

from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
lowerCAmelCase__						=	2_0

# Time between time steps in seconds
lowerCAmelCase__						=	INTERVAL / 1_0_0_0

class    a__  :
			"""simple docstring"""







			def __init__(      self , lowercase , lowercase , lowercase , lowercase , lowercase = 1.0 , lowercase = 1.0 , lowercase = "blue" , )					->  None:
							'''simple docstring'''






							A__            =     position_x
							A__            =     position_y
							A__            =     velocity_x
							A__            =     velocity_y
							A__            =     mass
							A__            =     size
							A__            =     color


			@property
			def        UpperCamelCase						(      self    )					->  tuple[float, float]:
							'''simple docstring'''






							return self.position_x, self.position_y


			@property
			def        UpperCamelCase						(      self    )					->  tuple[float, float]:
							'''simple docstring'''






							return self.velocity_x, self.velocity_y


			def        UpperCamelCase						(      self , lowercase , lowercase , lowercase    )					->  None:
							'''simple docstring'''






							self.velocity_x += force_x * delta_time
							self.velocity_y += force_y * delta_time


			def        UpperCamelCase						(      self , lowercase    )					->  None:
							'''simple docstring'''






							self.position_x += self.velocity_x * delta_time
							self.position_y += self.velocity_y * delta_time

class    a__  :
			"""simple docstring"""







			def __init__(      self , lowercase , lowercase = 1.0 , lowercase = 1.0 , lowercase = 0.0 , )					->  None:
							'''simple docstring'''






							A__            =     bodies
							A__            =     gravitation_constant
							A__            =     time_factor
							A__            =     softening_factor


			def __len__(      self    )					->  int:
							'''simple docstring'''






							return len(self.bodies    )


			def        UpperCamelCase						(      self , lowercase    )					->  None:
							'''simple docstring'''






							for bodya in self.bodies:
											A__            =     0.0
											A__            =     0.0
											for bodya in self.bodies:
															if bodya != bodya:
																			A__            =     bodya.position_x - bodya.position_x
																			A__            =     bodya.position_y - bodya.position_y

																			# Calculation of the distance using Pythagoras's theorem
																			# Extra factor due to the softening technique
																			A__            =     (dif_x**2 + dif_y**2 + self.softening_factor) ** (
																			    1 / 2
																			)

																			# Newton's law of universal gravitation.
																			force_x += (
																			    self.gravitation_constant * bodya.mass * dif_x / distance**3
																			)
																			force_y += (
																			    self.gravitation_constant * bodya.mass * dif_y / distance**3
																			)

            # Update the body's velocity once all the force components have been added
											bodya.update_velocity(lowercase , lowercase , delta_time * self.time_factor    )

							# Update the positions only after all the velocities have been updated
							for body in self.bodies:
											body.update_position(delta_time * self.time_factor    )



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							BodySystem		,    SCREAMING_SNAKE_CASE_:							float		,    SCREAMING_SNAKE_CASE_:							list[plt.Circle] )     -> None:




				'''simple docstring'''





				body_system.update_system(SCREAMING_SNAKE_CASE_ )

				# Update the positions of the patches
				for patch, body in zip(SCREAMING_SNAKE_CASE_		,    body_system.bodies ):
								A__            =     (body.position_x, body.position_y)



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							str		,    SCREAMING_SNAKE_CASE_:							BodySystem		,    SCREAMING_SNAKE_CASE_:							float = -1		,    SCREAMING_SNAKE_CASE_:							float = 1		,    SCREAMING_SNAKE_CASE_:							float = -1		,    SCREAMING_SNAKE_CASE_:							float = 1		,    )     -> None:




				'''simple docstring'''





				A__            =     plt.figure()
				fig.canvas.set_window_title(SCREAMING_SNAKE_CASE_ )
				A__            =     plt.axes(
				    xlim=(x_start, x_end)		,    ylim=(y_start, y_end) )  # Set section to be plotted
				plt.gca().set_aspect("equal" )  # Fix aspect ratio

				# Each body is drawn as a patch by the plt-function
				A__            =     [
				    plt.Circle((body.position_x, body.position_y)		,    body.size		,    fc=body.color )
				    for body in body_system.bodies
				]

				for patch in patches:
								ax.add_patch(SCREAMING_SNAKE_CASE_ )

