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lorentz.py
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from manimlib.imports import *
import math as mt
# DRIVER CODE
# python manim.py my_projects\For_Videos\Special_Relativity\lorentz.py -pl
class TwoFriends(CustomScene):
def construct(self):
man2 = self.add_man(file_name = "running.svg").to_edge(LEFT + 0.5*UP).shift(1*RIGHT)
man1 = self.add_man(velocity_label = "0",velocity_color = RED,frame_label = "S_{1}",add_vector = False).next_to(man2,DOWN,buff = 1)
self.write_by_index(man1,time_gap = 1,single_anim_time = 1)
self.write_by_index(man2,time_gap = 1,single_anim_time = 1)
self.wait(2)
self.play(man2.shift,4*RIGHT)
self.wait()
dot = Dot(radius = 0.5).to_edge(RIGHT).set_color(TEAL)
line1 = self.horizontal_distance_measuring_line(man1.get_bottom(),(dot.get_x() - man1.get_x())*RIGHT,label = "x_{1}").shift(0.5*DOWN)
line2 = self.horizontal_distance_measuring_line(man2.get_bottom(),(dot.get_x() - man2.get_x())*RIGHT,label = "x_{2}")
line3 = self.horizontal_distance_measuring_line(man2.get_top(),man1.get_x()*RIGHT - man2.get_x()*RIGHT,label = "vt")
self.play(ShowCreation(line3),run_time = 3)
self.wait(2)
self.play(FadeIn(dot))
self.wait(2)
self.play(ShowCreation(line1),run_time = 3)
self.wait(2)
self.play(ShowCreation(line2),run_time = 3)
self.wait(2)
galtran = TexMobject("x_{2}"," = ","x_{1}"," - ","vt").scale(2)
for i,color in [(0,RED),(2,RED),(-1,RED)]:
galtran[i].set_color(color)
galtran1 = TexMobject("t_{2}"," = ","t_{1}"," = ","t").next_to(galtran,DOWN,buff = 0.5).scale(2)
for i,color in [(0,RED),(-1,RED),(2,RED)]:
galtran1[i].set_color(color)
galtran_text = TextMobject("\"Galilean Transformation\"").scale(1.5).to_edge(UP).set_color_by_gradient(PINK,YELLOW,GREEN,RED)
self.play(*[FadeOut(x)
for x in [man1,man2,dot,line1,line2,line3]],
*[ReplacementTransform(x.copy(),galtran[j])
for x,j in [(line1[-1],2),(line2[-1],0),(line3[-1],-1)]],
Write(galtran[1]),Write(galtran[3]),
run_time = 3)
self.wait(2)
box = SurroundingRectangle(galtran,buff = 0.5,color = GREEN)
self.play(Write(box))
self.wait(2)
box_new = SurroundingRectangle(VGroup(galtran,galtran1),buff = 0.5,color = GREEN)
self.play(Transform(box,box_new),Write(galtran1),run_time = 2)
self.wait(2)
self.play(Write(galtran_text),run_time = 2)
self.wait(2)
self.play(*[FadeOut(x)
for x in [box,galtran,galtran1,galtran_text]])
self.wait(2)
self.play(*[FadeIn(x)
for x in [man1,man2,dot,line1,line2,line3]])
self.wait(2)
man2_new = self.add_man(file_name = "running.svg",velocity_label = "v \\approx c").to_edge(LEFT + 0.5*UP).shift(5*RIGHT)
self.play(Transform(man2[2],man2_new[2].shift(0.8*RIGHT)))
self.wait(2)
self.play(*[FadeOut(x)
for x in [man1,man2,dot,line1,line2,line3]])
self.wait()
def add_man(self,file_name = "standing.svg",vector_side = RIGHT,velocity_label = "v",velocity_color = GREEN,frame_label = "S_{2}",add_vector = True,flip_vertical_image = False,scale = 1):
man = SVGMobject("./extmedia/svg_images/" + file_name).rotate(PI,RIGHT).scale(scale)
if flip_vertical_image:
man.rotate(PI,axis = UP)
if add_vector:
vector = Vector(vector_side).set_color(velocity_color)
if (vector_side[0] == 1):
vector.set_left(man.get_right())
elif (vector_side[0] == -1):
vector.set_right(man.get_left())
else:
raise Exception("Not Implemented")
vector_label = TexMobject(velocity_label).move_to(vector.get_end() + 0.2*vector_side).set_color(velocity_color)
frame_label = TexMobject(frame_label,color = YELLOW).next_to(man,UP + vector_side,buff = 0.3)
if add_vector:
return VGroup(man,vector,vector_label,frame_label)
else:
return VGroup(man,frame_label)
def horizontal_distance_measuring_line(self,start,vector,label = None,label_direction = DOWN):
line = Line(start,start + vector).set_color(GREY)
tick_start = Line(-0.2*UP,0.2*UP).set_stroke(WHITE,5).move_to(start).set_color(GREY)
tick_end = Line(-0.2*UP,0.2*UP).set_stroke(WHITE,5).move_to(start + vector).set_color(GREY)
if label is not(None):
label = TexMobject(label)
label.next_to(line,label_direction,buff=0.2).set_color(RED)
return VGroup(line,tick_start,tick_end,label)
