mecanica_lagrangiana.pdf
TRANSCRIPT
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y= f(x)
y x z= F[f]
z f(x)
z= F[f] =
x2x1
dx [f(x)]2 .
f(x) [x1, x2]
y
xy
f(x1) =y1 f(x2) =y2
F[f] =
x2x1
dxg (f(x), f(x); x) ,
f(x) F
F
(x1, y1) (x2, y2)
f(x1) =y1, f(x2) =y2.
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(x1, y1) (x2, y2)
f(x)
xy
y= f(x)
F[f] =
21
dl.
dl=
dx2 +dy2 =dx
1 + (dy/dx)2 =dx
1 + [f(x)]2
F[f] =
x2x1
dx
1 + [f(x)]2,
f(x1) =y1
f(x2) =y2
f(x)
f(x) =y1+ x x1x2 x1 (y2 y1),
f(x)
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[x1, x2] f(x)
f(x1) =f(x2) = 0 F
f
F =
x2x1
dxg (f(x), f(x); x) =
x2x1
dxg (f(x), f(x); x) =
x2x1
dx
g
ff+
g
ff
.
f =(df/dx) = (d/dx)f x2x1
dxg
ff =
x2x1
dxg
fd
dxf=
g
ff(x)
x2
x1
x2x1
dxd
dx
g
f
f.
F =
x2x1
dx
g
f d
dx
g
f
f.
f(x) F = 0
g
f d
dx
g
f
= 0.
f(x)
F = 0
F = 0
2F
2F
f(x) x
df = 0 df
dx
df= df
dxdx= 0 df
dx= 0,
dx = 0
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d2f= 12d2f
dx2dx2
g(f, f
; x) =
1 +f
2 g
f d
dx g
f
= 0 d
dx f
1 +f2
= 0,
f1 +f2
=c f = c1 c2 a,
c a
f(x) = dxa= ax+b. a b
g
x
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xy
dl x
y dt= dl/v v
x
x= 0 0 = 12
mv2mgx v(x) = 2gx g g(f, f; x)
F[f] =
21
dl
v =
x2x1
dx
1 +f2
2gx .
g
f d
dx
g
f
= 0 g(f, f; x) =
1 +f2
2gx ,
f(x1) =y1
f(x2) =y2
g/f= 0
g
f=
f2gx(1 +f2)
=c,
c f
f2
2gx(1 +f2)=c2 f =
x
2a x, 2a 1
2gc2.
x
f(x) =
xdx
2ax x2 .
x= a(1 cos )
f(x) =
a2(1 cos )sin d2a2(1 cos ) a2(1 cos ) =a
d(1 cos ) =a( sin ) +b.
x= a(1 cos )y= a( sin ) +b
a, b x1 = y1 = 0
1 = 0 b = 0 a
y (x, y)
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xy xy a
y
O
a
g
x d
dx
g fg
f
= 0
g x g/x = 0
g fgf
= 0.
R (1, 1) (2, 2)
dl= R
d2 + sin2 d2 =Rd
sin2 +
d
d
2.
d
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()
F[] =R
21
d
sin2 +2,
(1) =1(2) =2
g(, ; ) =R
sin2 +2 g
g g
=R
sin2 +2 R2
sin2 +2
=aR,
a R R
sin2 +2
sin2 = a
sin2 +2.
d
d =
a
sin
sin2 a2
=
ad
sin
sin2 a2
.
=
+, =
1 a2a2
.
= sin ( ) =sin cos cos sin .
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R sin
R cos = B(R sin cos ) A (R sin sin ) ,
A = cos B = sin
Bx Ay= z,
A B
cos 1 = Bsin 1cos 1 A sin 1sin 1, cos 2= Bsin 2cos 2 A sin 2sin 2.
f1(x), f2(x), , fN(x) {fi(x)}
F[{fi}] = x2x1
dxg(f1, f
1, f2, f
2, , fN, fN; x),
fi(x1) = f(1)i fi(x2) = f
(2)i
N
g
fi d
dt
g
fi
= 0, i= 1, 2, , N.
