V. A. GROBOV

THE METHOD OF AVERAGING CANONICAL EQUATIONS CONTAINING A “QUASI-CYCLIC” ANGULAR COORDINATE

(Presented by Academician N. N. Bogolyubov, December 2, 1957)

Let us consider a dynamical system whose state is determined by (r) variables (q_1, q_2, \ldots, q_r) and an angular variable (\varphi). Suppose that the motion of the system under consideration is described by a canonical system of equations of the form

[
\frac{dq_k}{dt}=\frac{\partial H}{\partial p_k},
\tag{1a}
]

[
(k=1,2,\ldots,r)
]

[
\frac{dp_k}{dt}=-\frac{\partial H}{\partial q_k};
\tag{1б}
]

[
\frac{d\varphi}{dt}=\frac{\partial H}{\partial p_{r+1}},\qquad
\frac{dp_{r+1}}{dt}=-\mu\,\frac{\partial H_1}{\partial \varphi},
\tag{2}
]

where

[
H=H_0(q_1,\ldots,q_r,\ p_1,\ldots,p_r,\ p_{r+1})+
]

[
+\mu H_1(q_1,\ldots,q_r,\varphi,\ p_1,\ldots,p_{r+1})+\mu^2\ldots;
\tag{3}
]

[
H_0=\frac12\sum_{i=1}^{r}\sum_{k=1}^{r} a_{ik}q_iq_k
+\sum_{i=1}^{r}\sum_{k=1}^{r+1} b_{ik}q_ip_k
+\frac12\sum_{k=1}^{r+1} c_kp_k^2;
\tag{4}
]

(\mu) is a small parameter.

Systems of this type occur in the dynamics of turbogenerator rotors; moreover, the unperturbed Hamiltonian (H_0) is readily reduced to the form (4) by a suitable choice of generalized coordinates.

In equations (2) the derivative of the momentum coordinate (p_{r+1}), corresponding to the angular variable (\varphi), is proportional to the small parameter; therefore, according to N. N. Bogolyubov’s perturbation theory ((^1)), it is a slowly varying function of time, and the angular variable may be called “quasi-cyclic” (i.e., almost cyclic). From physical considerations it follows that the coordinates (q_1, q_2,\ldots,q_r) are periodic functions of the angle of rotation (\varphi) with period (2\pi).

Following the idea of the asymptotic methods of N. M. Krylov and N. N. Bogolyubov ((^2)), we assume for system (1), in the first approximation,

[
q_k^{(1)}=a_k\cos(\varphi+\psi_k),
\tag{5}
]

where (a_k) and (\psi_k) are regarded as slowly varying functions of time which, over the course of one period, may be considered constant; as a result we have

[
\dot q_k=-a_k\dot\varphi\sin(\varphi+\psi_k).
\tag{6}
]

Solving equation (1a) with respect to the momentum coordinates (p_1, p_2,\ldots,p_r), we obtain

[
p_k=-\frac{1}{c_k}a_k\dot{\varphi}\sin(\varphi+\psi_k)
-\frac{1}{c_k}\sum_{i=1}^{r} b_{ik}a_i\cos(\varphi+\psi_i)
-\frac{\mu}{c_k}\left(\frac{\partial H_1}{\partial p_k}\right)^{(1)} .
\tag{7}
]

Considering expressions (5) and (7) as formulas for a transformation of variables and differentiating them, taking into account the dependence of (a_k) and (\psi_k) on time, we obtain

[
\frac{da_k}{dt}\cos\theta_k-a_k\frac{d\psi_k}{dt}\sin\theta_k=0,
\tag{8}
]

[
\frac{da_k}{dt}\sin\theta_k+a_k\frac{d\psi_k}{dt}\cos\theta_k=
]

[
=\frac{c_k}{\dot{\varphi}}\left(\frac{\partial H}{\partial q_k}\right)^{(1)}
-\frac{a_k\ddot{\varphi}}{\dot{\varphi}}\sin\theta_k
-\frac{\mu}{\dot{\varphi}}\frac{d}{dt}\left(\frac{\partial H_1}{\partial p_k}\right)^{(1)}
-a_k\dot{\varphi}\cos\theta_k-
]

[
-\frac{1}{\dot{\varphi}}\sum_{i=1}^{r}
\left(\frac{db_{ik}}{dt}a_i\cos\theta_i-b_{ik}a_i\dot{\varphi}\sin\theta_i\right)
=
]

[
=F_k(a_1,\ldots,a_r,\varphi+\psi_1,\ldots,\varphi+\psi_r),
\tag{9}
]

where (\theta_k=\varphi+\psi_k). In the expressions for the derivatives (\partial H/\partial q_k) and (\partial H_1/\partial p_k) in equations (9), the values of (q_k) and (p_k) according to formulas (5) and (7) must be substituted.

