Abstract Generated abstract
The paper develops an averaging procedure for canonical Hamiltonian systems containing an angular coordinate whose conjugate momentum varies only by terms proportional to a small parameter, described as a quasi-cyclic coordinate. Using a Krylov-Bogolyubov style first approximation, the generalized coordinates are represented by slowly varying amplitudes and phases relative to the angular variable, and the canonical equations are transformed into averaged equations for these slow variables. The resulting system gives conditions for stationary motion through algebraic averaged equations, together with an averaged equation for the angular acceleration. The paper also indicates how the stability of such stationary motions may be examined by applying Lyapunov’s criterion to the quadratic part of the Hamiltonian expanded about the stationary solution.
Full Text
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.