F fi
[x1, x2]
F =N
i=1 x2x1
dx
g
fi d
dt
g
fi
fi(x) = 0,
fi
f(x, y)
(x, y) f
df=f
xdx+
f
ydy= 0.
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dx dy
df= 0
f
x = 0,
f
y = 0.
fi(x)
f1 f2 F[f1, f2]
G(f1, f2) = 0
G(f1, f2) = 0
f2 = H(f1) g(f1, f
1, f2, f
2; x)
g(f1, f
1, f2, f
2; x) =g(f1, f
1, H(f1), H(f1); x) = g(f1, f
1; x).
f2= H(f1)f
1 F[f1, f2] = F[f1]
F[f1, f2] = F[f1] =
x2x1
dxg(f1, f
1, f2, f
2; x) =
x2x1
dxg(f1, f
1; x).
g
f1 d
dt
g
f1
= 0.
f1(x) f2(x) = H(f1(x))
f2 f1
F =
x2x1
dx
g
f1 d
dx
g
f1
f1(x) +
g
f2 d
dx
g
f2
f2(x)
= 0.
f1 f2
G = G
f1f1(x) +
G
f2f2(x) = 0,
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f2(x)
f2(x) = G/f1G/f2
f1(x).
F= x2x1 dx
gf1
d
dx g
f1 gf2 ddx gf2
G/f1
G/f2
f1(x) = 0.
f1(x)
g
f1 d
dx
g
f1
G
f1
1
=
g
f2 d
dx
g
f2
G
f2
1
.
x
f1(x) f2(x)
(x) f1(x) f2(x) (x)
g
f1 d
dx
g
f1
+(x)
G
f1= 0,
g
f2 d
dx
g
f2
+(x)
G
f2= 0,
G(f1, f2) = 0.
N
fi(x)
i= 1, , N M Gj({fi}) = 0 j= 1, , M M < N
g
fi d
dx
g
fi
+
Mj=1
j(x)Gjfi
= 0, i= 1, , N
Gj({fi}) = 0, j = 1, , M
N+ M N+ M {fi(x)} j(x)
F[{fi}] =F[{fi}] +Mj=1
x2x1
dxj(x)Gj({fi}),
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{fi} {j(x)}
F[{fi}] = F[{fi}] +Mj=1
x2x1
dxj(x)Gj({fi})
= x2
x1
dxN
i=1 g
fi
d
dx g
f
i fi(x)
+
x2x1
dxMj=1
Gj({fi})j(x) +j(x)
Ni=1
G
fifi(x)
.
F[{fi}] = x2x1
dx
Ni=1
g
fi d
dx
g
fi
+
Mj=1
j(x)G
fi
fi(x)
+Mj=1
Gj({fi})j(x)
.
fi j
f(x, y)
h(x, y) = 0 dx dy
dh=h
xdx+
h
ydy dy= h/x
h/ydx
dy dx
df=f
xdx+
f
ydy=
f
x f
y
h/x
h/y
dx= 0.
dx
f
x
h
x
1
=f
y
h
y
1
.
(x, y)
f
x+
h
x= 0,
f
y +
h
y = 0, h(x, y) = 0,
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x, y
f
x= 0,
f
y = 0,
f
= 0,
f(x, y) f(x, y) +h(x, y)
F[f] =
21
dt=
21
ds
v =
1
c
21
nds,
n c xy
n n(x, y)
F =1
c
x2x1
dxn (x, f(x))
1 + [f(x)]2 = 0.
g(f, f; x) =n (x, f)
1 +f2
N xi = 1, , N i = 1, 2, 3 T = Ec U
S[{xi}] = t2t1
dtL ({xi, xi}) , L ({xi, xi}) T({xi}) U({xi}) ,
L S
t1< t2
xi(t1) =x(1)i, xi(t2) =x
(2)i
{x(1)i, x(2)i } xi(t)
S= 0.
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L
xi d
dt
L
xi
= 0,
xi(t1) =x
(1)i
xi(t2) =x(2)i
= 1, , Ni= 1, 2, 3.
m
k
L=1
2mx2 1
2kx2.
