code: 
  
  
  
  
  from __future__ import annotations
  
  import random
  
  from matplotlib import animation
  from matplotlib import pyplot as plt
  
  # Frame rate of the animation
  SCREAMING_SNAKE_CASE : List[Any]          =    20
  
  # Time between time steps in seconds
  SCREAMING_SNAKE_CASE : List[Any]          =    INTERVAL / 10_00
  
  
  
  
  
  class 					snake_case__  :
  
  
  
  
  
  	def __init__(  self				,    UpperCamelCase_				,    UpperCamelCase_				,    UpperCamelCase_				,    UpperCamelCase_				,    UpperCamelCase_ = 1.0				,    UpperCamelCase_ = 1.0				,    UpperCamelCase_ = "blue"				,    )		->			None:
  								"""simple docstring"""
  
  
  
  
  
  
  								a_       :   List[str]			=	position_x
  								a_       :   List[str]			=	position_y
  								a_       :   List[Any]			=	velocity_x
  								a_       :   Dict			=	velocity_y
  								a_       :   List[Any]			=	mass
  								a_       :   int			=	size
  								a_       :   Any			=	color
  
  
  
  
  
  	@property
  	def     A				(  self							)		->			tuple[float, float]:
  								"""simple docstring"""
  
  
  
  
  
  
  								return self.position_x, self.position_y
  
  
  
  
  
  	@property
  	def     A				(  self							)		->			tuple[float, float]:
  								"""simple docstring"""
  
  
  
  
  
  
  								return self.velocity_x, self.velocity_y
  
  
  
  
  
  	def     A				(  self				,    UpperCamelCase_				,    UpperCamelCase_				,    UpperCamelCase_							)		->			None:
  								"""simple docstring"""
  
  
  
  
  
  
  								self.velocity_x += force_x * delta_time
  								self.velocity_y += force_y * delta_time
  
  
  
  
  
  	def     A				(  self				,    UpperCamelCase_							)		->			None:
  								"""simple docstring"""
  
  
  
  
  
  
  								self.position_x += self.velocity_x * delta_time
  								self.position_y += self.velocity_y * delta_time
  
  
  
  
  
  class 					snake_case__  :
  
  
  
  
  
  	def __init__(  self				,    UpperCamelCase_				,    UpperCamelCase_ = 1.0				,    UpperCamelCase_ = 1.0				,    UpperCamelCase_ = 0.0				,    )		->			None:
  								"""simple docstring"""
  
  
  
  
  
  
  								a_       :   List[Any]			=	bodies
  								a_       :   int			=	gravitation_constant
  								a_       :   List[str]			=	time_factor
  								a_       :   Optional[int]			=	softening_factor
  
  
  
  
  
  	def __len__(  self							)		->			int:
  								"""simple docstring"""
  
  
  
  
  
  
  								return len(self.bodies							)
  
  
  
  
  
  	def     A				(  self				,    UpperCamelCase_							)		->			None:
  								"""simple docstring"""
  
  
  
  
  
  
  								for bodya in self.bodies:
  															a_       :   List[str]			=	0.0
  															a_       :   Dict			=	0.0
  															for bodya in self.bodies:
  																						if bodya != bodya:
  																													a_       :   Dict			=	bodya.position_x - bodya.position_x
  																													a_       :   List[str]			=	bodya.position_y - bodya.position_y
  
  																													# Calculation of the distance using Pythagoras's theorem
  																													# Extra factor due to the softening technique
  																													a_       :   Optional[Any]			=	(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(UpperCamelCase_				,    UpperCamelCase_				,    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]      ):
  
  
  
  
  
  
  							"""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_       :   int			=	(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	,	):
  
  
  
  
  
  
  							"""simple docstring"""
  
  
  
  							a_       :   Dict			=	plt.figure()
  							fig.canvas.set_window_title(SCREAMING_SNAKE_CASE_      )
  							a_       :   int			=	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_       :   Union[str, Any]			=	[
  							    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_       :   int			=	animation.FuncAnimation(  # noqa: F841
  							    SCREAMING_SNAKE_CASE_	,	SCREAMING_SNAKE_CASE_	,	interval=SCREAMING_SNAKE_CASE_	,	blit=SCREAMING_SNAKE_CASE_      )
  
  							plt.show()
  
  
  
  
  
  def 						_lowerCamelCase		(							):
  
  
  
  
  
  
  							"""simple docstring"""
  
  
  
  							a_       :   Optional[int]			=	0.9_700_436
  							a_       :   int			=	-0.24_308_753
  							a_       :   Tuple			=	0.466_203_685
  							a_       :   Tuple			=	0.43_236_573
  
  							a_       :   Any			=	[
  							    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		(							):
  
  
  
  
  
  
  							"""simple docstring"""
  
  
  
  							a_       :   Tuple			=	7.3476e22
  							a_       :   Union[str, Any]			=	5.972e24
  							a_       :   int			=	10_22
  							a_       :   Optional[Any]			=	3_84_39_90_00
  							a_       :   int			=	6.674e-11
  
  							# Calculation of the respective velocities so that total impulse is zero,
  							# i.e. the two bodies together don't move
  							a_       :   Optional[int]			=	earth_mass * velocity_dif / (earth_mass + moon_mass)
  							a_       :   Optional[int]			=	moon_velocity - velocity_dif
  
  							a_       :   Optional[Any]			=	Body(-earth_moon_distance	,	0	,	0	,	SCREAMING_SNAKE_CASE_	,	SCREAMING_SNAKE_CASE_	,	10_00_00_00	,	"""grey"""      )
  							a_       :   Optional[Any]			=	Body(0	,	0	,	0	,	SCREAMING_SNAKE_CASE_	,	SCREAMING_SNAKE_CASE_	,	50_00_00_00	,	"""blue"""      )
  							return BodySystem([earth, moon]	,	SCREAMING_SNAKE_CASE_	,	time_factor=1_00_00_00      )
  
