Optimize the method of leapfrog

1. Optimize the method of leapfrog
2. write new method of rkn, but regarding this method, I think it is not applicable to the problem in the project. Maybe the Tutor will inform me of the result(if it is suitable for the problem) tomorrow(17.01.2024)

tasks/src/integrators.py

+import numpy as np
+
+
+def leapfrog(initial_states, num_steps, dt, gravitational_constant):
+
+    positions, velocities = initial_states
+    traj = [positions]
+
+    for _ in range(num_steps):
+        # Leapfrog integration
+        accelerations = -gravitational_constant * positions / np.linalg.norm(positions, axis=1)[:, None]**3
+        velocities += accelerations * dt
+        positions += velocities * dt
+
+        traj.append(positions.copy())
+
+    return np.array(traj)
+
+
+def stormer_verlet(initial_states, num_steps, dt, gravitational_constant):
+    positions, velocities = initial_states
+    traj = [positions]
+
+    for _ in range(num_steps):
+        # Stormer-Verlet integration
+        half_velocities = velocities + 0.5 * dt * (-gravitational_constant * positions /
+                                                   np.linalg.norm(positions, axis=1)[:, None]**3)
+        positions += dt * half_velocities
+        velocities = half_velocities + 0.5 * dt * (-gravitational_constant * positions /
+                                                   np.linalg.norm(positions, axis=1)[:, None]**3)
+
+        traj.append(positions.copy())
+
+    return np.array(traj)