L
x =
kx,
d
dtL
x = d
dt(mx) =mx,
mx+km = 0,
m
l T = 12ms2
s
s = l T = 12ml22
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U=mgl(1cos )
L=1
2ml22 mgl(1 cos ).
L
d
dt
L
= 0.
L
= mgl sin , d
dt
L
=
d
dt
ml2
=ml2,
mgl sin +ml2= 0 + gl sin = 0.
N
s
s= 3N M M s
q1(t), q2(t), , qs(t) = {qj(t)}.
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q1(t), q2(t), , qs(t) = {qj(t)}.
xi= xi(q1, q2, , qs; t) =xi({qj}; t), xi= xi({qj, qj}; t).
qj =qj({xi}; t), qj = qj({xi, xi}; t).
S[{qj}] = t2t1
dtL ({qj, qj}; t) = 0,
L
L ({xi, xi}) = T({xi}) U({xi})= T({qj, qj}; t) U({qj}; t) L ({qj, qj}; t) .
L
qj d
dt L
qj
= 0, j = 1, , s.
s= 1
N = 1
M = 2 s = 3 1 2 = 1 q
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q=
x= l sin , y= l cos ,
O
x
x= l cos , y= l sin ,
L=1
2m
x2 + y2
=1
2ml22.
U=mgy = mgl cos .
L= T U=12
ml22 +mgl cos ,
L
d
dtL
= mgl sin d
dt
ml2
= mgl sin ml2= 0,
+ glsin = 0
(x, y)
L=1
2m
x2 + y2 mgy.
L/x= 0 x
L
y =mx=
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x y
L
y d
dt
L
y
= mg my= 0 y= g.
2 +g sin = 0,
2+ 2+g cos = 0.
L({qj , qj}) =T({qj}) U({qj}) L
qj d
dt
L
qj
= U
qj d
dt
T
qj
Qj
Qj = Uqj
T
mqj
mqj =Qj ,
U=U({qj}; t) T
qj U
qj d
dt
T
qj
= 0,
d
dt
T
qj
T
qj=Qj
Qj
Qj Uqj
.
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m
s = 2 (,,z)
= ztan 2 z
x= cos
y= sin
z= 2
x= cos sin y= sin + cos
z= 2
T =1
2
m x2 + y2 + z2 =12
m2 +22 + 2 2 2 .
1 + 2
2 = 1 +
1
tan2 2=
1
sin2 2= 2
2,
T
T =1
2m
2 2
2+22
.
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U=mgz= mg
2,
L=
1
2m
2
2
2 +2
2 mg 2 .
L/= 0
L
=m2= lz
z
L = mr v= m (e+zez) ( e+e+ zez)
=z e+m(z z)e+m2
ez
=2 2
e+m2ez.
L
d
dt
L
=m2 mg
2 m 2
2 = 0.
lz
l2zsin
2 2
m23 +
1
2g sin = 0.
a= s= 1
q=
x= 12
at2 +l sin
y= l cos
t0 = 0 O
x0 = 0 v0 = 0 x= at+l
cos
y= l sin
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t
L= T U=12
m
x2 + y2 mgy = 1
2m
at+l cos 2
+1
2m2l2 sin2
+mgl cos =1
2ma2t2 +
1
2ml22 +matl cos +mgl cos .
L
d
dt
L
=matl sin mgl sin d
dt
ml2+matl cos
= matl sin mgl sin ml2 mal cos +matl sin = 0.
+a
lcos +
g
lsin = 0.
a= 0
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= 0
e
a
lcos e+
g
l sin e= 0 tan e= a
g.
a > 0 a = 0
e
= e || 1
+a
lcos(+e) +
g
lsin(+e)
= +a
l(cos cos e sin sin e) + g
l(sin cos e+ cos sin e)
= + al
cos e+g
l
sin e cos + a
l
sin e+g
l
cos e sin = 0, sin
+a
l sin e g
lcos e
= 0.
a cos e+g sin e= 0
cos2 e+ sin2 e= 1
cos e= g
g2 +a2, sin e= a
g2 +a2
e < 0
a
l sin e g
lcos e =
g2 +a2
l ,
+2= 0, 2 =
g2 +a2
l .
g/l a= 0
a = 0
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M
M
l1 l2
O
x
q1 m1 q2
m2 xi i= 1, 2, 3
x1= q1, x2 = l1 q1+q2, x3 = l1 q1+l2 q2.
x1 = q1, x2 = q1+ q2, x3= q1 q2.
