cm: Improve precision

5 files changed, 212 insertions, 136 deletions

diff --git a/Year 4/Celestial mechanics/Exercise1.R b/Year 4/Celestial mechanics/Exercise1.R
index c76ce8a..6f9e3cb 100644
--- a/Year 4/Celestial mechanics/Exercise1.R
+++ b/Year 4/Celestial mechanics/Exercise1.R
@@ -10,7 +10,7 @@ m3x3 <- function(...) {
 
 # Принтира аргументът, така че всички цифри да имат 8 знака прецизност
 p <- function(x) {
-  formatC(x, format="f", digits = 8)
+  formatC(x, format="f", digits = 10)
 }
 
 # Матрица на въртене
@@ -103,7 +103,7 @@ planetData <- function(t, a2000, da, e2000, de, i2000, di, lambda2000, dlambda,
 # Връща координати, разстояние от слънцето, координатни скорости и скорост на всяка планета
 # от слънчевата система, използвайки моета рожденна дата и J2000 данни
 solarSystemAtBirth <- function() {
-  birthDate <- 0.02320328
+  birthDate <- 0.023203285421
   rbind(
     planetData(birthDate,    # Меркурии
                0.38709927,    0.00000037, 0.20563593, 0.00001906, 0.12225995, -0.00010380,
diff --git a/Year 4/Celestial mechanics/Exercise2.R b/Year 4/Celestial mechanics/Exercise2.R
index 5e98ada..7646b28 100644
--- a/Year 4/Celestial mechanics/Exercise2.R
+++ b/Year 4/Celestial mechanics/Exercise2.R
@@ -5,7 +5,7 @@ atDay <- function(epochConstant, rate, t) {
 
 # Принтира аргументът, така че всички цифри да имат 8 знака прецизност
 p <- function(x) {
-  formatC(x, format="f", digits = 8)
+  formatC(x, format="f", digits = 21)
 }
 
 # Аргумент на перихелия
@@ -47,30 +47,30 @@ delaunay <- function(t, mu, a2000, da, e2000, de, i2000, di, lambda2000, dlambda
 
 # Елементи на Делоне в моета дата на раждане
 delaunayAtBirth <- function() {
-  birthDate <- 0.02320328
+  birthDate <- 0.023203285421
   rbind(
-    delaunay(birthDate, 0.00000016,    # Меркурии
+    delaunay(birthDate, 0.00000016601367952719,    # Меркурии
                0.38709927,    0.00000037, 0.20563593, 0.00001906, 0.12225995, -0.00010380,
                4.40259868, 2608.79030501, 1.35189358, 0.00280085, 0.84353100, -0.00218761),
-    delaunay(birthDate, 0.00000244,    # Венера
+    delaunay(birthDate, 0.00000244783833966454,    # Венера
                0.72333566,    0.00000390, 0.00677672, -0.00004107, 0.05924827, -0.00001377,
                3.17613446, 1021.32854958, 2.29689636,  0.00004683, 1.33831572, -0.00484668),
-    delaunay(birthDate, 0.00000304,    # EM Bary
+    delaunay(birthDate, 0.0000030404326333266,     # EM Bary
                1.00000261,   0.00000562, 0.01671123, -0.00004392, -0.00000027, -0.00022596,
                1.75343756, 628.30757790, 1.79660147,  0.00564219,  0.0,         0.0),
-    delaunay(birthDate, 0.00000032,    # Марс
-               1.52371034,   0.00001847,  0.09339410, 0.00007882, 0.03228321, -0.00014192,
+    delaunay(birthDate, 0.00000032271514450539,    # Марс
+                1.52371034,   0.00001847,  0.09339410, 0.00007882, 0.03228321, -0.00014192,
                -0.07947238, 334.06130168, -0.41789517, 0.00775643, 0.86497713, -0.00510637),
-    delaunay(birthDate, 0.00095479,    # Юпитер
+    delaunay(birthDate, 0.00095479193842432214,    # Юпитер
                5.20288700, -0.00011607, 0.04838624, -0.00013253, 0.02276602, -0.00003206,
                0.60033114, 52.96631189, 0.25706047,  0.00370929, 1.75360053,  0.00357253),
-    delaunay(birthDate, 0.00028588,    # Сатурн
+    delaunay(birthDate, 0.0002858859806661029,     # Сатурн
                9.53667594, -0.00125060, 0.05386179, -0.00050991, 0.04338874,  0.00003379,
                0.87186604, 21.33653879, 1.61615531, -0.00731244, 1.98378354, -0.00503838),
-    delaunay(birthDate, 0.00004366,    # Уран
+    delaunay(birthDate, 0.00004366244043351564,    # Уран
                19.18916464, -0.00196176, 0.04725744, -0.00004397, 0.01348507, -0.00004240,
-               5.46703627,  7.47842217, 2.98371499,  0.00712187, 1.29183904,  0.00074012),
-    delaunay(birthDate, 0.00005151,    # Нептун
+                5.46703627,  7.47842217, 2.98371499,  0.00712187, 1.29183904,  0.00074012),
+    delaunay(birthDate, 0.00005151389020535497,    # Нептун
                30.06992276, 0.00026291, 0.00859048,  0.00005105, 0.03089309,  0.00000617,
                -0.96202600, 3.81283674, 0.78478315, -0.00562720, 2.30006864, -0.00008878)
   )
diff --git a/Year 4/Celestial mechanics/Project.pdf b/Year 4/Celestial mechanics/Project.pdf
index fa9fd7e..84c62af 100644
--- a/Year 4/Celestial mechanics/Project.pdf
+++ b/Year 4/Celestial mechanics/Project.pdf
diff --git a/Year 4/Celestial mechanics/Project.tex b/Year 4/Celestial mechanics/Project.tex
index c469110..62b6faa 100644
--- a/Year 4/Celestial mechanics/Project.tex
+++ b/Year 4/Celestial mechanics/Project.tex
@@ -60,8 +60,6 @@
 
