136
11 The Beginning of the Theory of Stability of Equilibrium
where
S =
a
2 γ sin γ −
1
γ
(sin γ − γ cos γ )
l,
a =
ρ
l
.
(11.38)
From the identity (11.37), it can be seen that the sign of the function L(f, ρ 2 , c ij )
or, what is the same, the value μ coincides with the sign of the expression
f S
2
+ (cos γ − 1)(cos γ − 1 + ηη).
(11.39)
Define the minimum positive values of γ ∗ (η), where the value (11.39) or (11.31)
goes from positive to negative values. Since in the expression (11.39), the first term
is non-negative, and the second for small γ , as we will see below, is positive,
the values we are interested in are γ ∗ (η) we will have when the value f S 2 is the
smallest. It is easy to make sure that the function (11.36) is always positive, and
f min < f (μ) f max ,
(11.40)
where
f min = lim
μ→0
f (μ) = 0; f max = f
μ=
1
ρ 2
=
1
4ρ 2 .
Since the relation (μ) of unknown friction coefficients can take any value from the
range (0, ∞), then the function f (μ) can be arbitrarily small. Since the value S(γ )
(11.38) is bounded at γ > 0, then in the expression (11.39), the value f S 2 can also
be infinitesimal. Hence, γ ∗ (η) are the smallest positive roots of the equation
(cos γ − 1)[cos γ − 1 + ηη(γ )] = 0.
The first multiplier can only reach zero, but it does not change the sign. Equating
the second multiplicand to zero and taking into account (11.25), we get
cos γ − 1 + η(2 − 2 cos γ − γ sin γ ) = 0.
The latter equation is easily converted to the form
− 2 sin
γ
2
sin
γ
2
+ η
γ cos
γ
2
− 2 sin
γ
2
= 0.
(11.41)
The smallest positive root of the equation sin
γ
2 = 0 is γ 1 = 2π and does not
depend on η. Equating to zero the multiplier in (11.41) in the square brackets, we
get
sin
γ
2
+ η
γ cos
γ
2
− 2 sin
γ
2
= 0.
(11.42)
11 The Beginning of the Theory of Stability of Equilibrium
where
S =
a
2 γ sin γ −
1
γ
(sin γ − γ cos γ )
l,
a =
ρ
l
.
(11.38)
From the identity (11.37), it can be seen that the sign of the function L(f, ρ 2 , c ij )
or, what is the same, the value μ coincides with the sign of the expression
f S
2
+ (cos γ − 1)(cos γ − 1 + ηη).
(11.39)
Define the minimum positive values of γ ∗ (η), where the value (11.39) or (11.31)
goes from positive to negative values. Since in the expression (11.39), the first term
is non-negative, and the second for small γ , as we will see below, is positive,
the values we are interested in are γ ∗ (η) we will have when the value f S 2 is the
smallest. It is easy to make sure that the function (11.36) is always positive, and
f min < f (μ) f max ,
(11.40)
where
f min = lim
μ→0
f (μ) = 0; f max = f
μ=
1
ρ 2
=
1
4ρ 2 .
Since the relation (μ) of unknown friction coefficients can take any value from the
range (0, ∞), then the function f (μ) can be arbitrarily small. Since the value S(γ )
(11.38) is bounded at γ > 0, then in the expression (11.39), the value f S 2 can also
be infinitesimal. Hence, γ ∗ (η) are the smallest positive roots of the equation
(cos γ − 1)[cos γ − 1 + ηη(γ )] = 0.
The first multiplier can only reach zero, but it does not change the sign. Equating
the second multiplicand to zero and taking into account (11.25), we get
cos γ − 1 + η(2 − 2 cos γ − γ sin γ ) = 0.
The latter equation is easily converted to the form
− 2 sin
γ
2
sin
γ
2
+ η
γ cos
γ
2
− 2 sin
γ
2
= 0.
(11.41)
The smallest positive root of the equation sin
γ
2 = 0 is γ 1 = 2π and does not
depend on η. Equating to zero the multiplier in (11.41) in the square brackets, we
get
sin
γ
2
+ η
γ cos
γ
2
− 2 sin
γ
2
= 0.
(11.42)