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variational.py
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variational.py
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from manimlib.imports import *
class EQScene(Scene):
def construct(self):
eq = TexMobject(self.eq)
eq.scale(2)
self.play(Write(eq))
self.wait()
class Intro(Scene):
def construct(self):
axes = Axes(
x_min=-2,
x_max=4,
y_min=-2,
y_max=4,
number_line_config={"include_tip": False}
)
f = FunctionGraph(
self.func,
x_min=-2,
x_max=4
)
s = [-0.904, 0.904, 3.528]
lbl = TexMobject("f(x)", color=YELLOW).move_to(f, RIGHT)
xlbl = TexMobject("x").shift(4.5 * RIGHT)
ylbl = TexMobject("y").shift(4.5 * UP)
lines = VGroup(*[self.get_lines(i) for i in s])
graph = VGroup(axes, f, lbl, xlbl, ylbl)
grp = VGroup(graph, lines)
grp.scale(0.9)
grp.move_to(3 * LEFT)
self.play(Write(graph))
self.wait()
head1 = TextMobject("Stationary Points", color=GOLD_B)
head1.shift(3.5 * RIGHT + 3 * UP)
head1.scale(1.25)
self.play(Write(head1))
eq1 = TexMobject("f'(x) = 0")
eq1.scale(1)
eq1.shift(3.5 * RIGHT + 1.5 * UP)
self.wait()
brect1 = BackgroundRectangle(
eq1, buff=0.25, stroke_opacity=1, fill_opacity=0, stroke_width=4, color=BLUE)
self.play(Write(eq1))
self.play(Write(brect1), Write(lines))
self.wait()
head2 = TextMobject("Local min/max", color=GREEN)
head2.shift(3.5 * RIGHT + 1 * DOWN)
head2.scale(1.25)
eq2 = TexMobject(r"\text{sign of } f''(x)")
eq2.shift(3.5 * RIGHT + 2.5 * DOWN)
brect2 = BackgroundRectangle(
eq2, buff=0.25, stroke_opacity=1, fill_opacity=0, stroke_width=4, color=PURPLE)
self.play(Write(head2))
self.play(Write(brect2), Write(eq2))
self.wait()
def func(self, x):
return x * np.sin(np.cos(x)) + 2
def get_lines(self, point):
ret = VGroup()
ret.add(DashedLine(ORIGIN, self.func(point) * UP,
dash_length=0.1, stroke_opacity=0.7).shift(point * RIGHT))
ret.add(Line(0.75 * LEFT, 0.75 * RIGHT,
color=BLUE).shift([point, self.func(point), 0]))
return ret
class Functional(Scene):
def construct(self):
eq1 = TexMobject("D(f(x))", tex_to_color_map={
"f": RED, "D": BLUE, "x": GREEN})
eq1.scale(3)
eq2 = TexMobject("D(f)", tex_to_color_map={
"f": RED, "D": BLUE, "x": GREEN})
eq2.scale(3)
self.play(Write(eq1))
self.wait()
self.play(Transform(eq1, eq2))
self.play(eq1.scale, 1.5/3)
self.play(eq1.shift, 3 * UP)
axes = Axes(
x_min=-3,
x_max=3,
y_min=-2,
y_max=2,
number_line_config={"include_tip": False}
)
tracker = ValueTracker(-1)
func = ParametricFunction(
self.f_r(theta=PI/4), color=RED, t_min=-2, t_max=2)
func.add_updater(lambda x: x.become(
ParametricFunction(
self.f_r(theta=PI/4, alpha=np.sin(tracker.get_value())), color=RED, t_min=-2, t_max=2)
).shift(0.5 * DOWN))
grp = VGroup(axes, func)
grp.shift(0.5 * DOWN)
p1 = Circle(radius=0.05, fill_opacity=1, color=YELLOW)
