Tartalomjegyzék
\[ \newenvironment{dcases}{\left\{\begin{array}{ll}}{\end{array}\right.} \]Meaning of involutive distribution
Teljes Matlab script kiegészítő függvényekkel.
File: involutive.m Directory: 4_gyujtemegy/11_CCS/_2_nonlin-pannon/2018 Author: Peter Polcz (ppolcz@gmail.com)
Created on 2018. July 27.
Inherited from
file: nonlin_distributions_coordinate_systems.m author: Peter Polcz <ppolcz@gmail.com>
Created on 2017.06.04. Sunday, 14:57:05
Requires
Polar coordinates
Canonic basis vectors in polar coordinates
syms x y real
Z = [x;y];
f_sym = [
-y
x
];
g_sym = [
x
y
] / sqrt(x^2 + y^2);
figure, hold on, grid on
Q1 = vekanal_quiver_sym(f_sym, Z, {0,8,9}, {-4,4,9}, 0.7);
Q2 = vekanal_quiver_sym(g_sym, Z, {0,8,9}, {-4,4,9}, 0.7);
title 'Canonic basis vectors in the polar coordinates'
Closed walk along the flow of f and g
br_fg = simplify(jacobian(g_sym,Z)*f_sym - jacobian(f_sym,Z)*g_sym)
f_fh = matlabFunction(f_sym,'vars',{'t' Z});
g_fh = matlabFunction(g_sym,'vars',{'t' Z});
T1 = [0.4 0.8 1.4];
T2 = [1 3 5];
phi0 = [
-10
-20
-40
] * pi / 180;
r0 = [
3
2
1
];
x0 = round([ r0.*cos(phi0) r0.*sin(phi0) ],2);
hold on
for i = 1:numel(T1)
fprintf('Path %d starting from from (%g,%g):\n', i, x0(i,:))
fprintf(' - %g seconds along the flow of f(x,y)\n', T1(i))
fprintf(' - %g seconds along the flow of g(x,y)\n', T2(i))
fprintf(' - %g seconds along the flow of f(x,y) (i.e. moving backwards)\n', -T1(i))
fprintf(' - %g seconds along the flow of g(x,y) (i.e. moving backwards)\n\n', -T2(i))
N = 1000;
[~,p1] = ode45(f_fh, linspace(0,T1(i),N), x0(i,:)');
[~,p2] = ode45(g_fh, linspace(0,T2(i),N), p1(end,:)');
[~,p3] = ode45(@(t,x) -f_fh(-t,x), linspace(0,T1(i),N), p2(end,:)');
[~,p4] = ode45(@(t,x) -g_fh(-t,x), linspace(0,T2(i),N), p3(end,:)');
plot(p1(:,1),p1(:,2), 'Color', Q1.Color, 'LineWidth', 5),
plot(p2(:,1),p2(:,2), 'Color', Q2.Color, 'LineWidth', 5),
plot(p3(:,1),p3(:,2), 'Color', Q1.Color, 'LineWidth', 5),
plot(p4(:,1),p4(:,2), 'Color', Q2.Color, 'LineWidth', 5),
plot(x0(i,1),x0(i,2),'ko', 'MarkerSize',10);
pcz_arrow(p1(end/2,1),p1(end/2,2),p1(end/2+1,1),p1(end/2+1,2),'HeadLength',14,'HeadWidth',14)
pcz_arrow(p2(end/2,1),p2(end/2,2),p2(end/2+1,1),p2(end/2+1,2),'HeadLength',14,'HeadWidth',14)
pcz_arrow(p3(end/2,1),p3(end/2,2),p3(end/2+1,1),p3(end/2+1,2),'HeadLength',14,'HeadWidth',14)
pcz_arrow(p4(end/2,1),p4(end/2,2),p4(end/2+1,1),p4(end/2+1,2),'HeadLength',14,'HeadWidth',14)
end
axis equal
axis tight
Output:
br_fg = 0 0 Path 1 starting from from (2.95,-0.52): - 0.4 seconds along the flow of f(x,y) - 1 seconds along the flow of g(x,y) - -0.4 seconds along the flow of f(x,y) (i.e. moving backwards) - -1 seconds along the flow of g(x,y) (i.e. moving backwards) Path 2 starting from from (1.88,-0.68): - 0.8 seconds along the flow of f(x,y) - 3 seconds along the flow of g(x,y) - -0.8 seconds along the flow of f(x,y) (i.e. moving backwards) - -3 seconds along the flow of g(x,y) (i.e. moving backwards) Path 3 starting from from (0.77,-0.64): - 1.4 seconds along the flow of f(x,y) - 5 seconds along the flow of g(x,y) - -1.4 seconds along the flow of f(x,y) (i.e. moving backwards) - -5 seconds along the flow of g(x,y) (i.e. moving backwards)
