chapitre2

< <0
<
[Ox),[Oy),[Oz)
{~ex~ey~ez}
~
OO0=a. ~ex+b. ~ey+c. ~ez
<0
[Ox0),[Oy0),[Oz0)
{~ e0
x~
e0
y~
e0
z}
~
O0M=x0.~
e0
x+y0.~
e0
y+z0.~
e0
z
~v(M/<)M<
~v(M/<0)M<0
~v(<0/<)<0<M< <0
~a(M/<)M<
~a(M/<0)M<0
~a(<0/<)<0<M<
<0
~v(<0/<)6=~v(O0/<)~a(<0/<)6=~a(O0/<)
O0
<0<
{~ex, ~ey, ~ez} { ~ e0
x,~
e0
y,~
e0
z}
{~ex, ~ey, ~ez}
{~ex, ~ey, ~ez}
{~ e0
x,~
e0
y,~
e0
z}

<0<
θ˙
θ= 0 <0<
~
OO0{~ex,~
e0
x} { ~ey,~
e0
y} { ~ez,~
e0
z}
~
OO0d
dt ~
OO0= 0 θ˙ θ6= 0 <0
<
~
OO0θ<0
<
~ez~
e0
z
~ω <0/<
~
OM =~ OO0+~
O0M
~
OM =~ OO0+~
O0M
~
OM = (a. ~ex+b. ~ey+c. ~ez) + x0.~
e0
x+y0.~
e0
y+z0.~
e0
z
{~ex, ~ey~ez}
d
dt <~
V~
V
<~ω (<0/<)<0<
~v(M/<)=d
dt <~
OM =d
dt <~
OO0+d
dt <~
O0M0
=d
dt <~
OO0+ dx0
dt ~
e0
x+x0d~
e0
x
dt +dy0 dt ~
e0
y+y0d~
e0
y
dt +dz0 dt ~
e0
z+z0d~
e0
z
dt !
=d
dt <~
OO0+x0d~
e0
x
dt +y0d~
e0
y
dt +z0d~
e0
z
dt +dx0 dt ~
e0
x+dy0
dt ~
e0
y
dz0
dt ~
e0
z
=d
dt <~
OO0+x0~ω (<0/<)∧~
e0
x+y0~ω (<0/<)∧~
e0
y+z0~ω (<0/<)∧~
e0
z+dx0
dt ~
e0
x+dy0
dt ~
e0
y
dz0
dt ~
e0
z
=d
dt <~
OO0+~ω (<0/<)∧x0~
e0
x+y0~
e0
y+z0~
e0
z
|{z }
~
O0M
+dx0
dt ~
e0
x+dy0
dt ~
e0
y+dz0
dt ~
e0
z
| {z }
d
dt |<0(~
O0M)
=d
dt <~
OO0+~ω (<0/<)∧~
O0M
|{z }
~v (<0/<)
+d
dt <0~
O0M
|{z }
~v (M /<0)
=~v(<0/<)+~v(M/<0)
d
dt <~
OO0~
OO0=a. ~ex+b. ~ey+c. ~ezd
dt <~
OO0=da
dt . ~ex+db dt . ~ey+dc dt . ~ezd ~e x dt
d ~e y
dt
d ~e z
dt ~ex~ey~ez