 \vfill
 
-\centerline{\textit{Всички изчисления ще са с точност до 8 знака след десетичния знак.}}
-
 Първо ще подготвим таблицата на орбиталните елементи.
 Уебсайтът на NASA \cite{nasa} предоставя следните константи за началото на епохата (1 Януари 2000, 00:00 часа):
 
@@ -142,8 +140,8 @@ Neptune & 30.06992276   &   0.00859048   & 0.03089309  & -0.96202600   & 0.78478
 
 Финално, ще изчислим рожденната ми дата във векове.
 Аз съм роден на 27 Април 2002 година.
-Това е 117-ят ден в годината, което е $\frac{117}{365.25} = 0.32032854$ Юлиански години.
-Общо излиза $2.32032854$ Юлиански години след началото на епохата, което са $\frac{2.32032854}{100} = 0.02320328$ Юлиански века.
+Това е 117-ят ден в годината, което е $\frac{117}{365.25} \approx 0.3203285421$ Юлиански години.
+Общо излиза $2.3203285421$ Юлиански години след началото на епохата, което са $\frac{2.3203285421}{100} = 0.023203285421$ Юлиански века.
 
 \paragraph{1. Решение}
 
@@ -201,14 +199,14 @@ atDay <- function(epochConstant, rate, t) {
 	\end{pmatrix}
 \end{align*}
 
-Дефинираме помощни функции \texttt{m3x3} и \texttt{p}. Първата приема редица елементи и ги преобразува в $3 \times 3$ матрица, втората изкарва подаденото на екрана, като всички числа са с 8 цифри прецизност:
+Дефинираме помощни функции \texttt{m3x3} и \texttt{p}. Първата приема редица елементи и ги преобразува в $3 \times 3$ матрица, втората изкарва подаденото на екрана, като всички числа са с 10 цифри прецизност:
 
 \begin{lstlisting}[language=R]
 m3x3 <- function(...) {
   matrix(c(...), nrow = 3, ncol = 3, byrow = TRUE)
 }
 p <- function(x) {
-  formatC(x, format="f", digits = 8)
+  formatC(x, format="f", digits = 10)
 }
 \end{lstlisting}
 
@@ -362,7 +360,7 @@ planetData <- function(t, a2000, da, e2000, de, i2000, di, lambda2000, dlambda,
 Финално дефинираме функцията \texttt{solarSystemAtBirth}, която извиква \texttt{planetData} за всяка планета с подадени константи:
 \begin{lstlisting}[language=R]
 solarSystemAtBirth <- function() {
-  birthDate <- 0.02320328
+  birthDate <- 0.023203285421
   rbind(
     planetData(birthDate,    # Меркурии
                0.38709927,    0.00000037, 0.20563593, 0.00001906, 0.12225995, -0.00010380,
@@ -399,25 +397,43 @@ p(solarSystemAtBirth())
 
 И получаваме следния резултат:
 
-\begin{sidewaystable}
-\begin{center}\begin{Large}
-\begin{tabular}{c||c|c|c|c|c|c|c|c} 
-& $x$ & $y$ & $z$ & $|r|$ & $v_x$ & $v_y$ & $v_z$ & $|v|$ \\
-\hline\hline
-Mercury & -0.25698453 & 0.21643004   & 0.04126659  & 0.33850545  & -1.38648452 & -1.18390004 & 0.03053503  & 1.82342837 \\
-Venus   & -0.11664062 & 0.70946840   & 0.01643289  & 0.71918043  & -1.16420814 & -0.19720358 & 0.06450187  & 1.18255247 \\
-EM Bary & -0.79848552 & -0.61311058  & 0.00000338  & 1.00671928  & 0.59280163  & -0.79699190 & 0.00000439  & 0.99328237 \\
-Mars    & 0.01018262  & 1.56865936   & 0.03261130  & 1.56903135  & -0.78255752 & 0.07431329  & 0.02078708  & 0.78635288 \\
-Jupiter & -1.81325044 & 4.88999123   & 0.02034819  & 5.21539121  & -0.41697241 & -0.13210905 & 0.00988286  & 0.43751167 \\
-Saturn  & 2.07612673  & 8.79599349   & -0.23579568 & 9.04076343  & -0.33293384 & 0.07398502  & 0.01195145  & 0.34126466 \\
-Uranus  & 16.49642718 & -11.30522661 & -0.25582952 & 20.00014268 & 0.12742638  & 0.17777408  & -0.00099186 & 0.21872835 \\
-Neptune & 18.92712374 & -23.39768921 & 0.04565204  & 30.09468321 & 0.14091098  & 0.11598482  & -0.00563534 & 0.18259282 \\
+\newpage
+
+\hfill
 
