Motion of the Frenet trihedron as a rigid body

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Problem 1.24.131 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Prove that for the curve $x = e^{t} \sin t, y = e^{t} \cos t, z = e^{t}$ , each edge of the accompanying Frenet trihedron forms a constant angle with the $z$ -axis.

Problem 1.24.140 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Prove that the acceleration vector $\mathbf{a}$ of a moving material point is perpendicular to the binormal of the trajectory of motion, and find its decomposition along the tangent vector and the principal normal vector. Prove that

\left|\mathbf{a}\right| = \sqrt{\frac{v^{4}}{R^{2}}+\left(\frac{d v}{d t}\right)^{2}}

where $v = \left|\mathbf{v}\right|, \mathbf{v} = d \mathbf{r} / d t, R$ is the radius of curvature of the trajectory of motion.

Problem 1.24.141 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

(1)(2)

Prove that for a three-times continuously differentiable curve $\Gamma = \left\{ M(s) ; 0 \leqslant s \leqslant S\right\}$ , where $s$ is the variable arc length:

(1)

the formula

\begin{aligned} \mathbf{r}(s+\Delta s) - & \mathbf{r}(s) = \left(\Delta s-\frac{1}{6} k^{2}(s) \Delta s^{3}\right) \tau (s)+\left(\frac{1}{2} k(s) \Delta s^{2}+\right. \\ & \left.+\frac{1}{6} k^{\prime }(s) \Delta s^{3}\right) \nu (s)+\frac{1}{6} k(s) \varkappa (s) \Delta s^{3} \beta (s)+\mathrm{o}\left(\Delta s^{3}\right), \Delta s \rightarrow 0 \end{aligned}

(2)

if by $\xi , \eta , \zeta$ we denote the coordinates of a point in the coordinate system given by the unit vectors $\mathbf{\tau }(s), \mathbf{\nu }(s), \mathbf{\beta }(s)$ , then in a neighborhood of the point $M(s)$ of the curve, the projection of the curve onto the osculating plane has the form

\eta = \frac{1}{2} k(s) \xi ^{2}+o\left(\xi ^{2}\right), \quad \xi \rightarrow 0

the projection onto the normal plane has the form

\eta = \sqrt[3]{\frac{9 k(s)}{2 \varkappa ^{2}(s)}} \zeta ^{2 / 3}+o\left(\zeta ^{2 / 3}\right), \quad \zeta \rightarrow 0, \quad \varkappa (s) \neq 0

the projection onto the rectifying plane has the form

\zeta = \frac{1}{6} k(s) \varkappa (s) \xi ^{3}+o\left(\xi ^{3}\right), \quad \xi \rightarrow 0.

Problem 1.24.142 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Prove that if, on a three-times continuously differentiable curve $\Gamma = \left\{ \mathbf{r}(s) ; 0 \leqslant s \leqslant S\right\}$ , where $s$ is the variable arc length, the parameter $s$ is regarded as time, then as it changes the accompanying Frenet trihedron moves along the curve like a rigid body with angular velocity $\omega = \varkappa \tau +k \beta$ . Thus the curvature is numerically equal to the speed of rotation of the Frenet trihedron about the binormal (this rotation always occurring from the tangent vector $\tau$ toward the principal normal vector $\nu$ ), and the torsion is the speed of its rotation about the tangent.

Problem 1.24.143 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Prove that the Frenet formulas can be written in the form

\frac{d \tau }{d s} = [\omega , \tau ], \quad \frac{d \nu }{d s} = [\omega , \nu ], \quad \frac{d \beta }{d s} = [\omega , \beta ]

(the vector $\omega$ is defined in the previous problem).

References

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.