~v (M /<0)~
O0M
d
dt <0<0
d
dt <0
d
dt <(OO0)
~ω (<0/<)∧~ O0M
~a(M/<)=d
dt <"~v(M/<)=d dt <"~v(<0/<)+d dt <"~v(M/<0)
=d2
dt2<~
OO0+d
dt x0d~
e0
x
dt +y0d~
e0
y
dt +z0d~
e0
z
dt +dx0 dt ~
e0
x+dy0
dt ~
e0
y
dz0
dt ~
e0
z!
=d2
dt2<~
OO0+ dx0
dt
d~ e0
x
dt +x0d2~
e0
x
dt2+dy0 dt
d~ e0
y
dt +y0d2~
e0
y
dt2+dz0 dt
d~ e0
z
dt +z0d2~
e0
z
dt2!
+ d2x0
dt2~
e0
x+dx0
dt
d~ e0
x
dt +d2y0 dt2~
e0
y+dy0
dt
d~ e0
y
dt +d2z0 dt2~
e0
z+dz0
dt
d~ e0
z
dt !
=d2
dt2<~
OO0+ 2dx0
dt
d~ e0
x
dt + 2 dy0 dt
d~ e0
y
dt + 2 dz0 dt
d~ e0
z
dt !+d2x0 dt2~
e0
x+d2y0
dt2~
e0
y+d2z0
dt2~
e0
z+ x0d2~
e0
x
dt2+y0d2~
e0
y
dt2+z0d2~
e0
z
dt2!
=d2
dt2<~
OO0+ 2dx0
dt
d~ e0
x
dt + 2 dy0 dt
d~ e0
y
dt + 2 dz0 dt
d~ e0
z
dt !+d2x0 dt2~
e0
x+d2y0
dt2~
e0
y+d2z0
dt2~
e0
z
+x0d
dt ~ω (<0/<)∧~
e0
x+y0d
dt ~ω (<0/<)∧~
e0
y+z0d
dt ~ω (<0/<)∧~
e0
z
=d2
dt2<~
OO0+ 2dx0
dt
d~ e0
x
dt + 2 dy0 dt
d~ e0
y
dt + 2 dz0 dt
d~ e0
z
dt !+d2x0 dt2~
e0
x+d2y0
dt2~
e0
y+d2z0
dt2~
e0
z
+#x0 d~ω (<0/<)
dt ∧~
e0
x+~ω (<0/<)∧d~
e0
x
dt !+y0 d~ω (<0/<) dt ∧~
e0
y+~ω (<0/<)∧d~
e0
y
dt !+z0 d~ω (<0/<) dt ∧~
e0
z+~ω (<0/<)∧d~
e0
z
dt !$
=d2
dt2<~
OO0+ 2dx0
dt
d~ e0
x
dt + 2 dy0 dt
d~ e0
y
dt + 2 dz0 dt
d~ e0
z
dt !+d2x0 dt2~
e0
x+d2y0
dt2~
e0
y+d2z0
dt2~
e0
z
+#d~ω (<0/<)
dt ∧x0.~
e0
x+y0.~
e0
y+z0.~
e0
z+~ω (<0/<)∧ x0.d~
e0
x
dt +y0.d~
e0
y
dt +z0.d~
e0
z
dt +!$
=d2
dt2<~
OO0+ 2~ω (<0/<)∧dx0
dt ~
e0
x+dy0
dt ~
e0
y+dz0
dt ~
e0
z
|{z }
d
dt |<0(~
O0M)
+d2x0
dt2~
e0
x+d2y0
dt2~
e0
y+d2z0
dt2~
e0
z
|{z }
d2
dt 2<0(~
O0M)
+d~ω (<0/<)
dt ∧x0.~
e0
x+y0.~
e0
y+z0.~
e0
z
|{z }
~
O0M
+~ω (<0/<)∧
~ω (<0/<)∧x0.~
e0
x+y0.~
e0
y+z0.~
e0
z
|{z }
~
O0M
=d2
dt2<~
OO0+d~ω (<0/<)
dt ∧~
O0M+~ω (<0/<)∧~ω (<0/<)∧~
O0M+ 2~ω (<0/<)∧d
dt <0~
O0M
|{z }
~v (M /<0)
+d2
dt2<0~
OM 0
|{z }
~a (M /<0)
=d2
dt2<~
OO0+d~ω (<0/<)
dt ∧~
O0M+~ω (<0/<)∧~ω (<0/<)∧~
O0M
|{z }
~a (<0/<)
+2~ω (<0/<)∧~v(M/<0)+~a(M/<0)
=~a(<0/<)+~ac+~a(M/<0)
d2
dt2<~
OO0~
OO0=a. ~ex+b. ~ey+c. ~ezd2
dt2<~
OO0=
d2a
dt2. ~ex+d2b dt2. ~ey+d2c dt2. ~ezd ~e x dt
d ~e y
dt
d ~e z
dt ~ex~ey~ez

~a(<0/<)~a(M /<0)
~ac
<0<
<0< <0<d
dt ~
e0
x=d
dt ~
e0
y=d
dt ~
e0
z= 0
O0<0
<~ω (<0/<)=~ 0
O0<0
<~ω (<0/<)=~ 0
~v (M /<)=d
dt ~
OO0+d
dt <0~
O0M
~a(M /<)=d2
dt2~
OO0+d2
dt2<0~
O0M
<0<
<0
<d
dt ~
e0
x6= 0 d
dt ~
e0
y6= 0 d
dt ~
e0
z6= 0 ~
OO0
d
dt ~
OO0=d2
dt2~
OO0= 0
~ω (<0/<)=~ Cte =ω . ~ez˙
~ω (<0/<)
~v (M /<)=~ω (<0/<)∧~ O0M+d
dt <0~
O0M
~a(M/<)=~ω (<0/<)∧~ω (<0/<)∧~ O0M+2~ω (<0/<)∧d
dt <0~
O0M+d2
dt2<0~
O0M
=~ω (<0/<)∧~ω (<0/<)∧~
O0M+"2~ω (<0/<)∧~v(M/<0)+d2
dt2<0~
O0M
~
OM 0=A .t. ~ e0
x~ω (<0/<)=ω . ~ez
~a(M /<)=ω . ~ez∧ω . ~ez∧A .t. ~ e0
x+2ω . ~ez∧d
dt (<0)A .t ~
e0
x+d2
dt2<0A .t ~
e0
x
~a(M /<)=ω . ~ez∧A .ω .t. ~ e0
y+2ω . ~ez∧A . ~
e0
x+ 0
~a(M /<)=−A .ω 2.t. ~ e0
x+ 2.ω .A . ~
e0
y