+\begin{table}[!h]
+\begin{center}\begin{large}
+\begin{tabular}{c||c|c|c|c} 
+& $x$ & $y$ & $z$ & $|r|$ \\
+\hline\hline
+Mercury & -0.2569892522 & 0.2164260053   & 0.0412666944  & 0.3385064720 \\
+Venus   & -0.1166445847 & 0.7094677286   & 0.0164331060  & 0.7191804113 \\
+EM Bary & -0.7984835049 & -0.6131132909  & 0.0000033801  & 1.0067193328 \\
+Mars    & 0.0101799533  & 1.5686596135   & 0.0326113752  & 1.5690315855 \\
+Jupiter & -1.8132518650 & 4.8899907784   & 0.0203482204  & 5.2153912785 \\
+Saturn  & 2.0761255995  & 8.7959937429   & -0.2357956393 & 9.0407634088 \\
+Uranus  & 16.4964276183 & -11.3052260072 & -0.2558295226 & 20.0001426991 \\
+Neptune & 18.9271242195 & -23.3976888193 & 0.0456520201  & 30.0946832084 \\
 \end{tabular}
-\end{Large}\end{center}
-\end{sidewaystable}
+\end{large}\end{center}
+\end{table}
 
-\newpage
+\begin{table}[!h]
+\begin{center}\begin{large}
+\begin{tabular}{c||c|c|c|c} 
+& $v_x$ & $v_y$ & $v_z$ & $|v|$ \\
+\hline\hline
+Mercury  & -1.3864619581 & -1.1839190472 & 0.0305314033  & 1.8234234939 \\
+Venus    & -1.1642070756 & -0.1972100811 & 0.0645017151  & 1.1825524945 \\
+EM Bary  & 0.5928042970  & -0.7969898554 & 0.0000043938  & 0.9932823185 \\
+Mars     & -0.7825575318 & 0.0743119103  & 0.0207870540  & 0.7863527531 \\
+Jupiter  & -0.4169723661 & -0.1321091661 & 0.0098828616  & 0.4375116648 \\
+Saturn   & -0.3329338533 & 0.0739849818  & 0.0119514504  & 0.3412646559 \\
+Uranus   & 0.1274263738  & 0.1777740834  & -0.0009918592 & 0.2187283458 \\
+Neptune  & 0.1409109730  & 0.1159848250  & -0.0056353427 & 0.1825928231 \\
+\end{tabular}
+\end{large}\end{center}
+\end{table}
 
 \paragraph{2. Решение}
 
@@ -449,14 +465,14 @@ GMs/GM8 & 19412.24    & $mass^{−1}$ for Neptune \\
 \begin{tabular}{c||c}
 & $\mu$ \\
 \hline\hline
-Mercury & 0.00000016 \\
-Venus   & 0.00000244 \\
-EM Bary & 0.00000304 \\
-Mars    & 0.00000032 \\
-Jupiter & 0.00095479 \\
-Saturn  & 0.00028588 \\
-Uranus  & 0.00004366 \\
-Neptune & 0.00005151 \\
+Mercury & 0.00000016601367952719 \\
+Venus   & 0.00000244783833966454 \\
+EM Bary & 0.0000030404326333266 \\
+Mars    & 0.00000032271514450539 \\
+Jupiter & 0.00095479193842432214 \\
+Saturn  & 0.0002858859806661029 \\
+Uranus  & 0.00004366244043351564 \\
+Neptune & 0.00005151389020535497 \\
 \end{tabular}
 \end{center}
 \end{table}
@@ -464,16 +480,20 @@ Neptune & 0.00005151 \\
 
 Входните данни на програмата ще бъдат орбиталните елементи в началото на епохата, рожденната ми дата (в Юлиански векове от началото на епохата) и съотношенията на масите $\mu$.
 
+\newpage
+
+\hfill
+
 \hfill
 
 Първо ще разгледаме елементите на Делоне: $L$, $G$, $\Theta$, $l$, $g$, $\theta$.
 
 Искаме да обновим величините от началото на епохата спрямо време $t$ и да изкарваме на екрана стойности с 8 цифри прецизност.
-Копираме функциите \texttt{atDay} и \texttt{p} от 1 задача:
+Копираме функциите \texttt{atDay} и \texttt{p} от 1 задача, като увеличаваме прецизността на 21 цифри след десетичната запетая:
 
 \begin{lstlisting}[language=R]
 atDay <- function(epochConstant, rate, t) { epochConstant + t * rate }
-p <- function(x) { formatC(x, format="f", digits = 8) }
+p <- function(x) { formatC(x, format="f", digits = 21) }
 \end{lstlisting}
 
 Величината $\theta$ е орбитален елемент.
@@ -515,14 +535,6 @@ G <- function(L, e) {
 Theta <- function(G, i) { G * cos(i) }
 \end{lstlisting}
 
-\newpage
-
-\hfill
-
-\hfill
-
-\hfill
-
 Обединявайки всичко, получаваме следната функция за намиране на елементите на Делоне по подадени време $t$, съотношение на маси $\mu$, орбитални елементи в началото на епохата и тяхното изменение за век:
 
 \begin{lstlisting}[language=R]
@@ -547,44 +559,44 @@ delaunay <- function(t, mu, a2000, da, e2000, de, i2000, di, lambda2000, dlambda
 }
 \end{lstlisting}
 
