ma4248 weeks 8-9. topics generalized forces, velocities, and momenta; lagrange’s,...
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MA4248 Weeks 8-9. Topics Generalized Forces, Velocities, and Momenta;Lagrange’s, Lagrange-Euler’s, and Hamilton’s Equations
1
D’Alembert’s Principle the sum of the work done bythe applied and inertial forces in a virtual disp. = 0
0 coninerappWWW
i
appi
apprNi
i FW
1
ii
i
inerrNi
i rmW
1
GENERALIZED FORCES
2
If the 3N particle coordinates are expressedin terms of a set of f generalized coordinates
N,...,1i),t;q,...,q(rr f1ii f1 q,...,q
N,...,1ii ,qf
1 q
irr
then
N,...,1ii ,r
q
f
1QW
app
where
f,...,1appi ,Ni
1i q
irFQ
are called generalized forces.
GENERALIZED VELOCITIES
3
Chain-rule gives derivatives of the particle velocities
and the kinetic energy
N,...,1i,0t
ir
q
irf
1qri
f,...,1;N,...,1i,q
ir
q
ir
q
f,...,1,q
irNi1i ii rm
qT
Ni1i iii2
1 rrmT
with respect to the generalized velocities
GENERALIZED MOMENTA
4
f,...,1,qTp
222 zyxmrrmT21
21
We define generalized momenta
Example (linear momenta) for unconstrained motionof a single particle with generalized coordinates x,y,z
,zmzT,ym
yT,xm
xT
are components of the linear momentum vector rm
and the generalized momenta
GENERALIZED MOMENTA
5
22mT21
Example (angular momenta) for motion of a particle in
a circle with radius with generalized coordinate
2mT
the generalized momentum
is the angular momentum
TIME DERIVATIVES OF GENERALIZED MOMENTA
6
torqueF2
mTdtd
Example (linear momenta)
.etc,netxFxmxm
dtd
xT
dtd
Example (angular momentum)
These consequences of Newton’s laws motivate us toconsiderer the quantities
qT
dtdp
TIME DERIVATIVES OF GEN. MOMENTA
7
The Eq. on page 3 and Leibniz’s formula yield
Ni
1i iiNi1i ii q
ir
q
ir
dtdrmrm
Ni
1i iiNi1i ii q
ir
q
irrmrm
qTNi
1i ii q
irrm
Ni
1ii
ii q
r
dtd
qT
dtdp rm
LAGRANGE’S EQUATIONS
8
Combining these equations yields
q
q
irNi
1irmW ii
f
1iner
q
qT
qTf
1 dtd
Therefore
0qqT
qTQ
f
1 dtd
f,...,1,QqT
qT
dtd
EULER-LAGRANGE EQUATIONS
9
If the applied forces are conservative, then
f,...,1,qV
Q
Therefore
f,...,1,0qL
qL
dtd
where )fq,...,1q(VV is the potential energy
where
is the Lagrangian (it may or may not depend on t)
)t,fq,...,1q,fq,...,1q(LVTL
EXAMPLE: PLANE PENDULUM
10
θsinmgθL
cosmgV potential energy
Lagrangian
θ g
m
kinetic energy 22m2
1T
cosmgm21L
22
θmθL 2
θsin-mgθm 2
EXAMPLE: SPHERICAL PENDULUM
11
θsinmg-cosθsinmθL22
Lagrangian
θ g
m
cosmgsinm21L 2222
θmθL2
0L
θsinmL22
θsinmg-cos θsinmθm222
0θsinmdtd 22
MOMENT OF INERTIA
12
For a particle with mass m andposition vector that rotateswith angular velocity abouta line through the origin
r
r
rrvhence with respect to orthonormal coordinates x, y, z
z
y
x
z
y
x
0xy
x0z
yz0
v
v
v
MOMENT OF INERTIA
13
Therefore the kinetic energy of the particle equals
where I is the 3 x 3 inertia matrix for the particle
IvvmvvmT TT
21
21
21
22
22
22
yxyzxz
yzzxxy
xzxyzy
mI
MOMENT OF INERTIA
14
For a rigid body with density (mass per volume)
dxdydz)z,y,x(
2y2xyzxz
yz2z2xxy
xzxy2z2y
I
If ze
dMr
21T 22then
where (see Calkin, page 38)
dxdydz)z,y,x(222 dM,yxr
CENTER OF MASS
15
The center of mass of a system of N particles withmasses and positions is N1 m,...,m
dxdydz)z,y,x(
M1R
z
y
x
MrmrmR NN11
N1 r,...,r
N1 mmM For a rigid body with density (mass per volume)
dxdydz)z,y,x(M
HAMILTONIAN
16
A function of
satisfies
f1f1 q,...,q,q,...,q
dtdLq
qLq
dtdH
ii
i ii q
Ldtd
The Hamiltonian
is said to be “conserved” if its time derivative is zero
LqLqH
i ii
tL
dtdLq
qLq
ii
i ii q
L
hence is conserved iff L does depend explicitly on t
HAMILTONIAN AND ENERGY
17
For scleronomic (time independent) holonomic constraints
N,...,1i),q,...,q(rr f1ii
N
1k kj
k
i
kij m
qr
qr
A
T is a quadratic form in the generalized velocities
f
1j,i ij21 AqqT ji
HAMILTONIAN AND ENERGY
18
Therefore, for scleronomic holonomic constraints
and the hamiltonian
i
f
1jij
iqA
qT
VT)VT(T2LqLqH
i ii
is the total energy
HAMILTONIAN AND ENERGY
19
For a bead with mass m on a horizontal wire rotating withangular speed and distance q from the center
therefore
)
222 qq(2mTL
and qm
qL
)222 qq(
2mH
is conserved but LEH This is an example of a rheonomic (time dependent)constraint in which the constraint (wire) can do work
EULER’S EQUATION
20
Let F be a function of 3 variables and for any smooth function
define R]1,0 tt[:q 1
0
t
tdt)t,q,q(F)q(I
Then for any smooth and small R]1,0 tt[:q
1
0
t
tdtq
qFq
qF)q(I)qq(I
1
0
1
0
ttq
F |qt
tqdt
qF
dtd
qF