T =1
2m1q
21+
1
2m2(q1 q2)2 +1
2m3(q1+ q2)
2 ,
U= m1gq1 m2g (l1 q1+q2) m3g (l1 q1+l2 q2) ,
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T = 1
2m1q
21+
1
2m2(q1 q2)2 +1
2m3(q1+ q2)
2
+ m1gq1+m2g (l1 q1+q2) +m3g (l1 q1+l2 q2) .
Lq1
ddt
Lq1
= (m1 m2 m3)g (m1+m2+m3)q1+ (m2 m3)q2= 0,
L
q2 d
dt
L
q2
= (m2 m3)g+ (m2 m3)q1 (m2+m3)q2 = 0.
(m1+m2+m3)q1+ (m3 m2)q2= (m1 m2 m3)g,(m2 m3)q1 (m2+m3)q2= (m3 m2)g,
q1 q2
fi(q1, q2, ; t) = 0, i= 1, 2, , M
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M s = 3N M s {qj} j = 1, 2, , s s
L
qj d
dt
L
qj
= 0, j= 1, 2, , s,
L
qj d
dt
L
qj
+
Mi=1
i(t)fiqj
= 0, j= 1, 2, , s.
L({qj, qj}) =T({qj}) U({qj})
L
qj d
dt L
qj+
M
i=1
i(t)fi
qj=
U
qj d
dtT
qj+
M
i=1
i(t)fi
qj
= Fj mqj+Mi=1
i(t)fiqj
= 0.
mqj =Fj+Mi=1
i(t)fiqj
.
QjMi=1
Qij , Qij i(t) fiqj
Qj j Qij i
j
Qij
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R M
x y
q1= x
q2 =
y = 0 x= 0
f(x, ) =x R = 0.
T =1
2Mx2 +
1
2I2,
I = MR2/2
U= Mgx sin
L=1
2Mx2 +
1
2I2 +M gx sin .
L
x d
dt
L
y
+(t)
f
x= 0 Mg sin Mx+= 0,
L
d
dtL
+(t)
f
= 0 1
2MR2 R= 0.
x(t) (t) (t) = x/R
1
2MR2
x
R
+R= 0 = 1
2Mx.
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Mg sin Mx 12
Mx= 0 x= 23
g sin .
= 13
Mg sin , = 2g
3Rsin .
t0 = 0 x0 = 0
0= x0= 0 0 = 0
x(t) =1
3gt2 sin , (t) =
1
3Rgt2 sin .
2/3
g sin = 90
I = MR2
x= g2sin = g2Rsin
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Qx = f
x= = 1
3Mg sin ,
Q = f
= R= R
3Mg sin .
Qx< 0
Q >0
= x/R x x
L=1
2mx2 +
1
2I
x
R
2+M gx sin =
3
4Mx2 +Mgx sin .
L
x d
dt
L
x
=M g sin 3
2Mx= 0 x= 2
3Mg sin .
fi(q1, q2, ) = 0, i= 1, 2, , M
fi({qj , qj}; t) = 0, i= 1, 2, , M,
j
Ajqj+B = 0,
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h({qj}; t)
Aj = h
qj, B=
h
t,
j
Ajqj+B =j
h
qj qj+
h
t =dh
dt = 0 h({qj}; t) c= 0,
c
x R
x x
R= 0
f(x, x,, ) = x R= ddt
(x R) = 0 x R= c.
c y y = 0
= 0 c= 0
T =1
2
i
mx2i.
xi= xi({qj}; t)
xi= j
xi
qjqj+
xi
t .
x2i=kj
xiqj
xiqk
qjqk+ 2j
xiqj
xit
qj+
xi
t
2,
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T =ijk
1
2m
xiqj
xiqk
qjqk+ij
mxiqj
xit
qj+i
1
2m
xi
t
2
=
jkajkqjqk+ j
bjqj+ c,
ajk =akj 12
i
mxiqj
xiqk
, bji
mxiqj
xit
, c=1
2
i
m
xi
t
2.
xi= xi({qj})
T = jk
ajkqjqk.