+\newpage
+
+\hfill
+
 Дефинираме функцията \texttt{delaunayAtBirth}, която връща всички елементи на Делоне за всички планети с подадени константи:
 
 \begin{lstlisting}[language=R]
 delaunayAtBirth <- function() {
-  birthDate <- 0.02320328
+  birthDate <- 0.023203285421
   rbind(
-    delaunay(birthDate, 0.00000016,    # Меркурии
+    delaunay(birthDate, 0.00000016601367952719,    # Меркурии
                0.38709927,    0.00000037, 0.20563593, 0.00001906, 0.12225995, -0.00010380,
                4.40259868, 2608.79030501, 1.35189358, 0.00280085, 0.84353100, -0.00218761),
-    delaunay(birthDate, 0.00000244,    # Венера
+    delaunay(birthDate, 0.00000244783833966454,    # Венера
                0.72333566,    0.00000390, 0.00677672, -0.00004107, 0.05924827, -0.00001377,
                3.17613446, 1021.32854958, 2.29689636,  0.00004683, 1.33831572, -0.00484668),
-    delaunay(birthDate, 0.00000304,    # EM Bary
+    delaunay(birthDate, 0.0000030404326333266,     # EM Bary
                1.00000261,   0.00000562, 0.01671123, -0.00004392, -0.00000027, -0.00022596,
                1.75343756, 628.30757790, 1.79660147,  0.00564219,  0.0,         0.0),
-    delaunay(birthDate, 0.00000032,    # Марс
+    delaunay(birthDate, 0.00000032271514450539,    # Марс
                 1.52371034,   0.00001847,  0.09339410, 0.00007882, 0.03228321, -0.00014192,
                -0.07947238, 334.06130168, -0.41789517, 0.00775643, 0.86497713, -0.00510637),
-    delaunay(birthDate, 0.00095479,    # Юпитер
+    delaunay(birthDate, 0.00095479193842432214,    # Юпитер
                5.20288700, -0.00011607, 0.04838624, -0.00013253, 0.02276602, -0.00003206,
                0.60033114, 52.96631189, 0.25706047,  0.00370929, 1.75360053,  0.00357253),
-    delaunay(birthDate, 0.00028588,    # Сатурн
+    delaunay(birthDate, 0.0002858859806661029,     # Сатурн
                9.53667594, -0.00125060, 0.05386179, -0.00050991, 0.04338874,  0.00003379,
                0.87186604, 21.33653879, 1.61615531, -0.00731244, 1.98378354, -0.00503838),
-    delaunay(birthDate, 0.00004366,    # Уран
+    delaunay(birthDate, 0.00004366244043351564,    # Уран
                19.18916464, -0.00196176, 0.04725744, -0.00004397, 0.01348507, -0.00004240,
                 5.46703627,  7.47842217, 2.98371499,  0.00712187, 1.29183904,  0.00074012),
-    delaunay(birthDate, 0.00005151,    # Нептун
+    delaunay(birthDate, 0.00005151389020535497,    # Нептун
                30.06992276, 0.00026291, 0.00859048,  0.00005105, 0.03089309,  0.00000617,
                -0.96202600, 3.81283674, 0.78478315, -0.00562720, 2.30006864, -0.00008878)
   )
 }
 \end{lstlisting}
 
-\newpage
-
-\hfill
-
 \hfill
 
 \hfill
@@ -623,6 +635,8 @@ poincare1 <- function() {
 	\end{matrix}
 \end{align*}
 
+\newpage
+
 Напълно аналогично на \texttt{poincare1}, дефинираме \texttt{poincare2}:
 
 \begin{lstlisting}[language=R]
@@ -657,62 +671,124 @@ p(poincare2())
 
 И получаваме следните резултати:
 
-\break
+\hfill
+
+\hfill
+
+\hfill
+
+\hfill
+
+\hfill
+
+\hfill
+
+\begin{table}[!h]
+\begin{center}
+\begin{tabular}{c|c|c|c}
+Делоне & $L$ & $G$ & $\Theta$ \\
+\hline\hline
+Mercury & 0.000000103289225349226 & 0.000000101081776576878 & 0.000000100327287123918 \\
+Venus   & 0.000002081863094111913 & 0.000002081815303335347 & 0.000002078162453508473 \\
+EM Bary & 0.000003040432177217194 & 0.000003040007655895516 & 0.000003040007655849318 \\
+Mars    & 0.000000398355249808993 & 0.000000396614057982431 & 0.000000396407441380291 \\
+Jupiter & 0.002176825400448131106 & 0.002174276008944916582 & 0.002173712615531286918 \\
+Saturn  & 0.000882731532142888770 & 0.000881450722537746524 & 0.000880621120751847700 \\
+Uranus  & 0.000191260833739243018 & 0.000191047155475361307 & 0.000191029787586196811 \\
+Neptune & 0.000282474574098807182 & 0.000282464148236046821 & 0.000282329368214719945 \\
+\end{tabular}
+\end{center}
+\end{table}
+
+\begin{table}[!h]
+\begin{center}
+\begin{tabular}{c|c|c|c}
+Делоне & $l$ & $g$ & $\theta$ \\
+\hline\hline
+Mercury & 63.583146161762698511666 & 0.508478328661191358862 & 0.843480240260780167461 \\
+Venus   & 24.577414857910831358367 & 0.958694185509240348608 & 1.338203261100615826251 \\
+EM Bary & 14.535505234845919275699 & 1.796732387344969428256 & 0.000000000000000000000 \\
+Mars    & 8.089562546332686565620 & -1.282573840780286689167 & 0.864858645439424700285 \\
+Jupiter & 1.572177054766796810981 & -1.496536886718685899211 & 1.753683424433265081177 \\
+Saturn  & -0.249041797926948138553 & -0.367680995663244303273 & 1.983666633030800374726 \\
+Uranus  & 2.656679993326903410633 & 1.692024027566735533412 & 1.291856213215605730937 \\
+Neptune & -1.658208241330383803103 & -1.515413999540041523062 & 2.300066580012320560655 \\
+\end{tabular}
+\end{center}
+\end{table}
+
+\begin{table}[!h]
+\begin{center}
+\begin{tabular}{c|c|c|c}
+Поанкаре 1 вид & $L$ & $L - G$ & $G - \Theta$ \\
+\hline\hline
+Mercury & 0.000000103289225349226 & 0.000000002207448772348 & 0.000000000754489452959 \\
+Venus   & 0.000002081863094111913 & 0.000000000047790776567 & 0.000000003652849826874 \\
+EM Bary & 0.000003040432177217194 & 0.000000000424521321678 & 0.000000000000000046198 \\
+Mars    & 0.000000398355249808993 & 0.000000001741191826562 & 0.000000000206616602140 \\
+Jupiter & 0.002176825400448131106 & 0.000002549391503214524 & 0.000000563393413629664 \\
+Saturn  & 0.000882731532142888770 & 0.000001280809605142246 & 0.000000829601785898824 \\
+Uranus  & 0.000191260833739243018 & 0.000000213678263881711 & 0.000000017367889164495 \\
+Neptune & 0.000282474574098807182 & 0.000000010425862760361 & 0.000000134780021326876 \\
+\end{tabular}
+\end{center}
+\end{table}
 