p1.shift([-np.sqrt(2), -np.sqrt(2) - 0.5, 0])
p2 = Circle(radius=0.05, fill_opacity=1, color=YELLOW)
p2.shift([np.sqrt(2), np.sqrt(2) - 0.5, 0])
lbl1 = TexMobject("A").move_to(p1).shift(0.5 * DOWN)
lbl2 = TexMobject("B").move_to(p2).shift(0.5 * UP)
points = VGroup(p1, p2, lbl1, lbl2)
self.play(Write(points))
self.wait()
self.bring_to_back(grp)
self.play(Write(grp))
self.wait()
self.play(tracker.increment_value, 8, rate_func=linear,
run_time=4*DEFAULT_ANIMATION_RUN_TIME)
self.wait()
head1 = TextMobject(r"Distance from \( A \) to \( B \) along \( f \)", tex_to_color_map={
r"\( f \)": RED, r"\( A \)": GREEN, r"\( B \)": GREEN})
head1.scale(1.25)
head1.shift(3 * UP + 1.75 * RIGHT)
self.play(eq1.shift, 5 * LEFT)
arrow = Arrow(4 * LEFT, 2.5 * LEFT, color=GOLD).shift(3 * UP)
self.play(Write(arrow), FadeInFromDown(head1))
self.wait()
def f(self, x, alpha=0):
return alpha * np.cos(PI/4 * x)
def f_r(self, theta=PI/4, alpha=0):
return lambda t: [
t * np.cos(theta) - self.f(t, alpha=alpha) * np.sin(theta),
t * np.sin(theta) + self.f(t, alpha=alpha) * np.cos(theta),
0]
def func(self, t):
return [t - self.f(t), t + self.f(t), 0]
class Difference(Scene):
def construct(self):
t1 = TextMobject("Traditional Calculus", color=GOLD)
t2 = TextMobject("Variational Calculus", color=PURPLE)
t1.shift(3.5 * LEFT + 3 * UP)
t2.shift(3.5 * RIGHT + 3 * UP)
t1.scale(1.25)
t2.scale(1.25)
l1 = Line(6 * UP, 6 * DOWN)
l2 = Line(10 * LEFT, 10 * RIGHT).shift(2 * UP)
grp = VGroup(t1, t2, l1, l2)
self.play(Write(grp))
self.wait()
f1 = TexMobject("f", color=RED)
i1 = TexMobject(r"\mathbb{R}", color=YELLOW).shift(1.5 * LEFT)
o1 = TexMobject(r"\mathbb{R}", color=YELLOW).shift(1.5 * RIGHT)
a1 = Arrow(1.25 * LEFT, 0.25 * LEFT, color=GREEN)
a2 = Arrow(0.25 * RIGHT, 1.25 * RIGHT, color=GREEN)
eq1 = VGroup(f1, i1, o1, a1, a2)
eq1.shift(3.5 * LEFT)
eq1.scale(1.5)
f2 = TexMobject("f", color=RED)
i2 = TexMobject(r"f(x)", color=BLUE).shift(1.5 * LEFT)
o2 = TexMobject(r"\mathbb{R}", color=YELLOW).shift(1.5 * RIGHT)
a3 = Arrow(1 * LEFT, 0 * LEFT, color=GREEN)
a4 = Arrow(0.25 * RIGHT, 1.25 * RIGHT, color=GREEN)
eq2 = VGroup(f2, i2, o2, a3, a4)
eq2.shift(3.5 * RIGHT)
eq2.scale(1.5)
self.play(Write(eq1))
self.play(Write(eq2))
self.wait()
class DistanceFunc(Scene):
def construct(self):
eq1 = TexMobject("D(f)", tex_to_color_map={
"f": RED, "D": BLUE, "x": GREEN})
eq1.scale(1.5)
eq1.shift(3 * UP + 5 * LEFT)
head1 = TextMobject(r"Distance from \( A \) to \( B \) along \( f \)", tex_to_color_map={
r"\( f \)": RED, r"\( A \)": GREEN, r"\( B \)": GREEN})
head1.scale(1.25)
head1.shift(3 * UP + 1.75 * RIGHT)
arrow = Arrow(4 * LEFT, 2.5 * LEFT, color=GOLD).shift(3 * UP)
self.play(Write(eq1), Write(arrow), FadeInFromDown(head1))
self.wait()
axes = Axes(
x_min=-3,
x_max=3,
y_min=-2,
y_max=2,