class ReferenceFrames(TwoFriends):
def construct(self):
man2 = self.add_man(file_name = "running.svg").to_edge(LEFT + 0.5*UP).shift(1*RIGHT)
man1 = self.add_man(velocity_label = "0",velocity_color = RED,frame_label = "S_{1}",add_vector = False).next_to(man2,DOWN,buff = 1)
man1_new,man2_new = self.flipped_reference(man1,man2)
graph0 = self.get_graphs([(lambda m:m/3,GREEN,-5,5)],x_label = "t_{1}",y_label ="x_{1}",scale = 0.5).shift(1.5*RIGHT)
graph1 = self.get_graphs([((lambda m:3*m),GREEN,-5/3,5/3),(self.light_path1,WHITE,-5,5),(self.light_path2,WHITE,-5,5)],
scale = 0.5).shift(1.5*RIGHT)
graph2 = self.get_graphs([((lambda m:-3*m),RED,-5/3,5/3),(self.light_path1,WHITE,-5,5),(self.light_path2,WHITE,-5,5)],
x_label = "x_{2}",y_label = "ct_{2}",frame_label = "S_{2}",scale=0.5).shift(1.5*RIGHT)
l1 = self.get_light_beam(man1.get_right())
l2 = self.get_light_beam(man2.get_right())
self.play(*[FadeIn(x)
for x in [man1,man2]])
self.wait(2)
self.scale_and_wait(man1,wait_time = 2)
self.wait(2)
self.play(Write(graph0[0][:3]),Write(graph0[-1]),run_time = 2)
self.wait(2)
self.highlight(man2[0])
self.wait(2)
self.play(Write(graph0[0][3]),run_time = 2)
self.wait(2)
self.play(*[ReplacementTransform(i,j)
for i,j in [(graph0[0][0],graph1[0][0]),(graph0[0][1],graph1[0][2]),(graph0[0][2],graph1[0][1]),(graph0[-1],graph1[-1])]],
FadeOut(graph0[0][-1]),
run_time = 3)
self.wait(2)
self.highlight(graph1[0][2])
self.wait(2)
self.highlight(man1[0])
self.wait(2)
man1_line1 = Line(-5*UP,5*UP,color = RED).move_to(graph1[0][0].get_center()).scale(0.5).rotate(PI,axis = RIGHT)
self.play(Write(man1_line1),run_time = 2)
self.wait(2)
self.highlight(man2[0])
self.wait(2)
self.play(Write(graph1[0][-1][0]),run_time = 2)
self.wait(2)
label_vect1 = Matrix(["v/c","1"]).set_color(GREEN).scale(0.5)
self.play(Write(label_vect1.next_to(graph1[0][-1][0],2*RIGHT + 1*UP,buff = 0.2)),run_time = 2)
self.wait(2)
self.play(FadeOut(label_vect1))
self.wait()
for i in range(3):
self.play(l1.copy().shift,20*RIGHT,run_time = 2)
self.wait(1)
self.play(Write(graph1[0][-1][1]))
self.play(Write(graph1[0][-1][-1]))
self.wait(2)
self.scale_and_wait(man2,wait_time = 2)
self.wait(2)
man2_line2 = Line(-5*UP,5*UP,color = GREEN).move_to(graph1[0][0].get_center()).scale(0.5)
self.play(*[ReplacementTransform(x,y)
for x,y in [(man1,man1_new),(man2,man2_new),
(graph1[0][0].copy(),graph2[0][0]),
(graph1[0][1].copy(),graph2[0][1]),
(graph1[0][2].copy(),graph2[0][2]),
(graph1[-1].copy(),graph2[-1]),
(graph1[0][-1][-1].copy(),graph2[0][-1][-1]),
(graph1[0][-1][-2].copy(),graph2[0][-1][-2]),
(graph1[0][-1][0].copy(),man2_line2),
(man1_line1.copy(),graph2[0][-1][0])]],
FadeOut(graph1),FadeOut(man1_line1)
,run_time = 3)
self.wait(2)
self.highlight_two_at_a_time(man2_new[0],man2_line2)
self.wait(2)
self.highlight_two_at_a_time(man1_new[0],graph2[0][-1][0])
self.wait(2)
label_vect2 = Matrix(["-v/c","1"]).set_color(RED).scale(0.5)
self.play(Write(label_vect2.next_to(graph2[0][-1][0],2*LEFT + 1*UP,buff = 0.2)),run_time = 2)
self.wait(2)
self.play(FadeOut(label_vect2))
self.wait()
self.highlight_two_at_a_time(graph2[0][-1][-1],graph2[0][-1][-2])
self.wait(2)
grp1 = VGroup(graph1.copy(),man1_line1.copy()).scale(0.5)
grp2 = VGroup(graph2.copy(),man2_line2.copy()).scale(0.5)
VGroup(grp2,grp1).arrange(DOWN,aligned_edge = RIGHT,buff = 0.5).to_edge(RIGHT,buff = 2)
self.play(*[ReplacementTransform(x,y)
for x,y in [(VGroup(graph1,man1_line1),grp1),
(VGroup(graph2,man2_line2),grp2)]],run_time = 3)
self.wait(2)
box1 = SurroundingRectangle(grp1,buff = 0.2)
box2 = SurroundingRectangle(grp2,buff = 0.2)
self.play(Write(box1),Write(box2),run_time = 2)
self.wait(2)
s1 = VGroup(grp1,box1)
s2 = VGroup(grp2,box2)
s1_new = VGroup(grp1,box1).copy().center().to_edge(LEFT,buff = 1.5).scale(1.5)
s2_new = VGroup(grp2,box2).copy().center().to_edge(RIGHT,buff = 1.5).scale(1.5)
self.play(*[FadeOut(x) for x in [man1_new,man2_new]],
*[Transform(x,y) for x,y in [(s1,s1_new),(s2,s2_new)]],
run_time = 3
)
self.wait(2)
arrow = Arrow(s1.get_right(),s2.get_left()).set_stroke(RED,5).set_color(RED)
self.highlight(s1)
self.wait(2)
