. position) or a mix of both (see anharmonic oscillator.md). To each eingenvalue is associate a eigen-wavefunction $\Psi_n$ which is also normalized.

```eq
LL3/3.1


@@ 270,7 270,7 @@ $$ \overline{\dot{f}}=\dot{\overline{f}}=\frac{\dd{}}{\dd{t}}\int\Psi^*\qop{f}\P
\int\Psi\qop{f}\frac{\partial\Psi}{\partial t}\dd{q}
$$

Substituting in `LL3/8.1`, the time derivative is simply the [Hamiltonian operator](hamiltonian-operator.md)
Substituting in `LL3/8.1`, the time derivative is simply the [Hamiltonian operator](LL3/hamiltonian-operator.md)
$$ \overline{\dot{f}}=
\int\Psi^*\frac{\partial\qop{f}}{\partial t}\Psi\dd{q}+
\frac{i}{\hbar}\int(\qop[*]{H}\Psi^*)\qop{f}\Psi\dd{q}-

R md/translation-operator.md => src/LL3/translation-operator.md +1 -1
@@ 3,7 3,7 @@ title: translation operator
type: problem
---

Express the [operator](quantum-operator.md) $\hat{T_a}$ of a parallel displacement over a finite distance $a$ in terms of the momentum operator.
Express the [operator](LL3/quantum-operator.md) $\hat{T_a}$ of a parallel displacement over a finite distance $a$ in terms of the momentum operator.

:::solution


R md/wave-function.md => src/LL3/wave-function.md +0 -0
A src/index.md => src/index.md +46 -0
@@ 0,0 1,46 @@
---
title: index
topics:
  - name: single variable calculus
    slug: 18.01x
    sources:
      - name: MIT 18.01x
        links:
          - homepage: https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/
  - name: multivariable calculus
    slug: 18.02sc
    sources:
      - name: MIT 18.02sc
        links:
          - homepage: https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/
  - name: ordinary differential equations
    slug: 18.03x
    sources:
      - name: MIT 18.03x
        links:
          - homepage: https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/
  - name: classical mechanics
    slug: LL1
    sources:
      - name: Course of Theoretical Physics Volume 1
        links:
          - pdf: https://archive.org/download/landau-and-lifshitz-physics-textbooks-series/Vol%201%20-%20Landau%2C%20Lifshitz%20-%20Mechanics%20%283rd%20ed%2C%201976%29.pdf
  - name: classical field theory
    slug: LL2
    sources:
      - name: Course of Theoretical Physics Volume 2
        links:
          - pdf: https://archive.org/download/landau-and-lifshitz-physics-textbooks-series/Vol%202%20-%20Landau%2C%20Lifshitz%20-%20The%20classical%20theory%20of%20fields%20%284th%2C%201994%29.pdf
      - name: LP353
        links:
          - pdf: https://archive.org/download/landau-and-lifshitz-physics-textbooks-series/Vol%202%20-%20Landau%2C%20Lifshitz%20-%20The%20classical%20theory%20of%20fields%20%284th%2C%201994%29.pdf
  - name: quantum mechanics
    slug: LL3
    sources:
      - name: Course of Theoretical Physics Volume 3
        links:
          - pdf: https://archive.org/download/landau-and-lifshitz-physics-textbooks-series/Vol%203%20-%20Landau%2C%20Lifshitz%20-%20Quantum%20mechanics%20-%20non-relativistic%20theory%20%283ed.%2C%201991%29.pdf
      - name: The Feynman Lectures on Physics, Volume III
        links:
          - html: https://www.feynmanlectures.caltech.edu/III_toc.html
---

A src/index.tpl => src/index.tpl +27 -0
@@ 0,0 1,27 @@
<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='_DEFAULT'>
$for(topics)$
  <h2>$it.name$</h2>

  $if(it.sources)$
    <h3>sources</h3>
    <ul>
  $endif$
  $for(it.sources)$
	  <li>$it.name$
	  $for(it.links)$
		(
		  $for(it/pairs)$
		  <a href="$it.value$">$it.key$</a>$sep$
		  $endfor$
		  )
	  $endfor$
	  </li>
  $endfor$
  </ul>

  <div style="margin:auto"><object data="$it.slug$/index.svg" type=image/svg+xml></object></div>
$endfor$
</body>

M tools/generate-graph.sh => tools/generate-graph.sh +16 -8
@@ 3,17 3,17 @@
GET=tools/get

generate_one_node() {
    node=$(basename ${1%.md})
    node=$1
    slug=${node#src/}
    slug=${slug%.md}
    
    type=$(./tools/get type $node)
    [ "$type" == "math" ] && return
    [ "$type" == "problem" ] && return
    # [ "$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" ] && continue
	[ "$type" == "problem" ] && continue
    [ -f graph/$slug.txt ] && for line in $(cat graph/$slug.txt); do
	$GET graph_node $line
	printf  '    "%s" -> "%s"\n' "$($GET title $line)" "$title"
    done
}


@@ 23,6 23,14 @@ echo '    bgcolor="#ffffff00"'
echo '    node  [style="rounded,filled", shape=box]'