number_line_config={"include_tip": False}
)
tracker = ValueTracker(-1)
func = ParametricFunction(
self.f_r(theta=PI/4), color=RED, t_min=-2, t_max=2)
func.add_updater(lambda x: x.become(
ParametricFunction(
self.f_r(theta=PI/4, alpha=np.sin(tracker.get_value())), color=RED, t_min=-2, t_max=2)
).shift(0.5 * DOWN))
grp = VGroup(axes, func)
grp.shift(0.5 * DOWN)
p1 = Circle(radius=0.05, fill_opacity=1, color=YELLOW)
p1.shift([-np.sqrt(2), -np.sqrt(2) - 0.5, 0])
p2 = Circle(radius=0.05, fill_opacity=1, color=YELLOW)
p2.shift([np.sqrt(2), np.sqrt(2) - 0.5, 0])
lbl1 = TexMobject("A").move_to(p1).shift(0.5 * DOWN)
lbl2 = TexMobject("B").move_to(p2).shift(0.5 * UP)
points = VGroup(p1, p2, lbl1, lbl2)
self.play(Write(points))
self.wait()
self.bring_to_back(grp)
self.play(Write(grp))
self.wait()
self.play(tracker.increment_value, 1 + 3 * PI, rate_func=linear,
run_time=4*DEFAULT_ANIMATION_RUN_TIME)
self.wait()
f2 = ParametricFunction(
self.f_r(theta=PI/4), color=RED, t_min=-2, t_max=2).shift(0.5 * DOWN)
graph = VGroup(axes, f2, points)
self.add(f2)
self.remove(func)
self.play(ApplyMethod(graph.shift, 3.5 * LEFT))
eq2 = TexMobject(r"D(f) ", r"= \int_{ x_1 }^{ x_2 } \text{d}s", tex_to_color_map={
"f": RED, "D": BLUE, r"\text{d}s": GREEN, "_{ x_1 }": YELLOW, "^{ x_2 }": YELLOW})
eq2.scale(1.25)
eq2.shift(1.25 * UP + 3 * RIGHT)
self.play(TransformFromCopy(eq1, eq2[:4]))
self.play(FadeInFromDown(eq2[4:]))
self.wait()
eq3 = TexMobject(r"=", r"\int_{ x_1 }^{ x_2 } \sqrt{1 +f'(x)^2 } \ \text{d}x", tex_to_color_map={
"_{ x_1 }": YELLOW, "^{ x_2 }": YELLOW, r"\text{d}x": GREEN})
eq3.scale(1)
eq3.shift(1 * DOWN + 3 * RIGHT)
self.play(FadeInFromDown(eq3))
self.wait()
rect = BackgroundRectangle(eq3[1:], stroke_width=DEFAULT_STROKE_WIDTH,
stroke_opacity=1, color=GOLD, fill_opacity=0, buff=0.1)
self.play(Write(rect))
txt = TextMobject("Minimize this", color=ORANGE)
txt.next_to(rect, DOWN)
self.play(Write(txt))
self.wait()
def f(self, x, alpha=0):
return alpha * np.cos(PI/4 * x)
def f_r(self, theta=PI/4, alpha=0):
return lambda t: [
t * np.cos(theta) - self.f(t, alpha=alpha) * np.sin(theta),
t * np.sin(theta) + self.f(t, alpha=alpha) * np.cos(theta),
0]
def func(self, t):
return [t - self.f(t), t + self.f(t), 0]
class Brac(Scene):
def construct(self):
br1 = ScreenRectangle(height=3, width=4).shift(3.5 * LEFT)
br2 = ScreenRectangle(height=3, width=4).shift(3.5 * RIGHT)
t1 = TextMobject("3blue1brown", color=BLUE).shift(
3.5 * LEFT + 2.5 * UP).scale(1.5)
t2 = TextMobject("VSauce", color=GREEN).shift(
3.5 * RIGHT + 2.5 * UP).scale(1.5)
self.play(Write(t1), Write(t2))
self.play(Write(br1), Write(br2))
self.wait()
class IntroToProblem(Scene):
def construct(self):
head = TextMobject("Variational Calculus",
color=GOLD).scale(1.5).shift(3 * UP)