  
  
  
  
  def 						_lowerCamelCase		(							):
  
  
  
  
  
  
  							"""simple docstring"""
  
  
  
  							a_       :   Dict			=	[]
  							for _ in range(10      ):
  														a_       :   Union[str, Any]			=	random.uniform(-0.5	,	0.5      )
  														a_       :   str			=	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	,	10	,	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_30_00_00_00,
  	    4_30_00_00_00,
  	    -4_30_00_00_00,
  	    4_30_00_00_00,
  	)
  	plot("Random system with many bodies", example_a(), -1.5, 1.5, -1.5, 1.5)
  
  
  
style_context: 
  
  
  from typing import TYPE_CHECKING
  
  from ...utils import (
      OptionalDependencyNotAvailable,
      _LazyModule,
      is_sentencepiece_available,
      is_tf_available,
      is_tokenizers_available,
      is_torch_available,
  )
  
  
  SCREAMING_SNAKE_CASE : Optional[Any]          =    {
      "configuration_camembert": ["CAMEMBERT_PRETRAINED_CONFIG_ARCHIVE_MAP", "CamembertConfig", "CamembertOnnxConfig"],
  }
  
  try:
  	if not is_sentencepiece_available():
  		raise OptionalDependencyNotAvailable()
  except OptionalDependencyNotAvailable:
  	pass
  else:
  	SCREAMING_SNAKE_CASE : Optional[int]          =    ["CamembertTokenizer"]
  
  try:
  	if not is_tokenizers_available():
  		raise OptionalDependencyNotAvailable()
  except OptionalDependencyNotAvailable:
  	pass
  else:
  	SCREAMING_SNAKE_CASE : Any          =    ["CamembertTokenizerFast"]
  
  try:
  	if not is_torch_available():
  		raise OptionalDependencyNotAvailable()
  except OptionalDependencyNotAvailable:
  	pass
  else:
  	SCREAMING_SNAKE_CASE : Union[str, Any]          =    [
  	    "CAMEMBERT_PRETRAINED_MODEL_ARCHIVE_LIST",
  	    "CamembertForCausalLM",
  	    "CamembertForMaskedLM",
  	    "CamembertForMultipleChoice",
  	    "CamembertForQuestionAnswering",
  	    "CamembertForSequenceClassification",
  	    "CamembertForTokenClassification",
  	    "CamembertModel",
  	    "CamembertPreTrainedModel",
  	]
  
  try:
  	if not is_tf_available():
  		raise OptionalDependencyNotAvailable()
  except OptionalDependencyNotAvailable:
  	pass
  else:
  	SCREAMING_SNAKE_CASE : List[Any]          =    [
  	    "TF_CAMEMBERT_PRETRAINED_MODEL_ARCHIVE_LIST",
  	    "TFCamembertForCausalLM",
  	    "TFCamembertForMaskedLM",
  	    "TFCamembertForMultipleChoice",
  	    "TFCamembertForQuestionAnswering",
  	    "TFCamembertForSequenceClassification",
  	    "TFCamembertForTokenClassification",
  	    "TFCamembertModel",
  	    "TFCamembertPreTrainedModel",
  	]
  
  
  if TYPE_CHECKING:
  	from .configuration_camembert import CAMEMBERT_PRETRAINED_CONFIG_ARCHIVE_MAP, CamembertConfig, CamembertOnnxConfig
  
  	try:
  		if not is_sentencepiece_available():
  			raise OptionalDependencyNotAvailable()
  	except OptionalDependencyNotAvailable:
  		pass
  	else:
  		from .tokenization_camembert import CamembertTokenizer
  
  	try:
  		if not is_tokenizers_available():
  			raise OptionalDependencyNotAvailable()
  	except OptionalDependencyNotAvailable:
  		pass
  	else:
  		from .tokenization_camembert_fast import CamembertTokenizerFast
  
  	try:
  		if not is_torch_available():
  			raise OptionalDependencyNotAvailable()
  	except OptionalDependencyNotAvailable:
  		pass
  	else:
  		from .modeling_camembert import (
  		    CAMEMBERT_PRETRAINED_MODEL_ARCHIVE_LIST,
  		    CamembertForCausalLM,
  		    CamembertForMaskedLM,
  		    CamembertForMultipleChoice,
  		    CamembertForQuestionAnswering,
  		    CamembertForSequenceClassification,
  		    CamembertForTokenClassification,
  		    CamembertModel,
  		    CamembertPreTrainedModel,
  		)
  
  	try:
  		if not is_tf_available():
  			raise OptionalDependencyNotAvailable()
  	except OptionalDependencyNotAvailable:
  		pass
  	else:
  		from .modeling_tf_camembert import (
  		    TF_CAMEMBERT_PRETRAINED_MODEL_ARCHIVE_LIST,
  		    TFCamembertForCausalLM,
  		    TFCamembertForMaskedLM,
  		    TFCamembertForMultipleChoice,
  		    TFCamembertForQuestionAnswering,
  		    TFCamembertForSequenceClassification,
  		    TFCamembertForTokenClassification,
  		    TFCamembertModel,
  		    TFCamembertPreTrainedModel,
  		)
  
  else:
  	import sys
  
  	SCREAMING_SNAKE_CASE : Any          =    _LazyModule(__name__, globals()["__file__"], _import_structure, module_spec=__spec__)
  
  
  