Merge pull request #811 from abhro/patch-1

Update bullet list in table in outer_solar_system.md

Author: ChrisRackauckas

docs/src/examples/outer_solar_system.md

@@ -4,13 +4,13 @@
 
 The chosen units are masses relative to the sun, meaning the sun has mass $1$. We have taken $m_0 = 1.00000597682$ to take account of the inner planets. Distances are in astronomical units, times in earth days, and the gravitational constant is thus $G = 2.95912208286 \cdot 10^{-4}$.
 
-| planet  | mass                         | initial position                                                     | initial velocity                                                     |
-|:------- |:---------------------------- |:-------------------------------------------------------------------- |:-------------------------------------------------------------------- |
-| Jupiter | $m_1 = 0.000954786104043$    | <ul><li>-3.5023653</li><li>-3.8169847</li><li>-1.5507963</li></ul>   | <ul><li>0.00565429</li><li>-0.00412490</li><li>-0.00190589</li></ul> |
-| Saturn  | $m_2 = 0.000285583733151$    | <ul><li>9.0755314</li><li>-3.0458353</li><li>-1.6483708</li></ul>    | <ul><li>0.00168318</li><li>0.00483525</li><li>0.00192462</li></ul>   |
-| Uranus  | $m_3 = 0.0000437273164546$   | <ul><li>8.3101420</li><li>-16.2901086</li><li>-7.2521278</li></ul>   | <ul><li>0.00354178</li><li>0.00137102</li><li>0.00055029</li></ul>   |
-| Neptune | $m_4 = 0.0000517759138449$   | <ul><li>11.4707666</li><li>-25.7294829</li><li>-10.8169456</li></ul> | <ul><li>0.00288930</li><li>0.00114527</li><li>0.00039677</li></ul>   |
-| Pluto   | $ m_5 = 1/(1.3 \cdot 10^8 )$ | <ul><li>-15.5387357</li><li>-25.2225594</li><li>-3.1902382</li></ul> | <ul><li>0.00276725</li><li>-0.00170702</li><li>-0.00136504</li></ul> |
+| planet  | mass                        | initial position                        | initial velocity                       |
+|:------- |:--------------------------- |:--------------------------------------- |:-------------------------------------- |
+| Jupiter | $m_1 = 0.000954786104043$   | [-3.5023653,   -3.8169847,  -1.5507963] | [0.00565429, -0.00412490, -0.00190589] |
+| Saturn  | $m_2 = 0.000285583733151$   | [9.0755314,    -3.0458353,  -1.6483708] | [0.00168318,  0.00483525,  0.00192462] |
+| Uranus  | $m_3 = 0.0000437273164546$  | [8.3101420,   -16.2901086,  -7.2521278] | [0.00354178,  0.00137102,  0.00055029] |
+| Neptune | $m_4 = 0.0000517759138449$  | [11.4707666,  -25.7294829, -10.8169456] | [0.00288930,  0.00114527,  0.00039677] |
+| Pluto   | $m_5 = 1/(1.3 \cdot 10^8 )$ | [-15.5387357, -25.2225594,  -3.1902382] | [0.00276725, -0.00170702, -0.00136504] |
 
 The data is taken from the book “Geometric Numerical Integration” by E. Hairer, C. Lubich and G. Wanner.
 
@@ -42,7 +42,11 @@ tspan = (0.0, 200_000.0)
 
 The N-body problem's Hamiltonian is
 
-$$H(p,q) = \frac{1}{2}\sum_{i=0}^{N}\frac{p_{i}^{T}p_{i}}{m_{i}} - G\sum_{i=1}^{N}\sum_{j=0}^{i-1}\frac{m_{i}m_{j}}{\left\lVert q_{i}-q_{j} \right\rVert}$$
+```math
+H(p,q) = \frac{1}{2}\sum_{i=0}^{N}\frac{p_i^T p_i}{m_i} - G\sum_{i=1}^N \sum_{j=0}^{i-1}\frac{m_i m_j}{\left\lVert q_i - q_j \right\rVert}
+```
+
+where each ``p_i`` and ``q_i`` is a 3-dimensional vector describing the planet's position and momentum, respectively.
 
 Here, we want to solve for the motion of the five outer planets relative to the sun, namely, Jupiter, Saturn, Uranus, Neptune, and Pluto.
 
@@ -61,11 +65,15 @@ potential = -G *
 
 `NBodyProblem` constructs a second order ODE problem under the hood. We know that a Hamiltonian system has the form of
 
-$$\dot{p} = -H_{q}(p,q)\quad \dot{q}=H_{p}(p,q)$$
+```math
+\dot{p} = -H_{q}(p,q), \quad \dot{q} = H_{p}(p,q)
+```
 
-For an N-body system, we can symplify this as:
+For an N-body system, we can simplify this as:
 
-$$\dot{p} = -\nabla{V}(q)\quad \dot{q}=M^{-1}p.$$
+```math
+\dot{p} = -\nabla V(q), \quad \dot{q} = M^{-1} p.
+```
 
 Thus, $\dot{q}$ is defined by the masses. We only need to define $\dot{p}$, and this is done internally by taking the gradient of $V$. Therefore, we only need to pass the potential function and the rest is taken care of.
 
@@ -84,3 +92,4 @@ for i in 1:N
 end
 Plots.plot!(plt; xlab = "x", ylab = "y", zlab = "z", title = "Outer solar system")
 ```
+