self.play(Write(arrow),run_time = 2)
self.wait(2)
self.highlight(s2)
self.wait(2)
self.play(arrow.rotate,PI,axis = UP,run_time = 2)
self.wait(2)
self.play(*[FadeOut(x) for x in [s1,s2,arrow]])
self.wait(2)
def get_graphs(self,funcol,x_min = -5,x_max = 5,y_min = -5,y_max = 5,x_label = "x_{1}",y_label = "ct_{1}",frame_label = "S_{1}",scale = 0.5):
axes = Axes(x_min = x_min,x_max = x_max, y_min = y_min ,y_max = y_max,color = GREY,axis_config = {"include_tip":True})
all_graphs = VGroup()
x_label = TexMobject(x_label).scale(1/scale).shift((x_max + 0.3/scale)*RIGHT)
y_label = TexMobject(y_label).scale(1/scale).shift((y_max + 0.3/scale)*UP)
for function,color,x_min,x_max in funcol:
all_graphs.add(FunctionGraph(function,x_min = x_min,x_max = x_max,color = color))
grp = VGroup(axes,x_label,y_label,all_graphs)
frame_label = TexMobject(frame_label,color = YELLOW).next_to(grp,UR,buff = 0).scale(1/scale).set_stroke(YELLOW,3)
return VGroup(grp,frame_label).scale(scale).center()
def light_path1(self,x):
return x
def light_path2(self,x):
return -x
def get_light_beam(self,start_point):
light1 = Line(-0.2*RIGHT,0.2*RIGHT).set_stroke(WHITE,6).move_to(start_point)
light2 = Line(-0.2*RIGHT,0.2*RIGHT).set_stroke(WHITE,5).move_to(start_point - 0.1*UR)
light3 = Line(-0.2*RIGHT,0.2*RIGHT).set_stroke(WHITE,5).move_to(start_point + 0.1*UL)
light4 = Line(-0.2*RIGHT,0.2*RIGHT).set_stroke(WHITE,4).move_to(start_point - 0.2*UR)
light5 = Line(-0.2*RIGHT,0.2*RIGHT).set_stroke(WHITE,4).move_to(start_point + 0.2*UL)
light = VGroup(light1,light2,light3,light4,light5)
return light
def flipped_reference(self,man1,man2):
man1_new = self.add_man(file_name = "running.svg",vector_side = LEFT,velocity_label = "v",velocity_color = RED,frame_label = "S_{1}",flip_vertical_image = True)
man2_new = self.add_man(file_name = "standing.svg",add_vector = False,velocity_label = "0",vector_side = LEFT)
man1_new.move_to(man1.get_center())
man2_new.move_to(man2.get_center())
return (man1_new,man2_new)
def highlight_two_at_a_time(self,object1,object2,half_angle = PI/24,scale_factor = 1.3,single_anim_time = 0.5):
self.play(object1.scale,scale_factor,
object2.scale,scale_factor,
run_time = single_anim_time)
self.play(object1.rotate,half_angle,
object2.rotate,half_angle,
run_time = single_anim_time)
self.play(object1.rotate,-2*half_angle,
object2.rotate,-2*half_angle,
run_time = single_anim_time)
self.play(object1.rotate,half_angle,
object2.rotate,half_angle,
run_time = single_anim_time)
self.play(object1.scale,1/scale_factor,
object2.scale,1/scale_factor,
run_time = single_anim_time)
class Postulates(CustomScene):
def construct(self):
title = TextMobject("Postulates of \\\\ Special Relativity").scale(1.5).to_edge(UP).set_color(GREEN)
l = BulletedList("Laws of physics are same in any Inertial Frame.","Speed of Light is constant in any Inertial Frame.").set_color(TEAL_E)
self.play(Write(title))
self.wait()
self.write_by_index(l,time_gap = 1,single_anim_time = 2)
self.wait(4)
cons = TextMobject("\"Constant Velocity\"").to_edge(DOWN).shift(1*UP).scale(2).set_color(BLUE)
self.play(Write(cons),run_time = 2)
self.wait()
self.play(FadeOut(cons))
self.wait(2)
class ShowTransformation1to2(LinearTransformationScene):
CONFIG = {"lorentz_matrix" : [[3/(2*mt.sqrt(2)),1/(2*mt.sqrt(2))],[1/(2*mt.sqrt(2)),3/(2*mt.sqrt(2))]],
"show_basis_vectors" : False,
}
def construct(self):
self.setup()
s1 = TexMobject("S_{1}",color = YELLOW).set_stroke(YELLOW,5).scale(3).to_corner(UR)
s2 = TexMobject("S_{2}",color = YELLOW).set_stroke(YELLOW,5).scale(3).to_corner(UR)
for s in [s1,s2]:
s.add_to_back(BackgroundRectangle(s))
light1 = FunctionGraph(lambda x : x,x_min = -10,x_max = 10,color = WHITE)
light2 = FunctionGraph(lambda x : -x,x_min = -10,x_max = 10,color = WHITE)
line1 = Line(-20*UP,20*UP,color = RED)
line2 = FunctionGraph(lambda x : 3*x,x_min = -10,x_max = 10,color = GREEN)
self.add_transformable_mobject(line1,line2,light1,light2)
self.remove(line1,line2,light1,light2)
x_label,y_label = self.get_reference_labels("x_{1}","c t_{1}")
self.play(ShowCreation(self.plane),run_time = 2)
self.wait(2)
