7.3 Solutions
311
Equation (9) describes the orbit of the planet (Kepler’s first law of planetary
motion)
7.16 T =
1
2
m 1 ˙
x
2
1 +
1
2
m 2 ˙
x
2
2
(1)
V =
1
2
k(x 1 − x 2 )
2
(2)
L =
1
2
m 1 ˙
x
2
1 +
1
2
m 2 ˙
x
2
2 −
1
2
k(x 1 − x 2 )
2
(3)
Equations of motion are
d
dt
∂ L
∂ ˙
x 1
−
∂ L
∂ x 1
= 0
( 4 )
d
dt
∂ L
∂ ˙
x 2
−
∂ L
∂ x 2
= 0
( 5 )
Using (3) in (4) and (5)
m 1 ¨
x 1 + k(x 1 − x 2 ) = 0
( 6 )
m 2 ¨
x 2 − k(x 1 − x 2 ) = 0
( 7 )
It is assumed that the motion is periodic and can be considered as superposition of harmonic components of various amplitudes and frequencies. Let one
of these harmonics be represented by
x 1 = A sin ωt, ¨
x 1 = −ω
2 A sin ωt
(8)
x 2 = B sin ωt, ¨
x 2 = −ω
2 B sin ωt
(9)
Substituting (8) and (9) in (6) and (7) we obtain
(k − m 1 ω
2
) A − k B = 0
− k A + (k − m 2 ω
2
)B = 0
The frequency equation is obtained by equating to zero the determinant
formed by the coefficients of A and B:
(k − m 1 ω 2 )
−k
−k
(k − m 2 ω 2 )
= 0
Expansion of the determinant gives
m 1 m 2 ω
4
− k(m 1 + m 2 )ω
2
= 0