T
ql=k
alkqk+j
ajlqj ,
l
qlT
ql=kl
alkqkql+kl
ajlqjql = 2kl
alkqkql= 2T,
l
qlT
ql= 2T
f(y1, y2, , yN) n
f(y1, y2, , yN) =nf(y1, y2, , yN),
Ni=1
yif
yi=nf
i
yif(y1, , yN)
(yi)
=1
=n n1f=1
= 1
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L= L({qj, qj}; t)
t0
L({qj, qj}; t) =L({qj, qj}; t+t0) Lt
= 0.
L
dL
dt =j
L
qjqj+
j
L
qjqj+
L
t =j
L
qjqj+
j
L
qjqj.
dL
dt =j
d
dt
L
qj
qj+
j
L
qjqj =
j
d
dt
qj
L
qj
.
d
dt
j
qjL
qj L
= 0 H
j
qjL
qj L=
H
U U=U({qj})
L
qj=
T
qj
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H=j
qjL
qj L=
j
qjT
qj T+U.
U=U({qj}) xi= xi({qj})
H= 2T T+U=T+U=E,
E
H
L H=E E
L H= E
U H = E
E L
H=E
H
qk
L(q1, , qk, , qs; t) =L(q1, , qk+c, , qs; t) Lqk
= 0,
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c L
qk L
qk qk
d
dt L
qk
= 0 pk L
qk =
pk
pk =pk(q1, , qk1, qk+1, , qs, q1, , qs; t) =
qk pk
qk pk
N
Fi = 0 ri
i
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i
Fi ri = 0 Fi= pi
Fi pi = 0 i
(Fi pi) ri= 0 Fi fi
i
(Fi+ fi pi) ri= i
Fi ri+ i
fi ri i
pi ri= 0.
i
Fi ri i
pi ri = 0
ri
ri = ri({qj}; t)
{qj}
j = 1, 2, , s
ri =j
riqj
qj ,
i
Fi ri i
pi ri =ij
Fi riqj
qj ij
miri riqj
qj = 0.
miri riqj
= d
dt
miri ri
qj
miri d
dt
riqj
.
d
dt
riqj
=
riqj
,
ri
qj=
ri
qj.
d
dt
riqj
=k
2riqkqj
qk+ 2riqjt
,
riqj
=
qj
k
riqk
qk+ri
t
=k
2riqkqj
qk+ 2riqjt
,
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riqj
=
qj
k
riqk
qk+ri
t
=
riqj
.
Qj =iFi
riqj ,
j
Qj
i
d
dt
miri ri
qj
miri ri
qj
qj = 0.
i
miri riqj
=
qj
1
2 i
mi |ri|2
= T
qj,
i
miri riqj
=
qj
1
2
i
mi |ri|2
= T
qj.
j
Qj
d
dt
T
qj
T
qj
qj = 0.
qj
d
dt
T
qj
T
qj=Qj, j = 1, 2, , s.
U
Fi =
riU
Qj =i
Fi riqj
= i
riUriqj
= Uqj
.
U {qj} d
dt
T
qj
T
qj=
d
dt
L
qj
T
qj= U
qj
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d
dt
L
qj
T
qj+
U
qj=
d
dt
L
qj
L
qj= 0,
S[{qj}] = t2t1
dt (T+V) = 0,
V
V =i
Fi ri.
S =
t2t1
dtT+
t2t1
dti
Fi ri = t2t1
dtT+
t2t1
dti
Fi ri
=
t2t1
dtT+
t2t1
dtj
Qjqj.
j Q
jqj =ijFi
ri
qj qj =iFi j
ri
qj qj =iFi ri.
T =T({qj, qj}; t)
t2t1
dtT =
t2t1
dtj
T
qj d
dt
T
qj
qj.
t2t1
dtj
Tqj
ddt
Tqj
+Qj
qj = 0
qj
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{qj} {qj} L= L({qj, qj}; t) {qj, pj}
H=j
qjL
qj L=
j
qjpj L.
pj = L/qj L {qj, qj}
{qj , pj}
pj = L
qj=pj(
{qj , qj
}; t)
qj = qj(
{qj , pj
}; t).