self.play(Write(x_label),Write(y_label),run_time = 2)
self.wait(2)
self.play(Write(s1))
self.wait(2)
self.play(Write(line1),run_time =2)
self.bring_to_front(x_label,y_label)
self.wait(2)
self.play(Write(line2),run_time =2)
self.wait(2)
x_label_new,y_label_new = self.get_reference_labels("x_{2}","c t_{2}")
self.bring_to_front(x_label,y_label)
self.apply_inverse_transpose(self.lorentz_matrix,
added_anims = [ReplacementTransform(x_label,x_label_new),Transform(y_label,y_label_new),ReplacementTransform(s1.copy(),s2),FadeOut(s1)])
self.bring_to_front(x_label_new,y_label_new)
self.wait(2)
x_label,y_label = self.get_reference_labels("x_{1}","c t_{1}")
self.wait(2)
self.apply_transposed_matrix(self.lorentz_matrix,
added_anims = [ReplacementTransform(x_label_new,x_label),ReplacementTransform(y_label_new,y_label),ReplacementTransform(s2.copy(),s1),FadeOut(s2)]
)
self.bring_to_front(x_label,y_label)
v1 = self.add_vectors_along(UR,BLUE,GOLD_E,ORANGE)
v2 = self.add_vectors_along(DR,BLUE,GOLD_E,ORANGE)
self.play(Write(v1),run_time = 2)
self.wait(2)
self.play(Write(v2),run_time = 2)
self.wait(2)
x_label_new,y_label_new = self.get_reference_labels("x_{2}","c t_{2}")
self.apply_inverse_transpose(self.lorentz_matrix,
added_anims = [ReplacementTransform(x_label,x_label_new),Transform(y_label,y_label_new),ReplacementTransform(s1.copy(),s2),FadeOut(s1)])
self.bring_to_front(x_label_new,y_label_new)
self.wait(2)
self.highlight(VGroup(light1,v1,))
self.wait(2)
self.highlight(VGroup(light2,v2))
self.wait(2)
matrix_mob = Matrix([["?","?"],["?","?"]],include_background_rectangle = True).shift(3*LEFT + 1*UP)
self.play(Write(matrix_mob))
self.wait(2)
def add_vectors_along(self,direction,*colors):
unit_vector = (direction[0]*RIGHT + direction[1]*UP + direction[2]*OUT)/self.magnitude(direction)
vectors = VGroup()
for i in range(-5,6,1):
if (i == 0):
vectors.add(self.get_vector(unit_vector))
vectors.add(self.get_vector(-1*unit_vector))
else:
vectors.add(self.get_vector(np.sign(i)*unit_vector).shift(i*unit_vector))
self.add_transformable_mobject(vectors.set_color_by_gradient(*colors))
self.remove(vectors)
return vectors
def magnitude(self,vector):
return (vector[0]**2 + vector[1]**2 + vector[2]**2)**0.5
def get_reference_labels(self,x_label,y_label,buff = 0.1):
x_label = TexMobject(x_label).to_edge(RIGHT,buff = buff)
y_label = TexMobject(y_label).to_edge(UP,buff = buff)
for lbl in [x_label,y_label]:
lbl.add_to_back(BackgroundRectangle(lbl))
return (x_label,y_label)
def get_special_mobjects(self, mob_list, *mobs_to_add):
for mobject in mobs_to_add:
if mobject not in mob_list:
mob_list.append(mobject)
return mob_list
def get_transformable_mobject(self,*mobjects):
self.get_special_mobjects(self.transformable_mobjects, *mobjects)
def add_to_moving_vectors(self, *mobjects):
for mobject in mobjects:
self.moving_vectors.append(mobject)
def remove_from_moving_vectors(self, *mobjects):
for mobject in mobjects:
self.moving_vectors.remove(mobject)
def remove_from_moving_mobjects(self, *mobjects):
for mobject in mobjects:
self.moving_mobjects.remove(mobject)
def add_moving_mobjects(self, *mobjects):
for mobject in mobjects:
self.add_moving_mobject(mobject)
class FindingTheMatrix(CustomScene):
def construct(self):
matrix = Matrix([["a","b"],["c","d"]]).set_color(BLUE)
vect1 = Matrix(["x_{1}","ct_{1}"]).set_color(RED)
vect2 = Matrix(["x_{2}","ct_{2}"]).set_color(GREEN)
eqn = self.get_matrix_equation(matrix,vect1,vect2)
brace_1 = Brace(eqn[1],UP,buff = 0.1)
brace_2 = Brace(eqn[-1],UP,buff = 0.1)
brace_1_tex = brace_1.get_tex("S_{1}")
brace_2_tex = brace_2.get_tex("S_{2}")
brace1 = VGroup(brace_1,brace_1_tex).set_color(YELLOW)
brace2 = VGroup(brace_2,brace_2_tex).set_color(YELLOW)
self.play(Write(eqn[1]))
self.wait(2)
self.play(FadeInFrom(brace1,UP))
self.wait(2)
self.play(Write(eqn[-1]))
self.wait(2)
self.play(FadeInFrom(brace2,UP))
self.wait(2)
self.play(Write(matrix),Write(eqn[-2]))
self.wait(2)
self.scale_and_wait(matrix,scale_factor = 1.1,wait_time = 2)
self.wait(2)
matrix_asign_eqn = self.get_matrix_assignment(matrix.copy(),"M")
self.play(*[FadeOut(x) for x in [brace1,brace2,eqn[1],eqn[-1]]],