self.play(Write(head))
self.wait()
p1 = TextMobject(r"Find some function \( f \) such that:",
tex_to_color_map={r"\( f \)": RED})
p1.shift(1 * UP)
self.play(Write(p1))
eq1 = TexMobject(r"I[f] = \int_{ x_1 }^{ x_2 } F({x}, f({x}), f'({x})) \ \text{d}x", tex_to_color_map={
r"f": RED, "_{ x_1 }": YELLOW, "^{ x_2 }": YELLOW, r"\text{d}x": GREEN, r"{x}": BLUE, "I": BLUE, "F": PURPLE})
eq1.scale(1.25)
eq1.shift(0.5 * DOWN)
self.play(Write(eq1))
p2 = TextMobject(r"is stationary")
p2.shift(2 * DOWN)
self.play(Write(p2))
self.wait()
class Proof(Scene):
def construct(self):
axes = Axes(
x_min=-1,
x_max=5,
y_min=0,
y_max=4,
number_line_config={"include_tip": False}
)
func1 = ParametricFunction(
self.f_r(alpha=1), color=TEAL, t_min=-2, t_max=2)
func2 = ParametricFunction(
self.f_r(alpha=0.6), color=RED, t_min=-2, t_max=2)
func3 = ParametricFunction(
self.f_r(alpha=0.2), color=TEAL, t_min=-2, t_max=2)
p1 = Circle(radius=0.05, fill_opacity=1, color=YELLOW)
p1.shift([-np.sqrt(3), -1, 0])
p2 = Circle(radius=0.05, fill_opacity=1, color=YELLOW)
p2.shift([np.sqrt(3), 1, 0])
points = VGroup(p1, p2)
f = VGroup(func1, func2, func3)
f.shift(2.5 * RIGHT + 2.5 * UP)
points.shift(2.5 * RIGHT + 2.5 * UP)
l1 = DashedLine((-np.sqrt(3) + 2.5) * RIGHT,
(-np.sqrt(3) + 2.5) * RIGHT + 1.5 * UP,
dash_length=0.075, stroke_opacity=0.75)
l2 = DashedLine((np.sqrt(3) + 2.5) * RIGHT,
(np.sqrt(3) + 2.5) * RIGHT + 3.5 * UP,
dash_length=0.075, stroke_opacity=0.75)
eq1 = TexMobject("q(x)", color=RED).scale(0.75)
eq1.move_to(p2).shift(0.75 * RIGHT)
eq2 = TexMobject(r"q(x) + s \cdot \delta q(x)", color=TEAL).scale(0.75)
eq2.move_to(func1).shift(1.5 * UP)
everything = VGroup(axes, func1, func2, func3, l1, l2,
points, eq1, eq2).center().scale(1.5)
self.play(Write(axes))
self.play(Write(func2), Write(eq1))
self.wait()
self.play(Write(func3), Write(func1), Write(eq2))
self.play(Write(points), Write(VGroup(l1, l2)))
self.wait()
def f(self, x, alpha=0):
return alpha * np.cos(PI/4 * x)
def f_r(self, theta=PI/6, alpha=0):
return lambda t: [
t * np.cos(theta) - self.f(t, alpha=alpha) * np.sin(theta),
t * np.sin(theta) + self.f(t, alpha=alpha) * np.cos(theta),
0]
def func(self, t):
return [t - self.f(t), t + self.f(t), 0]
class FEq(EQScene):
CONFIG = {
"eq": r"I[f] = \int_{ x_1 }^{ x_2 } F({x}, f({x}), f'({x})) \ \text{d}x"
}
class EulerLagrangeScene(Scene):
def get_eq(self):
eqq = TexMobject(r"{{\partial F}", r" \over {\partial y}}", r" -\frac{d}{d x} {{\partial F} \over {\partial y'}}=0",
tex_to_color_map={r"y": GREEN})
eqq.scale(1.25)
eqq2 = TexMobject(r"{{\partial {F}} \over ", r"{\partial y}}", r" -\frac{d}{d x} {{\partial F} \over {\partial y'}}=0",
tex_to_color_map={r"{F}": YELLOW})
eqq2.scale(1.25)
eqq3 = TexMobject(r"{{\partial F} \over ", r"{\partial y}}", r" -\frac{d}{d x} ", r"{{\partial {F}} ", r"\over {\partial y'}}=0",