H({qj , pj}; t) =j
qjpj L=j
qj({qj , pj}; t)pj L({qj, qj({qk, pk}; t)}; t).
H
{qj , pj}
L= L({qj , qj}; t) H=H({qj, pj}; t)
y = f(x) x p = f(x)
f(x) p
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x y
y = f(x)
(0, g)
y = g+ px g = y px y f(x) g(p) = f(x) px f(x) g(p) p=f(x) px
L {qj} H {pj} L qjpj H
y = f(x)
p g
(x, y) (g, p) g
y p
H
H
dH=j
H
qjdqj+
H
pjdpj
+
H
tdt.
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dH =j
(pjdqj+ qjdpj) j
L
qjdqj+
L
qjdqj
L
t
= j (pjdqj+ qjdpj) j ( pjdqj+pjdqj) L
tdt.
L/qj = pj
pj =L/qj
j
H
qjdqj+
H
pjdpj
+
H
tdt=
j
(qjdpj pjdqj) Lt
dt
qj =H
pj, pj =
H
pj, j = 1, 2,
, s.
H
t =L
t
2s
s
2s
{qj0, pj0} 2s
{qj0, qj0
} (qj, pj)
(qj, qj)
dH=j
H
qjdqj+
H
pjdpj
+
H
tdt=
j
( pjdqj+ qjdpj) + Ht
dt=H
tdt,
dHdt
= Ht
.
H H=
H L/t = 0
H/t= L/t
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{qj} j = 1, 2, , s
L= T U
pj =L/qj
qj = qj({qj, pj}; t)
H= jpjqjL
H
R
r
m
x2 +y2 =2 =R2 F = kr = R
q1 = q2 = z U(r)
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rU= Ur
er+1
r
U
e+
1
r sin
U
e= krer.
U
e= 0 U=U(r, ),
U
e = 0 U=U(r),
U
r =kr U(r) =k
r0
drr =1
2kr2 =
1
2k
R2 +z2
,
r= 0
x= cos = R cos , y= sin = R sin , z= z
x= R sin , y= R cos , z= z,
L= T U = 12
m
x2 + y2 + z2 1
2k
R2 +z2
= 1
2m
R22 + z21
2k
R2 +z2
.
p=L
=mR2, pz =
L
z =mz.
= p
mR2, z=
pz
m.
H = p+ zpz L=p2
mR2+
p2zm
12
m
R2 p
mR2
2+pz
m
2+
1
2k
R2 +z2
=p2
2mR2+
p2z2m
+1
2k
R2 +z2
.
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L H=E= T+ U= H=T+ U
+ T+U
p= H
= 0 (1) pz = Hz
= kz (3)
= H
p=
pmR2
(2) z=H
pz=
pzm
(4)
p= mR2=
z (t) =pt/mR2 + c c
z= pzm
= km
z z+20z= 0,
=
k/m z
H
qj H L
qj L
L= L(q1, q2, , qj1, qj+1, , qs; q1, q2, , qj1, qj , qj+1, , qs; t),
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s 1 pj =L/qj 2(s 1) + 1 = 2s 1 qj
pj = 0 pj =j = ,
qj =H
j(q1, q2, , qj1, qj+1, , qs;p1, p2, , pj1, , pj+1, , ps; t).
qj
2s2 2s2 qj
qj(t) =
dtj(q1, q2, , qj1, qj+1, , qs;p1, p2, , pj1, , pj+1, , ps; t).
s
pj =j =
qj(t) =
dtj =jt+j,
j
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O
mr +mR= F,
R O
O R(t) = 12at2ex
R= aex
F =Fg+ T = mgey Te,
T
ml2
e+mle= maex mgey Te
=
ma(sin e+ cos e)
mg(
cos e+ sin e)
Te
= 0 = 0 = + 34
ml
2 = ma sin +mg cos Tml= ma cos mg sin
T
pi
qj
= 0