 \begin{table}[!h]
 \begin{center}
-\begin{tabular}{c|c|c|c|c|c|c}
-Делоне & $L$ & $G$ & $\Theta$ & $l$ & $g$ & $\theta$ \\
+\begin{tabular}{c|c|c|c}
+Поанкаре 1 вид & $l + g + \theta$ & $-g - \theta$ & $-\theta$ \\
 \hline\hline
-Mercury & 0.00000010 & 0.00000010 & 0.00000010 & 63.58313202 & 0.50847833 & 0.84348024 \\
-Venus   & 0.00000208 & 0.00000208 & 0.00000207 & 24.57740932 & 0.95869419 & 1.33820326 \\
-EM Bary & 0.00000304 & 0.00000304 & 0.00000304 & 14.53550183 & 1.79673239 & 0.00000000 \\
-Mars    & 0.00000040 & 0.00000039 & 0.00000039 & 8.08956074 & -1.28257384 & 0.86485865 \\
-Jupiter & 0.00217682 & 0.00217427 & 0.00217371 & 1.57217677 & -1.49653689 & 1.75368342 \\
-Saturn  & 0.00088271 & 0.00088143 & 0.00088060 & -0.24904191 & -0.36768100 & 1.98366663 \\
-Uranus  & 0.00019125 & 0.00019104 & 0.00019102 & 2.65667995 & 1.69202403 & 1.29185621 \\
-Neptune & 0.00028245 & 0.00028244 & 0.00028231 & -1.65820826 & -1.51541400 & 2.30006658 \\
+Mercury & 64.935104730684670926166 & -1.351958568921971526322 & -0.843480240260780167461 \\
+Venus   & 26.874312304520689309584 & -2.296897446609856174859 & -1.338203261100615826251 \\
+EM Bary & 16.332237622190888259865 & -1.796732387344969428256 & -0.000000000000000000000 \\
+Mars    & 7.671847350991824576738 & 0.417715195340861988882 & -0.864858645439424700285 \\
+Jupiter & 1.829323592481375992946 & -0.257146537714579181966 & -1.753683424433265081177 \\
+Saturn  & 1.366943839440607932900 & -1.615985637367556071453 & -1.983666633030800374726 \\
+Uranus  & 5.640560234109244674983 & -2.983880240782341264349 & -1.291856213215605730937 \\
+Neptune & -0.873555660858104765509 & -0.784652580472279037593 & -2.300066580012320560655 \\
 \end{tabular}
 \end{center}
 \end{table}
 
 \begin{table}[!h]
 \begin{center}
-\begin{tabular}{c|c|c|c|c|c|c}
-Поанкаре 1 вид & $L$ & $L - G$ & $G - \Theta$ & $l + g + \theta$ & $-g - \theta$ & $-\theta$ \\
+\begin{tabular}{c|c|c|c}
+Поанкаре 2 вид & $L$ & $\xi$ & $p$ \\
 \hline\hline
-Mercury & 0.00000010 & 0.00000000 & 0.00000000 & 64.93509059 & -1.35195857 & -0.84348024 \\
-Venus   & 0.00000208 & 0.00000000 & 0.00000000 & 26.87430677 & -2.29689745 & -1.33820326 \\
-EM Bary & 0.00000304 & 0.00000000 & 0.00000000 & 16.33223422 & -1.79673239 & -0.00000000 \\
-Mars    & 0.00000040 & 0.00000000 & 0.00000000 & 7.67184554 & 0.41771520 & -0.86485865 \\
-Jupiter & 0.00217682 & 0.00000255 & 0.00000056 & 1.82932331 & -0.25714654 & -1.75368342 \\
-Saturn  & 0.00088271 & 0.00000128 & 0.00000083 & 1.36694372 & -1.61598564 & -1.98366663 \\
-Uranus  & 0.00019125 & 0.00000021 & 0.00000002 & 5.64056019 & -2.98388024 & -1.29185621 \\
-Neptune & 0.00028245 & 0.00000001 & 0.00000013 & -0.87355568 & -0.78465258 & -2.30006658 \\
+Mercury & 0.000000103289225349226 & 0.000014424827544034668  & 0.000025827152487161843  \\
+Venus   & 0.000002081863094111913 & -0.000006491251753606220 & 0.000019701746607823352  \\
+EM Bary & 0.000003040432177217194 & -0.000006527533112570199 & 0.000000009612286302040  \\
+Mars    & 0.000000398355249808993 & 0.000053937790938604713  & 0.000013187832795805843  \\
+Jupiter & 0.002176825400448131106 & 0.002183802985649161126  & -0.000193054628745658468 \\
+Saturn  & 0.000882731532142888770 & -0.000072301145995106139 & -0.000516837678338016642 \\
+Uranus  & 0.000191260833739243018 & -0.000645611834774382142 & 0.000051316013457963606  \\
+Neptune & 0.000282474574098807182 & 0.000102183215085003621  & -0.000345950770443006518 \\
 \end{tabular}
 \end{center}
 \end{table}
 
