code: from __future__ import annotations
  
  import random
  
  from matplotlib import animation
  from matplotlib import pyplot as plt
  
  # Frame rate of the animation
  a     :			Any					    =						2_0
  
  # Time between time steps in seconds
  a     :			Optional[Any]					    =						INTERVAL / 1_0_0_0
  
  class 				_lowercase						:
  
  
  
  
  
  
      '''simple docstring'''
  
  
  
  
      def __init__(						self					,	lowerCamelCase__					,	lowerCamelCase__					,	lowerCamelCase__					,	lowerCamelCase__					,	lowerCamelCase__ = 1.0					,	lowerCamelCase__ = 1.0					,	lowerCamelCase__ = "blue"					,	):
         lowerCAmelCase_:       List[str]         =  position_x
         lowerCAmelCase_:       Tuple         =  position_y
         lowerCAmelCase_:       List[str]         =  velocity_x
         lowerCAmelCase_:       Tuple         =  velocity_y
         lowerCAmelCase_:       int         =  mass
         lowerCAmelCase_:       Tuple         =  size
         lowerCAmelCase_:       Dict         =  color
  
  
  
  
      @property
      def     _a						(						self   ):
         return self.position_x, self.position_y
  
  
  
  
      @property
      def     _a						(						self   ):
         return self.velocity_x, self.velocity_y
  
  
  
  
      def     _a						(						self					,	lowerCamelCase__					,	lowerCamelCase__					,	lowerCamelCase__   ):
         self.velocity_x += force_x * delta_time
         self.velocity_y += force_y * delta_time
  
  
  
  
      def     _a						(						self					,	lowerCamelCase__   ):
         self.position_x += self.velocity_x * delta_time
         self.position_y += self.velocity_y * delta_time
  
  
  
  
  
  class 				_lowercase						:
  
  
  
  
  
  
      '''simple docstring'''
  
  
  
  
      def __init__(						self					,	lowerCamelCase__					,	lowerCamelCase__ = 1.0					,	lowerCamelCase__ = 1.0					,	lowerCamelCase__ = 0.0					,	):
         lowerCAmelCase_:       Optional[Any]         =  bodies
         lowerCAmelCase_:       Union[str, Any]         =  gravitation_constant
         lowerCAmelCase_:       Optional[Any]         =  time_factor
         lowerCAmelCase_:       Any         =  softening_factor
  
  
  
  
      def __len__(						self   ):
         return len(self.bodies   )
  
  
  
  
      def     _a						(						self					,	lowerCamelCase__   ):
         for bodya in self.bodies:
            lowerCAmelCase_:       List[Any]         =  0.0
            lowerCAmelCase_:       Union[str, Any]         =  0.0
            for bodya in self.bodies:
               if bodya != bodya:
                  lowerCAmelCase_:       int         =  bodya.position_x - bodya.position_x
                  lowerCAmelCase_:       str         =  bodya.position_y - bodya.position_y
  
                  # Calculation of the distance using Pythagoras's theorem
                  # Extra factor due to the softening technique
                  lowerCAmelCase_:       Dict         =  (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(lowerCamelCase__					,	lowerCamelCase__					,	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 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase				):
     body_system.update_system(lowercase				)
  
     # Update the positions of the patches
     for patch, body in zip(lowercase       ,							body_system.bodies				):
        lowerCAmelCase_:       Dict         =  (body.position_x, body.position_y)
  def 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase = -1       ,							lowercase = 1       ,							lowercase = -1       ,							lowercase = 1       ,							):
     lowerCAmelCase_:       List[Any]         =  plt.figure()
     fig.canvas.set_window_title(lowercase				)
     lowerCAmelCase_:       Dict         =  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
     lowerCAmelCase_:       int         =  [
         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(lowercase				)
  