echo '    layout=fdp'

for node in md/*.md; do generate_one_node $node& done | sort; wait 
input=$1
part=${input#www-data/}
part=${part%/index.svg}

for node in $(find src/$part -name "*.md"); do
	generate_one_node $node& 
done | sort

wait 

echo "}"

M tools/generate-local-graph.sh => tools/generate-local-graph.sh +12 -5
@@ 7,17 7,24 @@ echo '    bgcolor="#ffffff00"'
echo '    node  [style="rounded,filled", shape=box]'
echo '    rankdir=LR'

node=$(basename ${1%.md})
$GET graph_node $node
centre_title=$($GET title $node)
file=$1
slug=${file#graph/}
slug=${slug%.txt}

[ -f graph/$node ] && for line in $(cat graph/$node); do
$GET graph_node $slug
centre_title=$($GET title $slug)

echo "#inward links"
[ -f graph/$slug.txt ] && for line in $(cat graph/$slug.txt); do
    title=$($GET title $line)
    $GET graph_node $line
    printf  '    "%s" -> "%s"\n' "$title" "$centre_title"
done

for line in $(grep -oP '\(\K[^ ]*(?=\.md\))' md/$node.md | sort | uniq); do
echo "#outward links"
# match [xyz](abc)
# don't match ![xyz](abc)
for line in $(grep -oP '[^!]\[.*\]\(\K[^ ]*/[^ ]*(?=\))' src/$slug.md | sort | uniq); do
    title=$($GET title $line)
    $GET graph_node $line
    printf  '    "%s" -> "%s"\n' "$centre_title" "$title"

M tools/get => tools/get +18 -6
@@ 9,16 9,28 @@ colors[math]="#f5d6e5"		#pink
colors[DEFAULT]="#fff8e7"	#cosmic latte

_or_default() { grep ^ || echo DEFAULT; }
slug() { echo $1; }
title() { awk -F': ' '/title: / {print $2}' md/$1.md | _or_default; }
type() { awk -F': ' '/type: / {print $2}' md/$1.md | _or_default; }
title() { awk -F': ' '/title: / {print $2}' $md | _or_default; }
type() { awk -F': ' '/type: / {print $2}' $md | _or_default; }
color() { printf "${colors[$(type $1)]}"; }
graph_node() { printf '    "%s"[href="%s.html", target="_parent", fillcolor="%s"]\n' "$(title $1)" "$1" "$(color $1)"; }
graph() { grep "($1.md)" md/*.md | awk -F: '{print $1}' | uniq | sed s:md/::g | sed s:.md::g || true; }
graph_node() { 
	printf '    "%s"[href="%s", target="_parent", fillcolor="%s"]\n' "$(title)" "../$slug.html" "$(color)"; 
}
slug() { echo $slug; }
graph() { grep "$slug" src/*/*.md | awk -F: '{print $1}' | uniq | sed s:src/::g | sed s:.md::g || true; }
bg_css() {
    for type in ${!colors[@]}; do
	    echo "._$type{ background-color: ${colors[$type]}; }"
    done
}

$1 $(basename ${2%.*})
command=$1
input=$2
slug=${input#src/}
slug=${slug#graph/}
slug=${slug%.md}
slug=${slug%.txt}
md=src/$slug.md

export file
export slug
"$command"

D tools/index.html => tools/index.html +0 -7
@@ 1,7 0,0 @@
<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='_DEFAULT'>
<h1>Main Graph</h1>
<div style="margin:auto"><object data=graph.svg type=image/svg+xml></object></div>
</body>

M tools/math.tex => tools/math.tex +3 -0
@@ 16,6 16,9 @@
\DeclareMathOperator{\sh}{sh}
\DeclareMathOperator{\th}{th}

\DeclareMathOperator{\o}{o}
\DeclareMathOperator{\O}{O}

\DeclareMathOperator{\det}{det}

\newenvironment{nalign}{

M tools/pf-filter.py => tools/pf-filter.py +1 -1
@@ 12,7 12,7 @@ def action(elem, doc):
        return

    if isinstance(elem, pf.Link) and elem.url.endswith('.md'):
        elem.url = elem.url[:-3] + '.html'
        elem.url = '../' + elem.url[:-3] + '.html'
        return elem

    # reference

M tools/template.html => tools/template.html +4 -4
@@ 1,12 1,12 @@
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="$lang$" xml:lang="$lang$"$if(dir)$ dir="$dir$"$endif$>
<div align="left"><a href="index.html">back to graph</a></div>
<div align="left"><a href="../index.html">back to graph</a></div>
<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" type="text/javascript"></script>
$if(prod)$
<script async defer data-website-id="b9b10aa0-177f-42d5-8493-f3a8062db28e" src="https://umami.pourtan.eu/umami.js"></script>
$endif$
<link rel="stylesheet" type="text/css" href="colors.css">
<link rel="stylesheet" type="text/css" href="style.css">
<link rel="stylesheet" type="text/css" href="../colors.css">
<link rel="stylesheet" type="text/css" href="../style.css">
<head>
  <meta charset="utf-8" />
  <meta name="generator" content="pandoc" />


@@ 73,7 73,7 @@ $endfor$
$endif$
<hr>
<h3>See also</h3>
<center><object data=$slug$.svg type=image/svg+xml></object></center>
<center><object data=../$slug$.svg type=image/svg+xml></object></center>
</body>
</div>
</html>