elements of classical mechanics - mvputten.org · i. trajectories cartesian and polar ... three...
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Elements of classical mechanics
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Contents
I. Trajectories
Cartesian and polar coordinates
Rotations
Energy and forces
I. Trajectories
Cartesian and polar coordinates
Rotations
Energy and forces
II. Three classical mechanics problems
Hooke’s spring
Newton’s apple
The jumper
Particle trajectories
Cartesian and polar coordinates
Trajectories: bound and unbound
Energy and forces
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Cartesian coordinate system
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— Cartesian coordinates (x,y)
🍎abscissa
ordi
nate
x-axis
y-axis A(x,y)
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— polar coordinates
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— polar coordinates (r,phi)
r = le
ngth
penc
il
ϕpolar angle
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Basis vectors
A=(a,b)
x-axis
y-axis
b
a
i
j
{i,j} = orthonormal basis:
i,j have unit length
i,j are orthogonala = projection of A onto the x-axis
b = projection of A onto the y-axis
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Representations of a vector
A = a i + b j = a 10
⎛
⎝⎜⎞
⎠⎟+ b 0
1⎛
⎝⎜⎞
⎠⎟
= a0
⎛
⎝⎜⎞
⎠⎟+ 0
b⎛
⎝⎜⎞
⎠⎟= a
b⎛
⎝⎜⎞
⎠⎟
i = 10
⎛
⎝⎜⎞
⎠⎟, j = 0
1⎛
⎝⎜⎞
⎠⎟
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choice of basis vectors
Rotation of a coordinate system
x-axis
y-axis
b’ a’
Ay’-axis
x’-axis
{i,j} {i’,j’}
ϕ
Rotation of an ONB
In rotating a coordinate system, vectors A remain unchanged
a’ = projection of A onto the x’-axis
b’ = projection of A onto the y’-axis
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— projections
{i’, j’} from {i,j}:
a’ = projection of X onto the x’-axis
b’ = projection of X onto the y’-axis
x-axis
y-axis
Xunit circle
1
sinϕ
cosϕ
ϕi ' = cosϕ i + sinϕ jj ' = −sinϕ i + cosϕ j
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Rotation: a linear transformation
A = a 'i '+ b ' j '= a '(cosϕ i + sinϕ j) + b'(−sinϕ i + cosϕ j)= (a 'cosϕ − b 'sinϕ )i +(a'sinϕ + b 'cosϕ ) j≡ ai +bj
ab
⎛
⎝⎜⎞
⎠⎟= R a '
b '⎛
⎝⎜⎞
⎠⎟=
cosϕ −sinϕsinϕ cosϕ
⎛
⎝⎜
⎞
⎠⎟
a 'b '
⎛
⎝⎜⎞
⎠⎟
R = rotation matrix
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— example (I)
ϕ =ωt
ab
⎛
⎝⎜⎞
⎠⎟=
cosωt −sinωtsinωt cosωt
⎛
⎝⎜⎞
⎠⎟a 'b '
⎛
⎝⎜⎞
⎠⎟
In a co-rotating frame of the moon:
ω =2πP
angular velocity of circular motion with period P
Earth
Moon
a 'b '
⎛
⎝⎜⎞
⎠⎟= R
0⎛
⎝⎜⎞
⎠⎟
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:
— example (II)ab
⎛
⎝⎜⎞
⎠⎟= R cosωt −sinωt
sinωt cosωt⎛
⎝⎜⎞
⎠⎟10
⎛
⎝⎜⎞
⎠⎟
= R cosωtsinωt
⎛
⎝⎜⎞
⎠⎟ Earth
Moon
A = ab
⎛
⎝⎜⎞
⎠⎟,
dAdt
= da / dtdb / dt
⎛
⎝⎜⎞
⎠⎟= Rω −sinωt
cosωt⎛
⎝⎜⎞
⎠⎟
Earth
Moon
A
dA/dtvelocity vector
position vector
dAdt
=dadt
⎛⎝⎜
⎞⎠⎟
2
+dbdt
⎛⎝⎜
⎞⎠⎟
2
= Rω
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||A||=R d||A||/dt=0
:
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ddti ' =ω j '
ddtj ' = −ω i '
⎧
⎨⎪⎪
⎩⎪⎪
— example (III)
{i '(t), j '(t)}Same applied to the rotating ONB
i’(t)
i’(t+dt) di’j’(t)
j’(t+dt)
dj’
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dϕdt
=ω
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Trajectories in Newton’s gravitational field
Bound orbits
Unbound trajectories
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Trajectories in Newtonian gravity
Bound: elliptical orbits
Unbound: hyperbolic orbitsfocal points at ± p
ϕ
m
M
−ϕ0 <ϕ(t) <ϕ0
p-p
M
ml1 l2l1 + l2 = const.
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Elliptical trajectories (Kepler)
p-p
M
ml1 l2
l1 + l2 = const.
l1,2 = (x ± p)2 + y2
(l2 − 4 p2 )x2 + l2y2 = 14l2 (l2 − 4 p2 )
xa
⎛⎝⎜
⎞⎠⎟
2
+yb⎛⎝⎜
⎞⎠⎟
2
= 1
semi-major axis a semi-minor axis b
For a > b:
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Hyperbolic trajectories (“scattering”)
ϕ
m
M
−ϕ0 <ϕ(t) <ϕ0
r = rcosϕsinϕ
⎛
⎝⎜
⎞
⎠⎟ , r = r ϕ( )Polar coordinates:
Newton’s gravitational force F = −GMmr2
r̂, r̂ = rr,
Can show: r = const.sinϕ0 + sinϕ
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Open and closed according to total energy H
Bound orbits:
Unbound trajectories:
closed
open (reach out to infinity)
H < 0
H > 0
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Kinetic and potential energy
Kinetic energy:
Potential energy (Newtonian gravitational binding energy):
Ek =12mv2
U = −GMmr
v
r
Ek ∝ v2 ≥ 0
U ∝−Mmr
≤ 0
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Total energy H
Total energy H: kinetic energy plus potential energy
H = Ek +U =12mv2 − GMm
r
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One-dimensional “orbits”
v = drdt
m dvdt= F = − d
drU = −
GMmr2
M m
linear motion in one dimension
dHdt
= mv dvdt+GMmr2
drdt= v m dv
dt+GMmr2
⎛⎝⎜
⎞⎠⎟= 0
Newton’s law of gravity
Total energy H is conserved
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