*[ReplacementTransform(x,y) for x,y in [(eqn[2],matrix_asign_eqn[1]),(eqn[0],matrix_asign_eqn[-1])]],
Write(matrix_asign_eqn[0]),
run_time = 3
)
self.wait(2)
def get_matrix_equation(self,matrix,input_vect,output_vect,gap = 0.2):
input_vect.next_to(matrix,RIGHT,buff = gap)
eql = TexMobject(" = ").next_to(input_vect,RIGHT,buff = gap)
output_vect.next_to(eql,RIGHT,buff = gap)
VGroup(matrix,input_vect,eql,output_vect).center()
return VGroup(matrix,input_vect,eql,output_vect)
def get_matrix_equation_with_factor(self,matrix,input_vect,output_vect,factor = "\\lambda_{1}",gap = 0.2):
input_vect.next_to(matrix,RIGHT,buff = gap)
eql = TexMobject(" = ").next_to(input_vect,RIGHT,buff = gap)
factor = TexMobject(factor).next_to(eql,RIGHT,buff = gap).set_color(BLUE)
output_vect.next_to(factor,RIGHT,buff = gap)
VGroup(matrix,input_vect,eql,factor,output_vect).center()
return VGroup(matrix,input_vect,eql,factor,output_vect)
def get_matrix_assignment(self,matrix,assignment,gap = 0.2):
assignment = TexMobject(assignment).set_color(matrix.get_color())
eql = TexMobject(" = ").next_to(assignment,RIGHT,buff = gap)
matrix.next_to(eql,RIGHT,buff = gap)
VGroup(assignment,eql,matrix).center()
return VGroup(assignment,eql,matrix)
def get_vector_eqn_with_factor(self,vect1,vect2,factor = "\\lambda_{1}",gap = 0.2):
eql = TexMobject(" = ").next_to(vect1,RIGHT,buff = gap)
factor = TexMobject(factor).next_to(eql,RIGHT,buff = gap).set_color(BLUE)
vect2.next_to(factor,RIGHT,buff = gap)
VGroup(vect1,eql,factor,vect2).center()
return VGroup(vect1,eql,factor,vect2)
def get_vector_eqn(self,vect1,vect2,gap = 0.2):
eql = TexMobject(" = ").next_to(vect1,RIGHT,buff = gap)
vect2.next_to(eql,RIGHT,buff = gap)
VGroup(vect1,eql,vect2).center()
return VGroup(vect1,eql,vect2)
class UsingTheEigenVectors(FindingTheMatrix,ReferenceFrames):
def construct(self):
matrix = Matrix([["a","b"],["c","d"]]).set_color(BLUE)
matrix_new = Matrix([["a","b"],["b","a"]]).set_color(BLUE)
input_vect = Matrix(["x_{1}","ct_{1}"]).set_color(RED)
output_vect = Matrix(["x_{2}","ct_{2}"]).set_color(GREEN)
eigen_vect1 = Matrix([1,1]).set_color(RED)
eigen_vect2 = Matrix([1,-1]).set_color(RED)
eigen_vect1_new = Matrix(["a + b","c + d"]).set_color(PURPLE)
eigen_vect2_new = Matrix(["a - b","c - d"]).set_color(PURPLE)
matrix_asign_eqn = self.get_matrix_assignment(matrix_new.copy(),"M")
eqn = self.get_matrix_equation(matrix.copy(),input_vect.copy(),output_vect.copy())
eigen_eqn1 = self.get_matrix_equation_with_factor(matrix.copy(),eigen_vect1.copy(),eigen_vect1.copy().set_color(GREEN))
eigen_eqn2 = self.get_matrix_equation_with_factor(matrix.copy(),eigen_vect2.copy(),eigen_vect2.copy().set_color(GREEN),factor = "\\lambda_{2}")
eigen_eqn1_new = self.get_vector_eqn_with_factor(eigen_vect1_new.copy(),eigen_vect1.copy().set_color(GREEN))
eigen_eqn2_new = self.get_vector_eqn_with_factor(eigen_vect2_new.copy(),eigen_vect2.copy().set_color(GREEN),factor = "\\lambda_{2}")
#################################################################
self.play(Write(eqn.to_edge(UP,buff = 0.3)),run_time = 2)
self.wait(2)
eigen_eqn1.to_edge(LEFT,buff = 0.3).scale(0.8)
eigen_eqn2.to_edge(RIGHT,buff = 0.3).scale(0.8)
self.play(*[ReplacementTransform(eqn[i].copy(),eigen_eqn1[i])
for i in range(3)],ReplacementTransform(eqn[-1].copy(),eigen_eqn1[3:]),
run_time = 2)
self.wait(2)
self.play(*[ReplacementTransform(eqn[i].copy(),eigen_eqn2[i])
for i in range(3)],ReplacementTransform(eqn[-1].copy(),eigen_eqn2[3:]),
run_time = 2)
self.wait(2)
self.highlight_two_at_a_time(eigen_eqn1[-2],eigen_eqn2[-2])
self.wait(2)
self.highlight(eqn[0],scale_factor = 1.1)
self.wait(2)
eigen_eqn1_new.to_edge(LEFT,buff = 0.8).scale(0.8)
eigen_eqn2_new.to_edge(RIGHT,buff = 0.8).scale(0.8)
self.play(*[Transform(x,y) for x,y in [(eigen_eqn1,eigen_eqn1_new),(eigen_eqn2,eigen_eqn2_new)]],run_time = 3)
self.wait(2)
final = self.solving_eigen_equations(eigen_eqn1,eigen_eqn2)
self.play(Transform(eqn[0],matrix_new.move_to(eqn[0].get_center())),
run_time = 2)
self.wait(2)
self.play(*[FadeOut(x) for x in [final,eigen_eqn1,eigen_eqn2]])
self.wait(2)