     # Function called at each step of the animation
     def update(lowercase				) -> list[plt.Circle]:
        update_step(lowercase       ,							lowercase       ,							lowercase				)
        return patches
  
     lowerCAmelCase_:       Union[str, Any]         =  animation.FuncAnimation(  # noqa: F841
         lowercase       ,							lowercase       ,							interval=lowercase       ,							blit=lowercase				)
  
     plt.show()
  def 		snake_case__  (		):
     lowerCAmelCase_:       Tuple         =  0.9700436
     lowerCAmelCase_:       List[str]         =  -0.24308753
     lowerCAmelCase_:       List[Any]         =  0.466203685
     lowerCAmelCase_:       Any         =  0.43236573
  
     lowerCAmelCase_:       Tuple         =  [
         Body(lowercase       ,							lowercase       ,							lowercase       ,							lowercase       ,							size=0.2       ,							color="red"				),
         Body(-position_x       ,							-position_y       ,							lowercase       ,							lowercase       ,							size=0.2       ,							color="green"				),
         Body(0       ,							0       ,							-2 * velocity_x       ,							-2 * velocity_y       ,							size=0.2       ,							color="blue"				),
     ]
     return BodySystem(lowercase       ,							time_factor=3				)
  def 		snake_case__  (		):
     lowerCAmelCase_:       Tuple         =  7.3476E22
     lowerCAmelCase_:       List[Any]         =  5.972E24
     lowerCAmelCase_:       str         =  1022
     lowerCAmelCase_:       Optional[Any]         =  384399000
     lowerCAmelCase_:       str         =  6.674E-11
  
     # Calculation of the respective velocities so that total impulse is zero,
     # i.e. the two bodies together don't move
     lowerCAmelCase_:       int         =  earth_mass * velocity_dif / (earth_mass + moon_mass)
     lowerCAmelCase_:       Tuple         =  moon_velocity - velocity_dif
  
     lowerCAmelCase_:       Tuple         =  Body(-earth_moon_distance       ,							0       ,							0       ,							lowercase       ,							lowercase       ,							10000000       ,							"grey"				)
     lowerCAmelCase_:       str         =  Body(0       ,							0       ,							0       ,							lowercase       ,							lowercase       ,							50000000       ,							"blue"				)
     return BodySystem([earth, moon]       ,							lowercase       ,							time_factor=1000000				)
  def 		snake_case__  (		):
     lowerCAmelCase_:       Optional[Any]         =  []
     for _ in range(10				):
        lowerCAmelCase_:       Optional[int]         =  random.uniform(-0.5       ,							0.5				)
        lowerCAmelCase_:       int         =  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				)       ,							lowercase       ,							lowercase       ,							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(lowercase       ,							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_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)
style_context: from .balanced_parentheses import balanced_parentheses
  from .stack import Stack
  def 		snake_case__  (		lowercase				):
     return {"+": 1, "-": 1, "*": 2, "/": 2, "^": 3}.get(lowercase       ,							-1				)
  def 		snake_case__  (		lowercase				):
     if not balanced_parentheses(lowercase				):
        raise ValueError("Mismatched parentheses"				)
     lowerCAmelCase_:       Stack[str]         =  Stack()
     lowerCAmelCase_:       Union[str, Any]         =  []
     for char in expression_str:
        if char.isalpha() or char.isdigit():
           postfix.append(lowercase				)
        elif char == "(":
           stack.push(lowercase				)
        elif char == ")":
           while not stack.is_empty() and stack.peek() != "(":
              postfix.append(stack.pop()				)
           stack.pop()
        else:
           while not stack.is_empty() and precedence(lowercase				) <= precedence(stack.peek()				):
              postfix.append(stack.pop()				)
           stack.push(lowercase				)
     while not stack.is_empty():
        postfix.append(stack.pop()				)
     return " ".join(lowercase				)
  
  
  if __name__ == "__main__":
   from doctest import testmod
  
   testmod()
   a     :			int					    =						"""a+b*(c^d-e)^(f+g*h)-i"""
  
   print("""Infix to Postfix Notation demonstration:\n""")
   print("""Infix notation: """ + expression)
   print("""Postfix notation: """ + infix_to_postfix(expression))