Multiplying equation (8) successively by (\cos\theta_k) and (\sin\theta_k), and equation (9), respectively, by (\sin\theta_k) and (\cos\theta_k), we obtain, as the result of addition and subtraction, the following system of equations

[
\frac{da_k}{dt}
=
F_k(a_1,a_2,\ldots,a_r,\varphi+\psi_1,\ldots,\varphi+\psi_r)\sin\theta_k,
]

[
\frac{d\psi_k}{dt}
=
\frac{1}{a_k}
F_k(a_1,a_2,\ldots,a_r,\varphi+\psi_1,\ldots,\varphi+\psi_r)\cos\theta_k .
\tag{10}
]

In order to eliminate the “quasicyclic” variable from the right-hand sides of equations (10), let us average them over (\varphi+\psi_k) over a time equal to one period; we obtain

[
\frac{da_k}{dt}
=
-\frac{a_k\ddot{\varphi}}{2\dot{\varphi}}
-\frac{1}{2\dot{\varphi}}\sum_{i=1}^{r}
\left[
\frac{db_{ik}}{dt}a_i\sin(\psi_i-\psi_k)
-b_{ik}a_i\dot{\varphi}\cos(\psi_i-\psi_k)
\right]
+
]

[
+\frac{1}{2\pi\dot{\varphi}}\int_{0}^{2\pi}
\left[
c_k\left(\frac{\partial H}{\partial q_k}\right)^{(1)}
-\mu\frac{d}{dt}\left(\frac{\partial H_1}{\partial p_k}\right)^{(1)}
\right]\sin\theta_k\,d\theta_k
=
\Phi_k(a_1,\ldots,a_r,\psi_1,\ldots,\psi_r),
\tag{11}
]

[
\frac{d\psi_k}{dt}
=
-\frac{\dot{\varphi}}{2}
-\frac{1}{2\dot{\varphi}a_k}\sum_{i=1}^{r}
\left[
\frac{db_{ik}}{dt}a_i\cos(\psi_i-\psi_k)
-b_{ik}a_i\dot{\varphi}\sin(\psi_i-\psi_k)
\right]
+
]

[
+\frac{1}{2\pi\dot{\varphi}a_k}\int_{0}^{2\pi}
\left[
c_k\left(\frac{\partial H}{\partial q_k}\right)^{(1)}
-\mu\frac{d}{dt}\left(\frac{\partial H_1}{\partial p_k}\right)^{(1)}
\right]\cos\theta_k\,d\theta_k
=
\Psi_k(a_1,\ldots,a_r,\psi_1,\ldots,\psi_r).
]

Equating to zero the right-hand sides of equations (11) and averaging equations (2) for the “quasicyclic” coordinate, we obtain equations for determining

parameters of the stationary motion:

[
\Phi_k(a_1,\ldots,a_r,\psi_1,\ldots,\psi_r)=0,
]

[
\Psi(a_1,\ldots,a_r,\psi_1,\ldots,\psi_r)=0;
\tag{12}
]

[
\frac{d^2\varphi}{dt^2}
=
\frac{\mu}{2\pi}
\int_0^{2\pi}
\left[
-\,c_k\left(\frac{\partial H_1}{\partial\varphi}\right)^{(1)}
+
\frac{d}{dt}
\left(\frac{\partial H}{\partial p_{n+1}}\right)^{(1)}
\right]\,d\theta_k
=
0.
\tag{13}
]

Having found the values (a_k^0) and (\psi_k^0) from equations (12) and (13), and the expressions (\bar q_k^{(1)}, \bar p_k^{(1)}), which characterize the values of the generalized and momentum coordinates in stationary motion, we investigate its stability.

According to A. M. Lyapunov’s theory of stability ({}^{3}), a sufficient condition for the stability of the motion of a canonical system is the sign-definiteness of the quadratic form

[
H_2
=
\frac12
\sum_{i=1}^{r+1}
\sum_{k=1}^{r+1}
\left[
\left(\frac{\overline{\partial^2 H}}{\partial q_i\,\partial q_k}\right)\xi_i\xi_k
+
2\left(\frac{\overline{\partial^2 H}}{\partial q_i\,\partial p_k}\right)\xi_i\eta_k
+
\left(\frac{\overline{\partial^2 H}}{\partial p_i\,\partial p_k}\right)\eta_i\eta_k
\right]
\tag{14}
]

[
(i,k=1,2,\ldots,r+1),
]

formed from the lowest quadratic terms of the expansion of the Hamiltonian in a Taylor series in powers of the perturbations (\xi_i,\eta_i) of the generalized and momentum coordinates.

Received
18 XI 1957

CITED LITERATURE

({}^{1}) N. N. Bogolyubov, On certain statistical methods in mathematical physics, Kiev, 1945. ({}^{2}) N. M. Krylov, N. N. Bogolyubov, Introduction to nonlinear mechanics, Kiev, 1937. ({}^{3}) A. M. Lyapunov, The general problem of the stability of motion, 1950.