self.play(Write(matrix_asign_eqn[:2]))
self.play(ReplacementTransform(eqn[0].copy(),matrix_asign_eqn[2]),run_time = 2)
self.wait(2)
sym = TextMobject("\"Symmetric Matrix\"").to_edge(DOWN,buff = 1.5).set_color(GREEN).scale(2)
self.play(Write(sym),run_time = 2)
self.wait(2)
def eigen_value_eqn_scene(self,m):
eigen_val_eqn1 = TexMobject("det\\left(M - \\lambda I \\right) = 0").set_color(BLUE).to_edge(DOWN,buff = 1)
eigen_val_matrix = Matrix([["a-\\lambda","b"],["c","d-\\lambda"]],h_buff = 1.8).set_color(BLUE)
eigen_val_eqn2 = VGroup(eigen_val_matrix,get_det_text(eigen_val_matrix,determinant = 0)).move_to(eigen_val_eqn1.get_center())
eigen_val_eqn3 = TexMobject("1","\\lambda^2 ","- (d + a)","\\lambda + (ad - bc) = 0").set_color(BLUE).move_to(eigen_val_eqn1.get_center())
sumof_roots = TexMobject("\\lambda_{1}"," + ","\\lambda_{2}"," = ","-","{\\left(","-(d+a)","\\over","1","\\right)}")
sumof_roots1 = TexMobject("\\lambda_{1}"," + ","\\lambda_{2}"," = ","(d+a)")
for i in [0,2,6,8]:
sumof_roots[i].set_color(BLUE)
for i in [0,2,4]:
sumof_roots1[i].set_color(BLUE)
self.play(Write(eigen_val_eqn1),run_time = 2)
self.wait(2)
self.play(Transform(eigen_val_eqn1,eigen_val_eqn2),run_time = 2)
self.wait(2)
self.play(Transform(eigen_val_eqn1,eigen_val_eqn3[1:]),run_time = 2)
self.wait(2)
sumof_roots.next_to(eigen_val_eqn1,UP,buff = 0.3)
sumof_roots1.move_to(m.get_bottom() + 0.8*DOWN)
self.write_by_index(sumof_roots[:4],time_gap = 0.5,single_anim_time = 0.5)
self.play(Write(sumof_roots[4:6]),
ReplacementTransform(eigen_val_eqn3[2].copy(),sumof_roots[6]),
ReplacementTransform(eigen_val_eqn3[0],sumof_roots[-2]),
Write(sumof_roots[-3]),
Write(sumof_roots[-1]),
run_time = 4
)
self.wait(2)
self.play(Transform(sumof_roots,sumof_roots1),
FadeOut(eigen_val_eqn1),run_time = 2)
self.wait(2)
return sumof_roots
def solving_eigen_equations(self,ee1n,ee2n):
plus = TexMobject("+").scale(2)
sub = TexMobject("-").scale(2)
v1 = Matrix(["2a","2c"]).set_color(PURPLE)
v2 = Matrix(["\\lambda_{1} + \\lambda_{2}","\\lambda_{1} - \\lambda_{2}"]).set_color(BLUE)
v3 = Matrix(["2b","2d"]).set_color(PURPLE)
v4 = Matrix(["\\lambda_{1} - \\lambda_{2}","\\lambda_{1} + \\lambda_{2}"]).set_color(BLUE)
v5 = Matrix(["2d","2b"]).set_color(PURPLE)
v6 = Matrix(["\\lambda_{1} + \\lambda_{2}","\\lambda_{1} - \\lambda_{2}"]).set_color(BLUE)
add_eigen_eqn = self.get_vector_eqn(v1,v2).next_to(ee1n,DOWN,buff = 0.8).scale(0.8)
sub_eigen_eqn = self.get_vector_eqn(v3,v4).next_to(ee2n,DOWN,buff = 0.8).scale(0.8)
sub_eigen_eqn_flipped = self.get_vector_eqn(v5,v6).next_to(ee2n,DOWN,buff = 0.8).scale(0.8)
self.play(Write(plus))
self.wait(2)
self.play(ReplacementTransform(VGroup(ee1n.copy(),ee2n.copy()),add_eigen_eqn),run_time = 2)
self.wait(2)
self.play(FadeOut(plus))
self.wait(2)
self.play(Write(sub))
self.wait(2)
self.play(ReplacementTransform(VGroup(ee1n.copy(),ee2n.copy()),sub_eigen_eqn),run_time = 2)
self.wait(2)
self.play(FadeOut(sub))
self.wait(2)
self.play(Transform(sub_eigen_eqn,sub_eigen_eqn_flipped),run_time = 2)
self.wait(2)
sfinal = self.get_vector_eqn(Matrix(["a","c"]).set_color(PURPLE),Matrix(["d","b"]).set_color(PURPLE)).move_to(2.5*DOWN)
final = TexMobject("a"," = ","d","\\\\ ","b"," = ","c").move_to(sfinal.get_center())
for i in [0,2,4,6]:
final.set_color(PURPLE)
self.play(ReplacementTransform(VGroup(add_eigen_eqn,sub_eigen_eqn),sfinal),run_time = 2)
self.wait(2)
self.play(ReplacementTransform(sfinal,final),run_time = 2)
self.wait(2)
return final
class Interesting_Fact_About_Symmetric_Matrices(ShowTransformation1to2):
CONFIG = {
"symm_matrix" : [[3,1],[1,3]]
}
def construct(self):
self.setup()
self.play(ShowCreation(self.plane))
self.wait(2)
i_vector = self.get_vector(1*RIGHT).set_color(GREEN)
self.add_to_moving_vectors(i_vector)
self.play(Write(i_vector))
self.wait(2)
matrix_mob = Matrix(self.symm_matrix,include_background_rectangle = True).shift(3*LEFT + 1*UP)
self.play(Write(matrix_mob))
self.wait(2)
self.play(FadeOut(matrix_mob))
self.wait(2)
self.apply_transposed_matrix(self.symm_matrix)
self.wait(2)
self.apply_inverse_transpose(self.symm_matrix)
self.wait(2)
