15 done

1/14-motion-in-a-central-field.md

@@ -161,3 +161,11 @@ to $-1/r^n$ with $n > 2$.
 <!-- (m1+m2)x+m2) sin = constant, -->
 <!-- (2) -->
 <!-- which expresses the fact that the centre of mass of the system does not move horizontally. -->
+<!-- Using (1), we find the energy in the form -->
+<!-- E -->
+<!-- (3) -->
+<!-- Hence -->
+<!-- Expressing the co-ordinates X2 sin o, y = l cos of the particle m2 in terms of -->
+<!-- by means of (2), we find that its path is an arc of an ellipse with horizontal semi- -->
+<!-- axis lm1/(m1+m2) and vertical semi-axis l. As M1 8 we return to the familiar simple pen- -->
+<!-- dulum, which moves in an arc of a circle. -->

1/equations/15.1.tex

@@ -0,0 +1 @@
+U=-\alpha/r

1/equations/15.10.tex

@@ -0,0 +1 @@
+r=a(1-e\cos\xi),\qquad t=\sqrt{ma^3/\alpha}(\xi-e\sin\xi),

1/equations/15.11.tex

@@ -0,0 +1 @@
+x=a(\cos\xi-e),\qquad y=a\sqrt{q-e^2}\sin\xi

1/equations/15.12.tex

@@ -0,0 +1,4 @@
+\begin{align}
+	r &= a(e\ch\xi-1), & t&=\sqrt{ma^3/\alpha}(e\sh\xi-\xi), \\
+	x &= a(e-\ch\xi)), & t&=a\sqrt{e^2-1}\sh\xi,
+\end{align}

1/equations/15.13.tex

@@ -0,0 +1 @@
+U=\alpha/r,\quad(\alpha\gt0)

1/equations/15.14.tex

@@ -0,0 +1 @@
+p/r=-1+e\cos\phi

1/equations/15.15.tex

@@ -0,0 +1 @@
+r_\min=p/(e-1)=a(e+1)

1/equations/15.16.tex

@@ -0,0 +1,4 @@
+\begin{align}
+	r &= a(e\ch\xi+1), & t&=\sqrt{ma^3/\alpha}(e\sh\xi+\xi), \\
+	x &= a(\ch\xi+e)), & t&=a\sqrt{e^2-1}\sh\xi,
+\end{align}

1/equations/15.17.tex

@@ -0,0 +1 @@
+\v{v}\times\v{M}+\alpha\v{r}/r

1/equations/15.2.tex

@@ -0,0 +1 @@
+U_\text{eff}=-\frac{\alpha}{r}+\frac{M^2}{2mr^2}

1/equations/15.3.tex

@@ -0,0 +1 @@
+U_{\text{eff,min}}=-m\alpha^2/2M^2

1/equations/15.4.tex

@@ -0,0 +1 @@
+p=M^2/m\alpha,\qquad e=\sqrt{1+(2EM^2/m\alpha^2)}

1/equations/15.5.tex

@@ -0,0 +1 @@
+p/r=1+e\cos\phi

1/equations/15.6.tex

@@ -0,0 +1 @@
+a=p/(1-e^2)=\alpha/2|E|,\qquad b=p/\sqrt{1-e^2}=M/\sqrt{2m|E|}

1/equations/15.7.tex

@@ -0,0 +1 @@
+r_\min=p/(1+e)=a(1-e),\qquad r_\max=p/(1-e)=a(1+e)

1/equations/15.8.tex

@@ -0,0 +1,4 @@
+\begin{align}
+	T&=2\pi a^{3/2}\sqrt{m/\alpha} \\
+	 &=\pi\alpha\sqrt{m/2|E|^3}
+\end{align}

1/equations/15.9.tex

@@ -0,0 +1 @@
+r_\min=p/(e+1)=a(e-1)

1/index.md

@@ -23,12 +23,12 @@ III.  INTEGRATION OF THE EQUATIONS OF MOTION
 12. [Determination of the potential energy from the period of oscillation](12-determination-of-the-potential-energy-from-the-period-of-oscillation.html)
 13. [The reduced mass](13-the-reduced-mass.html)
 14. [Motion in a central field](14-motion-in-a-central-field.html)
+15. [Kepler's problem](15-keplers-problem.html)
 
 <span style="background-color: yellow; color: white: width: 100%;">
 🚧 WORK IN PROGRESS BELOW THIS POINT 🚧
 </span>
 
-15. [Kepler's problem](15-keplers-problem.html)
 IV.  COLLISION BETWEEN PARTICLES
 16. [Disintegration of particles](16-disintegration-of-particles.html)
 17. [Elastic collisions](17-elastic-collisions.html)

tools/math.tex

@@ -14,6 +14,7 @@
 \newcommand{\ptag}[2][]{\label{#1/#2}\tag*{(#2)}}
 \newcommand{\min}{\text{min}}
 \newcommand{\max}{\text{max}}
+\newcommand{\constant}{\text{constant}}
 
 \DeclareMathOperator{\ch}{ch}
 \DeclareMathOperator{\sh}{sh}

tools/pf-filter.py

@@ -40,7 +40,7 @@ def action(elem, doc):
             return elem
         else:
             # section ref
-            _pnum = text
+            pnum = text
             try:
                 slug = [f for f in os.listdir(book) if f.startswith(pnum)][0]
             except: