from typing import List, Tuple
import matplotlib.pyplot as plt


def euler(
    initial_q: float, initial_p: float, n: int = 10000, delta: float = 0.01
) -> Tuple[List[float], List[float]]:
    q_list = []
    p_list = []
    q = initial_q
    p = initial_p

    for _ in range(n):
        q_list.append(q)
        p_list.append(p)

        _q = q + delta * p
        _p = p - delta * q

        q = _q
        p = _p

    return (q_list, p_list)


def symplectic_euler(
    initial_q: float, initial_p: float, n: int = 10000, delta: float = 0.01
) -> Tuple[List[float], List[float]]:
    q_list = []
    p_list = []
    q = initial_q
    p = initial_p

    for _ in range(n):
        q_list.append(q)
        p_list.append(p)

        _q = q + delta * p
        _p = p - delta * _q

        q = _q
        p = _p

    return (q_list, p_list)


fig = plt.figure()
axes = fig.subplots(2)

euler_result = euler(1.0, 0.0)
axes[0].scatter(euler_result[0], euler_result[1], s=1)

symplectic_euler_result = symplectic_euler(1.0, 0.0)
axes[1].scatter(symplectic_euler_result[0], symplectic_euler_result[1], s=1)

plt.savefig("graph.png")