Document initial diff sim stable orbit study

_code/notes/diffsim/glut_grapher.cpp

@@ -64,9 +64,9 @@ void GlutGrapher::display() const {
 void GlutGrapher::idle() {
     bool const wait_to_start = false;
     if (wait_to_start && frame_ == 0) {
-        std::cout << "Press enter to start a 20 second countdown" << std::endl;
+        std::cout << "Press enter to start a 10 second countdown" << std::endl;
         do {} while (std::cin.get() != '\n');
-        for (uint32_t i = 20; i > 0; --i) {
+        for (uint32_t i = 10; i > 0; --i) {
             std::cout << i << "... " << std::flush;
             std::this_thread::sleep_for(std::chrono::seconds(1));
         }

_notes/differentiable_simulation.md

@@ -80,3 +80,19 @@ particular particle.
 $$
 a_i = \sum_{j \neq i} G m_j \frac{x_j - x_i}{|x_j - x_i|^3}
 $$
+
+To start, let's consider two particles. I'll make one 100 times heavier than the other, and our goal
+will be to find a stable circular orbit by varying the initial velocity of the satellite. As a
+baseline, here's a really bad trajectory, in which the satellite gains escape velocity and doesn't
+end up orbiting at all.
+
+<video width="480" height="320" controls="controls" muted plays-inline>
+<source type="video/mp4" src="../assets/mp4/diffsim_bad_orbit.mp4">
+</video>
+
+Differentiating through the simulation makes it very easy to find the desired solution. This orbit
+was found by gradient descent, but Nelder Mead finds the same optimum.
+
+<video width="480" height="320" controls="controls" muted plays-inline>
+<source type="video/mp4" src="../assets/mp4/diffsim_good_orbit.mp4">
+</video>