Champs-conservatifs-cours.ipynb

Kernel: SageMath 9.2

Indépendance de la trajectoire ou trajectoires fermées

On considère le champ de vecteurs $\mathbf{F}(x,y) = (3x^2 - 3, 12-3y^2)$, et on s'intéresse au travail effectué le long d'une trajectoire fermée $\mathcal{C}$ (ou le long de deux trajctoires avec mêmes origine / but)

%display typeset
var('x,y')
f(x,y) = x^3 -3*x-y^3 +12*y 
xmin, xmax, ymin, ymax = -2.5, 2.5, -2.5, 2.5 
Champ = plot_vector_field(f.gradient(), (x,xmin,xmax), (y,xmin,xmax), color= "grey", plot_points = 12)
C=contour_plot(f(x,y), (x,xmin,xmax), (y,xmin,xmax), cmap = "Spectral", 
               fill = False, contours = 15, colorbar = True, axes = True,  
               label_inline=True, label_fontsize=8, 
               axes_labels=["$x$","$y$"])
C1  = parametric_plot([-1,t], (t,0,1), color = "red",thickness = 3)
C2 = parametric_plot([cos(t), 1+sin(t)], (t,0,pi), color = "red",thickness = 3)
C3 = parametric_plot([1,t], (t,-2,1), color = "red",thickness = 3)
C4 = parametric_plot([2*t-1,-2*t], (t,0,1), color = "black", thickness = 3 )
show(Champ + C+ C1 + C2 + C3 +C4, aspect_ratio = 1)

C1  = parametric_plot([-1,t], (t,0,1), color = "red")
C2 = parametric_plot([cos(t), 1+sin(t)], (t,0,pi), color = "red")
C3 = parametric_plot([1,t], (t,-1,1), color = "red")
show(C1 + C2 + C3)

cm = colormaps.Spectral
def c(x,y) : 
    return float((f(x,y) + 24 )/48)
S = plot3d(f, (x,xmin,xmax), (y,xmin,xmax), color = (c,cm), viewer = "threejs", opacity = 0.75)
T1 = parametric_plot3d([-1,t,0], (t,0,1), color = "red", thickness = 1, linestyle = "dotted") 
T2 = parametric_plot3d([cos(t), 1+sin(t),0], (t,0,pi), color = "red", thickness = 1)  
T3 = parametric_plot3d([1,t,0], (t,-2,1), color = "red", thickness = 1)
T4 = parametric_plot3d([-1,t,f(-1,t)], (t,0,1), color = "red", thickness = 4)
T5 = parametric_plot3d([cos(t), 1 + sin(t), f(cos(t),1+sin(t))], (t,0,pi), color = "red", thickness = 4)
T6 = parametric_plot3d([1, t, f(1,t)], (t,-2,1), color = "red", thickness = 4)
T7 = parametric_plot3d([2*t-1,-2*t, 0], (t,0,1), color = "black", thickess=1)
T8 = parametric_plot3d([2*t-1,-2*t, f(2*t-1,-2*t)], (t,0,1),color = "black", thickess=4)
show(S+ T4+T5  + T6  + T8 , aspect_ratio = [4,4,1])

var('x,y,t')
Champ=plot_vector_field( (-y/(x^2+y^2), x/(x^2+y^2)), (x, -1.2, 1.2), (y, -1.2, 1.2),color="blue",aspect_ratio=1,plot_points=10,figsize=6)
C1=parametric_plot([cos(t),sin(t)],(t,0,pi), color="green", thickness=3)
C2=parametric_plot([cos(t),-sin(t)],(t,0,pi), color="red", thickness=3)
show(Champ + C1 + C2)

parametric_plot3d?

parametric_plot3d([-1,t,0], (t,0,1), color = "darkgreen",thickness = 3)