Pair of non-involutive vector fields
Vector fields
syms x y real
Z = [x;y];
f_sym = [
-y
x^3
];
g_sym = [
x
y
] / sqrt(x^2 + y^2);
figure, hold on, grid on
Q1 = vekanal_quiver_sym(f_sym, Z, {0,8,9}, {-4,4,9}, 0.7);
Q2 = vekanal_quiver_sym(g_sym, Z, {0,8,9}, {-4,4,9}, 0.7);
title 'Canonic basis vectors in the cycloid coordinates'
The walk along the flow of f and g is not closed
br_fg = simplify(jacobian(g_sym,Z)*f_sym - jacobian(f_sym,Z)*g_sym)
f_fh = matlabFunction(f_sym,'vars',{'t' Z});
g_fh = matlabFunction(g_sym,'vars',{'t' Z});
T1 = [0.05 ];
T2 = [2 3 1];
phi0 = [
-10
-20
-40
] * pi / 180;
r0 = [
3
2
1
];
x0 = round([ r0.*cos(phi0) r0.*sin(phi0) ],2);
hold on
for i = 1:numel(T1)
fprintf('Path %d starting from from (%g,%g):\n', i, x0(i,:))
fprintf(' - %g seconds along the flow of f(x,y)\n', T1(i))
fprintf(' - %g seconds along the flow of g(x,y)\n', T2(i))
fprintf(' - %g seconds along the flow of f(x,y) (i.e. moving backwards)\n', -T1(i))
fprintf(' - %g seconds along the flow of g(x,y) (i.e. moving backwards)\n\n', -T2(i))
N = 1000;
[~,p1] = ode45(f_fh, linspace(0,T1(i),N), x0(i,:)');
[~,p2] = ode45(g_fh, linspace(0,T2(i),N), p1(end,:)');
[~,p3] = ode45(@(t,x) -f_fh(-t,x), linspace(0,T1(i),N), p2(end,:)');
[~,p4] = ode45(@(t,x) -g_fh(-t,x), linspace(0,T2(i),N), p3(end,:)');
plot(p1(:,1),p1(:,2), 'Color', Q1.Color, 'LineWidth', 5),
plot(p2(:,1),p2(:,2), 'Color', Q2.Color, 'LineWidth', 5),
plot(p3(:,1),p3(:,2), 'Color', Q1.Color, 'LineWidth', 5),
plot(p4(:,1),p4(:,2), 'Color', Q2.Color, 'LineWidth', 5),
plot(x0(i,1),x0(i,2),'ko', 'MarkerSize',10);
pcz_arrow(p1(end/2,1),p1(end/2,2),p1(end/2+1,1),p1(end/2+1,2),'HeadLength',14,'HeadWidth',14)
pcz_arrow(p2(end/2,1),p2(end/2,2),p2(end/2+1,1),p2(end/2+1,2),'HeadLength',14,'HeadWidth',14)
pcz_arrow(p3(end/2,1),p3(end/2,2),p3(end/2+1,1),p3(end/2+1,2),'HeadLength',14,'HeadWidth',14)
pcz_arrow(p4(end/2,1),p4(end/2,2),p4(end/2+1,1),p4(end/2+1,2),'HeadLength',14,'HeadWidth',14)
end
axis equal
axis tight
Output:
br_fg =
-(x^2*y*(x^2 - 1))/(x^2 + y^2)^(3/2)
-(2*x^5 + 3*x^3*y^2 - x*y^2)/(x^2 + y^2)^(3/2)
Path 1 starting from from (2.95,-0.52):
- 0.05 seconds along the flow of f(x,y)
- 2 seconds along the flow of g(x,y)
- -0.05 seconds along the flow of f(x,y) (i.e. moving backwards)
- -2 seconds along the flow of g(x,y) (i.e. moving backwards)
syms x1 x2 x3 real
x = [x1;x2;x3];
tau1 = [
cos(x3)
sin(x3)
0
];
tau2 = [
0
0
1
];
jacobian(tau2,x)*tau1 - jacobian(tau1,x)*tau2
f_sym = @(t,x) [
sin(x(1))
cos(x(2))
];
g_sym = @(t,x) [
-cos(x(1))
sin(x(2))
];
figure, hold on
for h = linspace(0.05,0.5,10)
x0 = [1,1];
N = 1000;
[~,p1] = ode45(f_sym, linspace(0,h,N), x0);
[~,p2] = ode45(g_sym, linspace(0,h,N), p1(end,:)');
[~,p3] = ode45(@(t,x) -f_sym(-t,x), linspace(0,h,N), p2(end,:)');
[~,p4] = ode45(@(t,x) -g_sym(-t,x), linspace(0,h,N), p3(end,:)');
plot(p1(:,1),p1(:,2)),
plot(p2(:,1),p2(:,2)),
plot(p3(:,1),p3(:,2)),
plot(p4(:,1),p4(:,2)),
end
Output:
ans =
sin(x3)
-cos(x3)
0
