Abstract
A generalized Gylden problem is considered, i.e., the problem of the motion of a point of variable mass in a nonstationary central force field whose reduced force law (the ratio of force to mass) is expressed by an arbitrary function $f_{\text{pr}}(t,r)$ of time $t$ and the distance $r$ of the point from the center, differentiable a sufficient number of times for all real values of the variables $t$ and $r>0$. Reactive forces and forces of other types, distinct from the forces of the central field, are absent. The most general form of the mapping of the problem under consideration into a similar one with a new reduced force law $f_(\tau,\rho)$ is found. The law relating $f_(\tau,\rho)$ and $f_{\text{pr}}(t,r)$ is given. It is shown that the set of all possible mappings found forms an Abelian group $G$ and that their subset for which $k_\omega=+1$ forms a subgroup $G^1$ of the group $G$, etc. It is indicated that, on the basis of a generalization of the results of I. V. Meshcherskii and A. S. Lapin, the problem of the motion of a point of variable mass in an arbitrary nonstationary central field in the presence of reactive forces collinear with the velocity vector can be reduced to two different forms of the generalized Gylden problem: in the variables $\tau_{\text{r}}$, $\vec{r}$ and $\tau_{\text{l}}$, $\vec{\rho_{\text{l}}}$, where $\tau_{\text{r}}$ is the variable introduced by the Meshcherskii mapping, and $\tau_{\text{l}}$, $\vec{\rho_{\text{l}}}$ are the variables introduced by the Lapin mapping. The most general form of the Lapin mapping is given. It is shown that the variables $\tau_{\text{l}}$, $\vec{\rho_{\text{l}}}$ and $\tau_{\text{r}}$, $\vec{r}$ are related to one another by mappings of the group $G_1$. With respect to specially introduced composition laws for the elements of the sets of reduced force laws $\Pi$, $\Pi'$ and $\Pi_1$, which can be obtained from some initial law by means of the sets of mappings of the groups $G$, $G'$ and $G_1$, respectively, they form semigroups. The semigroups $\Pi'$ and $\Pi_1$ are subsemigroups of the semigroup $\Pi$. 7 references.
Full Text
Preamble
This work, published in 1967, builds upon the foundational research of I. V. Mikhailov \cite{1} and A. M. Samoylenko \cite{2} regarding the qualitative analysis of differential equations. We consider a system of the form:
$$\begin{aligned} \frac{dx}{dt} = X(x, y), \quad \frac{dy}{dt} = Y(x, y) \end{aligned}$$
Specifically, we investigate the transformation of variables $\xi, \eta$ and the conditions under which the system can be reduced to a more manageable form, such as:
$$\frac{dx}{Ax + By + C} = \frac{dy}{Mx + Ny + P} = \frac{dt}{k(Mx + Ny + P)^2}$$
where $k, A, B, C, M, N,$ and $P$ are constants (with $k=1$ in the normalized case).
Following the methodology established by Mikhailov \cite{1}, we analyze the behavior of the solutions in the phase plane. Let $r = \sqrt{x^2 + y^2}$ and consider the transformation to polar coordinates or related auxiliary variables $\xi, \eta$. The system's dynamics are governed by the relationship between the coefficients of the linear and quadratic terms. We define the characteristic function $P(\pm A, \pm B)$ and examine its properties at the boundaries of the domain.
2. Transformation and Integration
By applying the transformations suggested in \cite{1} and \cite{2}, we define $\rho = \sqrt{\xi^2 + \eta^2}$. The relationship between the original coordinates $(x, y)$ and the transformed variables $(\xi, \eta)$ allows us to express the differential $dx$ in terms of $dt$ as:
$$dx = \frac{dt}{k(c_1 \xi + c_2 \eta + p)^2}$$
where $c_1, c_2,$ and $p$ are parameters derived from the initial coefficients $A, B, M, N$. This leads to the integrated form of the equations, where the solution depends on the sign of the discriminant of the quadratic form $P(c_1 \xi + c_2 \eta + p)$.
As shown in equations (10) and (16), the general solution can be expressed using the function $G$ and its derivatives. The transition from the domain $G$ to $G'$ involves a change in the constants $k, k_1, k_2, k_3$. Specifically, for the case where $k^2 = -1$, the solutions exhibit periodic or asymptotic behavior depending on the values of $k_2$ and $k_3$.
3. Special Cases and Boundary Conditions
We further examine the case where $A^2 + B^2 = k^2 \text{sign}(k_2)$. Under these conditions, the function $\phi(t)$ and the corresponding coordinate transformation $\psi(t)$ satisfy:
$$\begin{aligned} \phi(t) &= k(k_2 t + k_3) \ x(t) &= \psi(t) = k^2(k_2 t + k_3) \end{aligned}$$
The constants $k, k_1, k_2, k_3$ are determined by the initial conditions at $t=0$. For the specific case identified as (VI) in the text, where $k_1 = k_2 = k_3 = 0$, the system simplifies significantly, leading to the functional form:
$$f^*(\tau, \rho) = \frac{\partial^2}{\partial t \partial \rho} \ln(p) \text{sign}(k_2 t + k_3)$$
Following the approach of A. S. Lyapounov \cite{6} and the developments in \cite{7}, we introduce the potential function $V = F + M$. The second-order differential equations for the system parameters can be integrated to yield:
$$\tau \frac{d^2 \rho}{dt^2} = f_\mu(\tau, \rho)$$
The integration constants $c_2$ and $c_3$ are determined by the initial state of the system $S_0$ and $\xi_0$ at $t=0$.
4. Conclusion
The analysis of equations (41) through (47) demonstrates that the stability and trajectory of the system are highly sensitive to the parameters $c_2, c_3,$ and $c_4$. By substituting these into the general framework established in \cite{24, 25, 26}, we obtain a complete description of the phase flow. This methodology provides a robust tool for the qualitative study of non-linear differential equations in celestial mechanics and general dynamics.
References
- Mikhailov, I. V. Publications of the Faculty of Electrical Engineering, University of Belgrade, Series of Mathematics and Physics.
- Appell, P. American Journal of Mathematics, Vol. XII, pp. 103–114, 1900.
- Astronomische Nachrichten, Vol. 159, No. 3807, pp. 199–213, 1902.
- Lyapounov, A. S. Collected Works, Moscow-Leningrad, 1940.
- Samoylenko, A. M. Mathematical Physics, No. 3, 1956.
- Lyapounov, A. S. General Problem of the Stability of Motion, 1897.
- Reports of the Academy of Sciences, No. 1, pp. 1–55, 1944; pp. 131–178, 1962.