j_vector = self.get_vector(1*UP).set_color(RED)
self.add_to_moving_vectors(j_vector)
self.play(Write(j_vector))
self.wait(2)
self.apply_transposed_matrix(self.symm_matrix)
self.wait(2)
light1 = FunctionGraph(lambda x:x,x_min = -10,x_max = 10,color = PINK)
light2 = FunctionGraph(lambda x:-x,x_min = -10,x_max = 10,color = PINK)
self.play(Write(light1))
self.wait(2)
self.apply_inverse_transpose(self.symm_matrix,run_time = 2)
self.wait(2)
self.play(FadeOut(i_vector),FadeOut(j_vector))
self.wait(2)
self.remove_from_moving_vectors(i_vector,j_vector)
v1 = self.add_vectors_along(RIGHT,BLUE,GREEN,PURPLE)
v2 = self.add_vectors_along(UP,BLUE,GREEN,PURPLE)
self.play(Write(v1),Write(v2),run_time = 2)
self.wait(2)
self.apply_transposed_matrix(self.symm_matrix)
self.wait(2)
self.play(Write(light2))
self.wait(2)
class SymmetricMatrixInOurCase(ShowTransformation1to2,FindingTheMatrix):
def construct(self):
self.setup()
s1 = TexMobject("S_{1}",color = YELLOW).set_stroke(YELLOW,5).scale(3).to_corner(DL)
s2 = TexMobject("S_{2}",color = YELLOW).set_stroke(YELLOW,5).scale(3).to_corner(DL)
for s in [s1,s2]:
s.add_to_back(BackgroundRectangle(s))
x1_label,ct1_label = self.get_reference_labels("x_{1}","c t_{1}",buff = 0.5)
x1_line = Line(-20*RIGHT,20*RIGHT,color = RED)
ct1_line = Line(-20*UP,20*UP,color = RED)
self.add_transformable_mobject(x1_line,ct1_line)
self.play(ShowCreation(self.plane),ShowCreation(x1_line),ShowCreation(ct1_line),run_time =2)
self.wait(2)
self.play(Write(s1))
self.wait(2)
self.add_moving_mobjects(x1_label,ct1_label)
self.play(Write(x1_label),Write(ct1_label),run_time = 2)
self.wait(2)
self.apply_inverse_transpose(self.lorentz_matrix,added_anims = [FadeOut(s1),ReplacementTransform(s1.copy(),s2)],run_time = 2)
self.wait(2)
light1 = DashedLine(20*DL,-20*DL,color = PINK)
self.play(Write(light1),run_time = 2)
self.wait(2)
x2_label,ct2_label = self.get_reference_labels("x_{2}","c t_{2}",buff = 0.5)
x2_line = Line(-20*RIGHT,20*RIGHT,color = GREEN)
ct2_line = Line(-20*UP,20*UP,color = GREEN)
self.add_transformable_mobject(x2_line,ct2_line)
self.play(ShowCreation(x2_line),ShowCreation(ct2_line),run_time =2)
self.wait(2)
self.add_moving_mobjects(x2_label,ct2_label)
self.play(Write(x2_label),Write(ct2_label),run_time = 2)
self.wait(2)
self.apply_transposed_matrix(self.lorentz_matrix,added_anims = [FadeOut(s2),ReplacementTransform(s2.copy(),s1)],run_time = 2)
self.wait(2)
light2 = DashedLine(20*DR,-20*DR,color = PINK)
self.play(Write(light2),run_time = 2)
self.wait(2)
vect1 = self.get_vector(1*UP).set_color(PURPLE)
self.add_to_moving_vectors(vect1)
sym_matrix = Matrix([["a","b"],["b","a"]]).set_color(BLUE)
vect_ip = Matrix(["0","1"]).set_color(PURPLE)
vect_op = Matrix(["-v/c","1"]).set_color(PURPLE)
eqn1 = self.get_matrix_equation_with_factor(sym_matrix.copy(),vect_ip.copy(),vect_op.copy(),factor = "k")
eqn1.to_edge(DOWN)
br_eqn1 = BackgroundRectangle(eqn1)
self.add(br_eqn1)
self.play(Write(vect1),Write(eqn1[1]),run_time = 2)
self.wait(2)
self.apply_inverse_transpose(self.lorentz_matrix,added_anims = [Write(eqn1[0])],run_time = 2)
self.wait(2)
self.write_by_index(eqn1[2:],time_gap = 0.5,single_anim_time = 1)
self.wait(2)
eqn2 = self.get_vector_eqn(Matrix(["b","a"]).set_color(PURPLE),Matrix(["-kv/c","k"]).set_color(PURPLE))
eqn2.move_to(eqn1.get_center())
br_eqn2 = BackgroundRectangle(eqn2)
self.play(*[ReplacementTransform(x,y)
for x,y in [(br_eqn1,br_eqn2),(eqn1,eqn2)]],run_time = 2)
self.wait(2)
eqn3 = TexMobject("b "," = ","-{av \\over c}").move_to(eqn2.get_center())
for i in [0,2]:
eqn3[i].set_color(BLUE)
br_eqn3 = BackgroundRectangle(eqn3)
self.play(*[ReplacementTransform(x,y)
for x,y in [(br_eqn2,br_eqn3),(eqn2,eqn3)]],run_time = 2)
self.wait(2)
matrix_asign_eqn1 = self.get_matrix_assignment(sym_matrix,"M").shift(3*LEFT + 1*UP)
br1 = BackgroundRectangle(matrix_asign_eqn1)
matrix_sf = TexMobject("\\begin{bmatrix} a & {-a{v \\over c}} \\\\ {-a{v \\over c}} & a \\end{bmatrix}").set_color(BLUE)
matrix_asign_eqn2 = self.get_matrix_assignment(matrix_sf.copy(),"M").shift(3*LEFT + 1*UP)
br2 = BackgroundRectangle(matrix_asign_eqn2)