				# Function called at each step of the animation
				def update(SCREAMING_SNAKE_CASE_:							int ) -> list[plt.Circle]:
								update_step(SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_ )
								return patches

				A__            =     animation.FuncAnimation(  # noqa: F841
				    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    interval=SCREAMING_SNAKE_CASE_		,    blit=SCREAMING_SNAKE_CASE_ )

				plt.show()



def    lowerCAmelCase__							(		)     -> BodySystem:




				'''simple docstring'''





				A__            =     0.9700436
				A__            =     -0.24308753
				A__            =     0.466203685
				A__            =     0.43236573

				A__            =     [
				    Body(SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    size=0.2		,    color="red" ),
				    Body(-position_x		,    -position_y		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    size=0.2		,    color="green" ),
				    Body(0		,    0		,    -2 * velocity_x		,    -2 * velocity_y		,    size=0.2		,    color="blue" ),
				]
				return BodySystem(SCREAMING_SNAKE_CASE_		,    time_factor=3 )



def    lowerCAmelCase__							(		)     -> BodySystem:




				'''simple docstring'''





				A__            =     7.3_476e22
				A__            =     5.972e24
				A__            =     1_0_2_2
				A__            =     3_8_4_3_9_9_0_0_0
				A__            =     6.674e-11

				# Calculation of the respective velocities so that total impulse is zero,
				# i.e. the two bodies together don't move
				A__            =     earth_mass * velocity_dif / (earth_mass + moon_mass)
				A__            =     moon_velocity - velocity_dif

				A__            =     Body(-earth_moon_distance		,    0		,    0		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    1_0_0_0_0_0_0_0		,    "grey" )
				A__            =     Body(0		,    0		,    0		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    5_0_0_0_0_0_0_0		,    "blue" )
				return BodySystem([earth, moon]		,    SCREAMING_SNAKE_CASE_		,    time_factor=1_0_0_0_0_0_0 )



def    lowerCAmelCase__							(		)     -> BodySystem:




				'''simple docstring'''





				A__            =     []
				for _ in range(1_0 ):
								A__            =     random.uniform(-0.5		,    0.5 )
								A__            =     random.uniform(-0.5		,    0.5 )

								# Bodies are created pairwise with opposite velocities so that the
								# total impulse remains zero
								bodies.append(
								    Body(
								        random.uniform(-0.5		,    0.5 )		,    random.uniform(-0.5		,    0.5 )		,    SCREAMING_SNAKE_CASE_		,    SCREAMING_SNAKE_CASE_		,    size=0.05		,    ) )
								bodies.append(
								    Body(
								        random.uniform(-0.5		,    0.5 )		,    random.uniform(-0.5		,    0.5 )		,    -velocity_x		,    -velocity_y		,    size=0.05		,    ) )
				return BodySystem(SCREAMING_SNAKE_CASE_		,    0.01		,    1_0		,    0.1 )


if __name__ == "__main__":
	plot("""Figure-8 solution to the 3-body-problem""", example_a(), -2, 2, -2, 2)
	plot(
	    """Moon's orbit around the earth""",
	    example_a(),
	    -4_3_0_0_0_0_0_0_0,
	    4_3_0_0_0_0_0_0_0,
	    -4_3_0_0_0_0_0_0_0,
	    4_3_0_0_0_0_0_0_0,
	)
	plot("""Random system with many bodies""", example_a(), -1.5, 1.5, -1.5, 1.5)




import math



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							int )     -> bool:




				'''simple docstring'''





				if 1 < number < 4:
								# 2 and 3 are primes
								return True
				elif number < 2 or number % 2 == 0 or number % 3 == 0:
								# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
								return False

				# All primes number are in format of 6k +/- 1
				for i in range(5		,    int(math.sqrt(SCREAMING_SNAKE_CASE_ ) + 1 )		,    6 ):
								if number % i == 0 or number % (i + 2) == 0:
												return False
				return True



def    lowerCAmelCase__							(		SCREAMING_SNAKE_CASE_:							int = 2_0_0_0_0_0_0 )     -> int:




				'''simple docstring'''





				return sum(num for num in range(3		,    SCREAMING_SNAKE_CASE_		,    2 ) if is_prime(SCREAMING_SNAKE_CASE_ ) ) + 2 if n > 2 else 0


if __name__ == "__main__":
	print(f"""{solution() = }""")