 \begin{table}[!h]
 \begin{center}
-\begin{tabular}{c|c|c|c|c|c|c}
-Поанкаре 2 вид & $L$ & $\xi$ & $p$ & $l + g + \theta$ & $\eta$ & $q$ \\
+\begin{tabular}{c|c|c|c}
+Поанкаре 2 вид & $l + g + \theta$ & $\eta$ & $q$ \\
 \hline\hline
-Mercury & 0.00000010 & 0.00001416 & 0.00002536 & 64.93509059 & -0.00006367 & 0.00002849 \\
-Venus   & 0.00000208 & -0.00000648 & 0.00001967 & 26.87430677 & -0.00000730 & 0.00008304 \\
-EM Bary & 0.00000304 & -0.00000653 & 0.00000001 & 16.33223422 & -0.00002840 & 0.00000000 \\
-Mars    & 0.00000040 & 0.00005371 & 0.00001313 & 7.67184554 & 0.00002384 & 0.00001540 \\
-Jupiter & 0.00217682 & 0.00218380 & -0.00019305 & 1.82932331 & -0.00057427 & 0.00104380 \\
-Saturn  & 0.00088271 & -0.00007230 & -0.00051683 & 1.36694372 & -0.00159886 & 0.00117985 \\
-Uranus  & 0.00019125 & -0.00064559 & 0.00005131 & 5.64056019 & -0.00010267 & 0.00017917 \\
-Neptune & 0.00028245 & 0.00010218 & -0.00034594 & -0.87355568 & -0.00010203 & 0.00038713 \\
+Mercury & 64.935104730684670926166 & -0.000064860017692109048 & 0.000029016152403852770 \\
+Venus   & 26.874312304520689309584 & -0.000007310622668714660 & 0.000083171755027473018 \\
+EM Bary & 16.332237622190888259865 & -0.000028397780808021289 & 0.000000000000000000000 \\
+Mars    & 7.671847350991824576738  & 0.000023939472880300536  & 0.000015469785726697444 \\
+Jupiter & 1.829323592481375992946  & -0.000574271300256992776 & 0.001043799184507826754 \\
+Saturn  & 1.366943839440607932900  & -0.001598872025701959104 & 0.001179865410141265995 \\
+Uranus  & 5.640560234109244674983  & -0.000102673689729541247 & 0.000179171552127487335 \\
+Neptune & -0.873555660858104765509 & -0.000102030956457410801 & 0.000387140939560313784 \\
 \end{tabular}
 \end{center}
 \end{table}
 
 \vfill
 
-\begin{thebibliography}{9}
+\renewcommand{\refname}{Източници}
+\begin{thebibliography}{2}
 \bibitem{nasa} Solar System Dynamics. (2025, Май 1). \textit{Approximate Positions of the Planets.} Jet Propultion Laboratory, National Aeronautics and Space Administration. \url{https://ssd.jpl.nasa.gov/planets/approx_pos.html}
 \bibitem{mass} Williams, James G., \& Standish, Myles E. (2006). Orbital Ephemerides of the Sun, Moon, and Planets. In Seidelmann, P. K. (Eds.), \textit{Explanatory Supplement to the Astronomical Almanac}. \url{https://www.researchgate.net/publication/232203657_Orbital_Ephemerides_of_the_Sun_Moon_and_Planets}
 \end{thebibliography}
diff --git a/Year 4/Celestial mechanics/results.txt b/Year 4/Celestial mechanics/results.txt
index d25eda0..d4c166a 100644
--- a/Year 4/Celestial mechanics/results.txt
+++ b/Year 4/Celestial mechanics/results.txt
@@ -1,44 +1,44 @@
 Изход след изпълнение на Exercise1.R
 