self.add(br1)
self.play(Write(matrix_asign_eqn1),run_time = 2)
self.play(ReplacementTransform(br1,br2),ReplacementTransform(matrix_asign_eqn1,matrix_asign_eqn2),run_time = 2)
self.wait(2)
self.play(FadeOut(VGroup(br_eqn3,eqn3)))
self.wait(2)
class Finding_Gamma_Rigorous(FindingTheMatrix,ReferenceFrames):
def construct(self):
matrix_1 = TexMobject("a","\\begin{bmatrix} 1 & {-{v \\over c}} \\\\ {-{v \\over c}} & 1 \\end{bmatrix}").set_color(BLUE)
matrix_2 = TexMobject("a","\\begin{bmatrix} 1 & {{v \\over c}} \\\\ {{v \\over c}} & 1 \\end{bmatrix}").set_color(BLUE)
matrix_3 = TexMobject("{a \\over det([M(-v)]_{S_{1}}^{S_{2}})}","\\begin{bmatrix} 1 & {-{v \\over c}} \\\\ {-{v \\over c}} & 1 \\end{bmatrix}").set_color(BLUE)
eqn_1 = self.get_matrix_assignment(matrix_1.copy(),TexMobject("\\left[","M(v)","\\right]_{S_{1}}^{S_{2}}"))
eqn_2 = self.get_matrix_assignment(matrix_2.copy(),TexMobject("\\left[","M(-v)","\\right]_{S_{1}}^{S_{2}}"))
eqn_3 = self.get_matrix_assignment(matrix_2.copy(),TexMobject("\\left[","M(-v)","\\right]_{S_{1}}^{S_{2}}"))
eqn_4 = self.get_matrix_assignment(matrix_3.copy(),TexMobject("\\left[","M(-v)","\\right]_{S_{2}}^{S_{1}}"))
box1 = self.box1()
box2 = self.box2(box1)
box3 = self.box3()
box4 = self.box4()
m1 = eqn_1[0].copy().next_to(box1,DOWN,buff= 0.8)
m2 = eqn_2[0].copy().next_to(box2,DOWN,buff= 0.8)
m3 = eqn_3[0].copy().next_to(box3,DOWN,buff= 0.8)
m4 = eqn_4[0].copy().next_to(box4,DOWN,buff= 0.8)
self.play(ShowCreation(box1),run_time = 4)
self.wait(2)
eqn_1.next_to(box1,RIGHT,buff = 1)
self.play(Write(eqn_1[0][1]),Write(eqn_1[1:]),run_time = 2)
self.wait(2)
self.play(Write(eqn_1[0][0]),Write(eqn_1[0][2:]),run_time = 2)
self.wait(2)
self.play(ReplacementTransform(eqn_1[0].copy(),m1),FadeOut(eqn_1),run_time = 2)
self.wait(2)
self.play(ReplacementTransform(box1.copy(),box2),ReplacementTransform(m1.copy(),m2),run_time = 3)
self.wait(2)
self.play(ReplacementTransform(box2.copy(),box3),ReplacementTransform(m2.copy(),m3),run_time = 3)
self.wait(2)
self.highlight_two_at_a_time(box1,box3,scale_factor = 1.1)
self.play(box3.scale,1.1)
self.wait(2)
self.play(ReplacementTransform(box3,box4.scale(1.1)),ReplacementTransform(m3,m4),run_time = 3)
self.wait(2)
self.play(box4.scale,1/1.1)
self.wait(2)
self.scale_and_wait2(box1,box4)
self.wait(2)
feqn = TexMobject("\\left[M(v)\\right]_{S_{1}}^{S_{2}}"," = ","\\left[M(-v)\\right]_{S_{2}}^{S_{1}}")
feqn.set_color(BLUE)
feqn[1].set_color(WHITE)
self.play(*[FadeOut(x) for x in [box1,box2,box4,m2]],ReplacementTransform(m1,feqn[0]),ReplacementTransform(m4,feqn[2]),Write(feqn[1]),run_time = 3)
self.wait(2)
feqn1 = TexMobject("\\left[M(v)\\right]_{S_{1}}^{S_{2}}"," = ","\\left(","\\left[M(-v)\\right]_{S_{1}}^{S_{2}}","\\right)^{-1}")
for i in [0,3]:
feqn1[i].set_color(BLUE)
self.play(*[ReplacementTransform(feqn[i],feqn1[j])
for i,j in [(0,0),(1,1),(2,3)]],Write(feqn1[2]),Write(feqn1[-1]),
run_time = 2)
self.wait(2)
feqn2 = TexMobject("a \\begin{bmatrix} 1 & {-{v \\over c}} \\\\ {-{v \\over c}} & 1 \\end{bmatrix}",
" = ",
"{a \\over det([M(-v)]_{S_{1}}^{S_{2}})}\\begin{bmatrix} 1 & {-{v \\over c}} \\\\ {-{v \\over c}} & 1 \\end{bmatrix}")
for i in [0,2]:
feqn2[i].set_color(BLUE)
self.play(ReplacementTransform(feqn1[0],feqn2[0]),run_time = 2)
self.wait(2)
self.play(ReplacementTransform(feqn1[2:],feqn2[-1]),ReplacementTransform(feqn1[1],feqn2[1]),run_time = 2)
self.wait(2)
feqn3 = TexMobject("{det([M(v)]_{S_{1}}^{S_{2}})"," = ","1")
feqn3[0].set_color(BLUE),feqn3[-1].set_color(BLUE)
feqn4 = TexMobject("det \\left( \\begin{bmatrix} a & {-a{v \\over c}} \\\\ {-a{v \\over c}} & a \\end{bmatrix} \\right)"," = ","1")
feqn4[0].set_color(BLUE),feqn4[-1].set_color(BLUE)
feqn5 = TexMobject("a"," = ","{1 \\over {\\sqrt{1-{v^2 \\over c^2}}}}"," = ","\\gamma")
feqn5[0].set_color(RED),feqn5[-1].set_color(RED),feqn5[-3].set_color(RED)
feqn6 = TexMobject("\\gamma"," = ","{1 \\over {\\sqrt{1-{v^2 \\over c^2}}}}")
feqn6[0].set_color(RED),feqn6[-1].set_color(RED)
self.play(ReplacementTransform(feqn2,feqn3),run_time = 2)
self.wait(2)
self.play(ReplacementTransform(feqn3,feqn4),run_time = 2)