tex_to_color_map={r"{F}": YELLOW})
eqq3.scale(1.25)
eqq4 = TexMobject(r"{{\partial F} \over ", r"{\partial y}}", r" -\frac{d}{d x} ", r"{{\partial {F}} ", r"\over {\partial y'}}=0",
tex_to_color_map={r"x": BLUE})
eqq4.scale(1.25)
grp = VGroup()
grp.add(eqq[1:], eqq2[:3], eqq3[-3:-2], eqq4[3])
return grp
class EulerLagrange(EulerLagrangeScene):
def construct(self):
eq = self.get_eq()
eq.scale(0.75)
eq.shift([1.25, 0.75, 0])
self.play(FadeIn(eq))
self.wait()
self.play(eq.shift, [-1.25, -0.75, 0])
self.play(eq.scale, 1.5 * 1.333)
self.wait()
title = TextMobject("Euler-Lagrange Equation", color=GOLD)
title.scale(1.5)
title.shift(2.5 * UP)
self.play(FadeInFromDown(title))
self.wait()
class DistanceFunc2(EulerLagrangeScene):
def construct(self):
eq2 = TexMobject(r"D(y) ", r"=", r"\int_{ x_1 }^{ x_2 } \sqrt{1 +y'(x)^2 } ", r"\ \text{d}x", tex_to_color_map={
"y": GREEN, "D": BLUE, r"\text{d}s": GREEN, "_{ x_1 }": YELLOW, "^{ x_2 }": YELLOW}).scale(1.5)
eq3 = TexMobject(r"F = ", tex_to_color_map={"F": YELLOW}).scale(1.5)
eq3.shift(1.5 * LEFT)
self.play(Write(eq2))
self.wait()
self.play(Transform(eq2[:8], eq3), Uncreate(eq2[-1]))
self.wait()
eq = self.get_eq()
self.play(VGroup(*self.mobjects).center)
self.play(VGroup(*self.mobjects).shift, 2.5 * UP)
self.play(VGroup(*self.mobjects).scale, 0.75)
self.wait()
self.play(FadeIn(eq))
self.wait()
eq2 = self.get_eq2()
self.play(Transform(eq, eq2))
self.wait()
def get_eq2(self):
eqq = TexMobject(r"0", r" -\frac{d}{d x} {{\partial F} \over {\partial y'}}=0",
tex_to_color_map={r"y": GREEN})
eqq.scale(1.25)
eqq2 = TexMobject(r"0", r" -\frac{d}{d x} {{\partial F} \over {\partial y'}}=0",
tex_to_color_map={r"F": YELLOW})
eqq2.scale(1.25)
eqq4 = TexMobject(r"0", r" -\frac{d}{d x} ", r"{{\partial {F}} ", r"\over {\partial y'}}=0",
tex_to_color_map={r"x": BLUE})
eqq4.scale(1.25)
grp = VGroup()
grp.add(eqq[1:], eqq2[:3], eqq4[2])
return grp
class Wolfram(Scene):
def construct(self):
img = ImageMobject("./img/wolfram.png")
img.scale(2.75)
img.shift(1.25 * UP)
self.play(ShowCreation(img))
self.wait()
rect1 = Rectangle(height=1, width=2, color=RED)
rect1.shift(2.75 * LEFT + 0.75 * DOWN)
eq1 = TexMobject("= 0", color=BLACK).scale(1.5)
eq1.shift(0.75 * DOWN + 1 * LEFT)
self.play(Write(rect1))
self.play(Write(eq1))
self.wait()
eq2 = TexMobject(r"f''(x) = 0", tex_to_color_map={
r"f": RED, "x": BLUE})
eq2.scale(1.5)
eq2.shift(2.5 * DOWN + 3.5 * LEFT)
self.play(FadeInFromDown(eq2))
self.wait()
eq3 = TexMobject(r"f(x) = c_0 + c_1 x",
tex_to_color_map={r"f": RED, "x": BLUE})
eq3.scale(1.5)
eq3.shift(2.5 * DOWN + 4 * RIGHT)
arrow = Arrow(LEFT, RIGHT, color=GREEN)
arrow.shift(2.5 * DOWN)
self.play(Write(arrow))
self.play(FadeInFromDown(eq3))
self.wait()