-     [,1]          [,2]           [,3]          [,4]          [,5]          [,6]          [,7]          [,8]
-[1,] "-0.25698453" "0.21643004"   "0.04126659"  "0.33850545"  "-1.38648452" "-1.18390004" "0.03053503"  "1.82342837"
-[2,] "-0.11664062" "0.70946840"   "0.01643289"  "0.71918043"  "-1.16420814" "-0.19720358" "0.06450187"  "1.18255247"
-[3,] "-0.79848552" "-0.61311058"  "0.00000338"  "1.00671928"  "0.59280163"  "-0.79699190" "0.00000439"  "0.99328237"
-[4,] "0.01018262"  "1.56865936"   "0.03261130"  "1.56903135"  "-0.78255752" "0.07431329"  "0.02078708"  "0.78635288"
-[5,] "-1.81325044" "4.88999123"   "0.02034819"  "5.21539121"  "-0.41697241" "-0.13210905" "0.00988286"  "0.43751167"
-[6,] "2.07612673"  "8.79599349"   "-0.23579568" "9.04076343"  "-0.33293384" "0.07398502"  "0.01195145"  "0.34126466"
-[7,] "16.49642718" "-11.30522661" "-0.25582952" "20.00014268" "0.12742638"  "0.17777408"  "-0.00099186" "0.21872835"
-[8,] "18.92712374" "-23.39768921" "0.04565204"  "30.09468321" "0.14091098"  "0.11598482"  "-0.00563534" "0.18259282"
+     [,1]            [,2]             [,3]            [,4]            [,5]            [,6]            [,7]            [,8]
+[1,] "-0.2569892522" "0.2164260053"   "0.0412666944"  "0.3385064720"  "-1.3864619581" "-1.1839190472" "0.0305314033"  "1.8234234939"
+[2,] "-0.1166445847" "0.7094677286"   "0.0164331060"  "0.7191804113"  "-1.1642070756" "-0.1972100811" "0.0645017151"  "1.1825524945"
+[3,] "-0.7984835049" "-0.6131132909"  "0.0000033801"  "1.0067193328"  "0.5928042970"  "-0.7969898554" "0.0000043938"  "0.9932823185"
+[4,] "0.0101799533"  "1.5686596135"   "0.0326113752"  "1.5690315855"  "-0.7825575318" "0.0743119103"  "0.0207870540"  "0.7863527531"
+[5,] "-1.8132518650" "4.8899907784"   "0.0203482204"  "5.2153912785"  "-0.4169723661" "-0.1321091661" "0.0098828616"  "0.4375116648"
+[6,] "2.0761255995"  "8.7959937429"   "-0.2357956393" "9.0407634088"  "-0.3329338533" "0.0739849818"  "0.0119514504"  "0.3412646559"
+[7,] "16.4964276183" "-11.3052260072" "-0.2558295226" "20.0001426991" "0.1274263738"  "0.1777740834"  "-0.0009918592" "0.2187283458"
+[8,] "18.9271242195" "-23.3976888193" "0.0456520201"  "30.0946832084" "0.1409109730"  "0.1159848250"  "-0.0056353427" "0.1825928231"
 
