adding new java implementation for verlet and some images to FFT

Author: leios

chapters/computational_mathematics/cooley_tukey.md

@@ -43,6 +43,14 @@ Where $$F(\xi)$$ represents a function in frequency space and $$\xi$$ represents
 
 If we take a sinusoidal function like $$\sin(\omega t)$$ or $$\cos(\omega t)$$, we find a curve that goes from $$\pm1$$, shown in FIGURE1a. Both of these curves can be described by their corresponding frequencies, $$\omega$$. So, instead of representing these curves as seen in FIGURE1a, We could instead describe them as shown in FIGURE1b. Here, FIGURE1a is in real space and FIGURE1b is in frequency space. 
 
+![Complicated Sinusoidal Function](sinusoid.png)
+
+*FIGURE1a: Complicated Sinusoidal Function*
+
+![FFT of FIGURE1a](fft.png)
+
+*FIGURE1b: abs(fft(FIGURE1a))*
+
 Now, how does this relate to the transformations above? Well, the easiest way is to substitute in the following relation:
 
 $$e^{2 \pi i \theta} = \cos(2 \pi \theta) + i \sin(2 \pi \theta)$$

chapters/computational_mathematics/fft.png

@@ -181,3 +181,74 @@ int main(){
 }
 
 ```
+
+#### Java
+```java
+
+// Submitted by lolatomroflsinnlos
+// Thanks a lot!
+static void verlet(double pos, double acc, double dt){
+
+    // Note that we are using a temp variable for the previous position
+    double prev_pos, temp_pos, time;
+    prev_pos = pos;
+    time = 0;
+
+    while (pos > 0){
+        time += dt;
+        temp_pos = pos;
+        pos = pos*2 - prev_pos + acc * dt * dt;
+        prev_pos = temp_pos;
+    }
+
+    System.out.println(time);
+
+}
+
+// Simple function for stormer-verlet
+static void stormer_verlet(double pos, double acc, double dt){
+
+    // Note that we are using a temp variable for the previous position
+    double prev_pos, temp_pos, time, vel;
+    prev_pos = pos;
+    vel = 0;
+    time = 0;
+    while (pos > 0){
+        time += dt;
+        temp_pos = pos;
+        pos = pos*2 - prev_pos + acc * dt * dt;
+        prev_pos = temp_pos;
+ 
+       // The acceleration is constant, so the velocity is straightforward
+       vel += acc*dt;
+    }
+ 
+    System.out.println(time);
+ 
+}
+ 
+// Simple function for velocity-verlet
+static void velocity_verlet(double pos, double acc, double dt){
+ 
+    // Note that we are using a temp variable for the previous position
+    double time, vel;
+    vel = 0;
+    time = 0;
+    while (pos > 0){
+        time += dt;
+        pos += vel*dt + 0.5*acc * dt * dt;
+        vel += acc*dt;
+    }
+ 
+    System.out.println(time);
+ 
+}
+ 
+public static void main(String[] args) {
+ 
+    verlet(5.0, -10, 0.01);
+    stormer_verlet(5.0, -10, 0.01);
+    velocity_verlet(5.0, -10, 0.01);
+ 
+}
+```