Post on 07-May-2018
(c)2017 van Putten 2
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
(c)2016 van Putten 3
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
(c)2017 van Putten8
Representations of a vector
A = a i + b j = a 10
⎛
⎝⎜⎞
⎠⎟+ b 0
1⎛
⎝⎜⎞
⎠⎟
= a0
⎛
⎝⎜⎞
⎠⎟+ 0
b⎛
⎝⎜⎞
⎠⎟= a
b⎛
⎝⎜⎞
⎠⎟
i = 10
⎛
⎝⎜⎞
⎠⎟, j = 0
1⎛
⎝⎜⎞
⎠⎟
(c)2017 van Putten9
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
(c)2017 van Putten10
— 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
(c)2017 van Putten12
— 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⎛
⎝⎜⎞
⎠⎟
(c)2017 van Putten13
:
— 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
(c)2017 van Putten14
||A||=R d||A||/dt=0
:
15
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’
(c)2017 van Putten
dϕdt
=ω
16
Trajectories in Newton’s gravitational field
Bound orbits
Unbound trajectories
(c)2017 van Putten
17
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.
(c)2017 van Putten
18
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:
(c)2017 van Putten
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
(c)2017 van Putten19
20
Open and closed according to total energy H
Bound orbits:
Unbound trajectories:
closed
open (reach out to infinity)
H < 0
H > 0
(c)2017 van Putten
21
Kinetic and potential energy
Kinetic energy:
Potential energy (Newtonian gravitational binding energy):
Ek =12mv2
U = −GMmr
v
r
Ek ∝ v2 ≥ 0
U ∝−Mmr
≤ 0
(c)2017 van Putten
22
Total energy H
Total energy H: kinetic energy plus potential energy
H = Ek +U =12mv2 − GMm
r
(c)2017 van Putten