 Изход след изпълнение на Exercise2.R
 
 > p(delaunayAtBirth())
-     [,1]         [,2]         [,3]         [,4]          [,5]          [,6]
-[1,] "0.00000010" "0.00000010" "0.00000010" "63.58313202" "0.50847833"  "0.84348024"
-[2,] "0.00000208" "0.00000208" "0.00000207" "24.57740932" "0.95869419"  "1.33820326"
-[3,] "0.00000304" "0.00000304" "0.00000304" "14.53550183" "1.79673239"  "0.00000000"
-[4,] "0.00000040" "0.00000039" "0.00000039" "8.08956074"  "-1.28257384" "0.86485865"
-[5,] "0.00217682" "0.00217427" "0.00217371" "1.57217677"  "-1.49653689" "1.75368342"
-[6,] "0.00088271" "0.00088143" "0.00088060" "-0.24904191" "-0.36768100" "1.98366663"
-[7,] "0.00019125" "0.00019104" "0.00019102" "2.65667995"  "1.69202403"  "1.29185621"
-[8,] "0.00028245" "0.00028244" "0.00028231" "-1.65820826" "-1.51541400" "2.30006658"
+     [,1]                      [,2]                      [,3]                      [,4]                       [,5]                       [,6]
+[1,] "0.000000103289225349226" "0.000000101081776576878" "0.000000100327287123918" "63.583146161762698511666" "0.508478328661191358862"  "0.843480240260780167461"
+[2,] "0.000002081863094111913" "0.000002081815303335347" "0.000002078162453508473" "24.577414857910831358367" "0.958694185509240348608"  "1.338203261100615826251"
+[3,] "0.000003040432177217194" "0.000003040007655895516" "0.000003040007655849318" "14.535505234845919275699" "1.796732387344969428256"  "0.000000000000000000000"
+[4,] "0.000000398355249808993" "0.000000396614057982431" "0.000000396407441380291" "8.089562546332686565620"  "-1.282573840780286689167" "0.864858645439424700285"
+[5,] "0.002176825400448131106" "0.002174276008944916582" "0.002173712615531286918" "1.572177054766796810981"  "-1.496536886718685899211" "1.753683424433265081177"
+[6,] "0.000882731532142888770" "0.000881450722537746524" "0.000880621120751847700" "-0.249041797926948138553" "-0.367680995663244303273" "1.983666633030800374726"
+[7,] "0.000191260833739243018" "0.000191047155475361307" "0.000191029787586196811" "2.656679993326903410633"  "1.692024027566735533412"  "1.291856213215605730937"
+[8,] "0.000282474574098807182" "0.000282464148236046821" "0.000282329368214719945" "-1.658208241330383803103" "-1.515413999540041523062" "2.300066580012320560655"
 > p(poincare1())
-     [,1]         [,2]         [,3]         [,4]          [,5]          [,6]
-[1,] "0.00000010" "0.00000000" "0.00000000" "64.93509059" "-1.35195857" "-0.84348024"
-[2,] "0.00000208" "0.00000000" "0.00000000" "26.87430677" "-2.29689745" "-1.33820326"
-[3,] "0.00000304" "0.00000000" "0.00000000" "16.33223422" "-1.79673239" "-0.00000000"
-[4,] "0.00000040" "0.00000000" "0.00000000" "7.67184554"  "0.41771520"  "-0.86485865"
-[5,] "0.00217682" "0.00000255" "0.00000056" "1.82932331"  "-0.25714654" "-1.75368342"
-[6,] "0.00088271" "0.00000128" "0.00000083" "1.36694372"  "-1.61598564" "-1.98366663"
-[7,] "0.00019125" "0.00000021" "0.00000002" "5.64056019"  "-2.98388024" "-1.29185621"
-[8,] "0.00028245" "0.00000001" "0.00000013" "-0.87355568" "-0.78465258" "-2.30006658"
+     [,1]                      [,2]                      [,3]                      [,4]                       [,5]                       [,6]
+[1,] "0.000000103289225349226" "0.000000002207448772348" "0.000000000754489452959" "64.935104730684670926166" "-1.351958568921971526322" "-0.843480240260780167461"
+[2,] "0.000002081863094111913" "0.000000000047790776567" "0.000000003652849826874" "26.874312304520689309584" "-2.296897446609856174859" "-1.338203261100615826251"
+[3,] "0.000003040432177217194" "0.000000000424521321678" "0.000000000000000046198" "16.332237622190888259865" "-1.796732387344969428256" "-0.000000000000000000000"
+[4,] "0.000000398355249808993" "0.000000001741191826562" "0.000000000206616602140" "7.671847350991824576738"  "0.417715195340861988882"  "-0.864858645439424700285"
+[5,] "0.002176825400448131106" "0.000002549391503214524" "0.000000563393413629664" "1.829323592481375992946"  "-0.257146537714579181966" "-1.753683424433265081177"
+[6,] "0.000882731532142888770" "0.000001280809605142246" "0.000000829601785898824" "1.366943839440607932900"  "-1.615985637367556071453" "-1.983666633030800374726"
+[7,] "0.000191260833739243018" "0.000000213678263881711" "0.000000017367889164495" "5.640560234109244674983"  "-2.983880240782341264349" "-1.291856213215605730937"
+[8,] "0.000282474574098807182" "0.000000010425862760361" "0.000000134780021326876" "-0.873555660858104765509" "-0.784652580472279037593" "-2.300066580012320560655"
 > p(poincare2())
-     [,1]         [,2]          [,3]          [,4]          [,5]          [,6]
-[1,] "0.00000010" "0.00001416"  "0.00002536"  "64.93509059" "-0.00006367" "0.00002849"
-[2,] "0.00000208" "-0.00000648" "0.00001967"  "26.87430677" "-0.00000730" "0.00008304"
-[3,] "0.00000304" "-0.00000653" "0.00000001"  "16.33223422" "-0.00002840" "0.00000000"
-[4,] "0.00000040" "0.00005371"  "0.00001313"  "7.67184554"  "0.00002384"  "0.00001540"
-[5,] "0.00217682" "0.00218380"  "-0.00019305" "1.82932331"  "-0.00057427" "0.00104380"
-[6,] "0.00088271" "-0.00007230" "-0.00051683" "1.36694372"  "-0.00159886" "0.00117985"
-[7,] "0.00019125" "-0.00064559" "0.00005131"  "5.64056019"  "-0.00010267" "0.00017917"
-[8,] "0.00028245" "0.00010218"  "-0.00034594" "-0.87355568" "-0.00010203" "0.00038713"
+     [,1]                      [,2]                       [,3]                       [,4]                       [,5]                       [,6]
+[1,] "0.000000103289225349226" "0.000014424827544034668"  "0.000025827152487161843"  "64.935104730684670926166" "-0.000064860017692109048" "0.000029016152403852770"
+[2,] "0.000002081863094111913" "-0.000006491251753606220" "0.000019701746607823352"  "26.874312304520689309584" "-0.000007310622668714660" "0.000083171755027473018"
+[3,] "0.000003040432177217194" "-0.000006527533112570199" "0.000000009612286302040"  "16.332237622190888259865" "-0.000028397780808021289" "0.000000000000000000000"
+[4,] "0.000000398355249808993" "0.000053937790938604713"  "0.000013187832795805843"  "7.671847350991824576738"  "0.000023939472880300536"  "0.000015469785726697444"
+[5,] "0.002176825400448131106" "0.002183802985649161126"  "-0.000193054628745658468" "1.829323592481375992946"  "-0.000574271300256992776" "0.001043799184507826754"
+[6,] "0.000882731532142888770" "-0.000072301145995106139" "-0.000516837678338016642" "1.366943839440607932900"  "-0.001598872025701959104" "0.001179865410141265995"
+[7,] "0.000191260833739243018" "-0.000645611834774382142" "0.000051316013457963606"  "5.640560234109244674983"  "-0.000102673689729541247" "0.000179171552127487335"
+[8,] "0.000282474574098807182" "0.000102183215085003621"  "-0.000345950770443006518" "-0.873555660858104765509" "-0.000102030956457410801" "0.000387140939560313784"