Abstract
Full Text
MECHANICS
Bl. DOLAPCHIEV
ON THE NIELSEN–TSENOV EQUATIONS AND THEIR APPLICATION TO NONHOLONOMIC SYSTEMS WITH NONLINEAR CONSTRAINTS
(Presented by Academician L. I. Sedov on May 3, 1966)
1. We start from the identity \((^1)\)
\[ \frac{d}{dt}\frac{\partial T}{\partial \dot q_\chi} = \frac{1}{n}\left( \frac{\partial \overset{(n)}{T}}{\partial \overset{(n)}{q}_\chi} - \frac{\partial T}{\partial q_\chi} \right) \quad (\chi=1,2,\ldots,k), \]
\[ \overset{(n)}{T}=\frac{d^nT}{dt^n},\qquad \overset{(n)}{q}_\chi=\frac{d^nq_\chi}{dt^n} \quad (n=1,2,\ldots), \tag{1} \]
where \(T=T(t,\ldots,q_\chi,\ldots,\dot q_\chi,\ldots)\) is the kinetic energy; \(q_\chi,\dot q_\chi\) are, respectively, the generalized coordinates and generalized velocities of the rheonomic-holonomic mechanical systems under consideration with \(k\) degrees of freedom, or from the generalized d’Alembert principle \((^2)\)
\[ \sum_{\nu=1}^{N} \left(\mathbf F_\nu-m_\nu\mathbf w_\nu\right)\cdot \delta \overset{(n)}{\mathbf r}_\nu=0, \]
\[ \delta t=0,\qquad \delta\mathbf r_\nu=\delta\dot{\mathbf r}_\nu= \delta\ddot{\mathbf r}_\nu=\cdots= \delta\overset{(n-1)}{\mathbf r}_\nu=0,\qquad \delta\overset{(n)}{\mathbf r}_\nu\ne0, \tag{2} \]
where \(\mathbf F_\nu\) is the resultant of the active forces acting on the \(\nu\)-th material point \(P_\nu\) with mass \(m_\nu\). We obtain \((^{1-3})\) the generalized Lagrange equation
\[ \frac{1}{n}\left( \frac{\partial \overset{(n)}{T}}{\partial \overset{(n)}{q}_\chi} -(n+1)\frac{\partial T}{\partial q_\chi} \right) = Q_\chi \quad (\chi=1,2,\ldots,k;\ n=1,2,\ldots), \tag{3} \]
where \(Q_\chi\) is the generalized force corresponding to \(q_\chi\).
For \(n=1\) we obtain the Nielsen equations \((^{4,5})\)
\[ \frac{\partial \dot T}{\partial \dot q_\chi} - 2\frac{\partial T}{\partial q_\chi} = Q_\chi \quad (\chi=1,2,\ldots,k); \tag{4} \]
for \(n=2\), the Tsenov equations \((^6)\)
\[ \frac{1}{2}\left( \frac{\partial \ddot T}{\partial \ddot q_\chi} - 3\frac{\partial T}{\partial q_\chi} \right) = Q_\chi \quad (\chi=1,2,\ldots,k). \tag{5} \]
Equations (4) also follow directly from the Jourdain principle \(((2),\, n=1)\), and equations (5) from the Gauss principle \(((2),\, n=2)\). It is shown directly that the Nielsen equations are obtained at once also from the Lagrange equations of the second kind
\[ \frac{d}{dt}\frac{\partial T}{\partial \dot q_\chi} - \frac{\partial T}{\partial q_\chi} = Q_n \quad (\chi=1,2,\ldots,k), \tag{6} \]
associated with the d’Alembert–Lagrange principle \(((2),\, n=0)\). For this purpose it is sufficient to differentiate \(\partial T/\partial \dot q_\chi\) \((^5)\).
2. Suppose that in the function \(T\) the generalized velocities \(\dot q_\chi\) are fixed, and in this case denote \(T\) by \(T_0\); then
\[ \frac{\partial T}{\partial q_\chi} = \frac{\partial \dot T_0}{\partial \dot q_\chi} = \frac{\partial \ddot T_0}{\partial \ddot q_\chi} = \cdots \qquad (\chi=1,2,\ldots,k). \tag{7} \]
Following Tzenov’s method \((^6)\), using the transformations
\[ R_1=T-2\dot T_0,\qquad \text{respectively}\qquad R_2=\frac12\bigl(\ddot T-3\ddot T_0\bigr), \tag{8} \]
we reduce equations (4) and (5) to the equations
\[ \frac{\partial R_1}{\partial \dot q_\chi}=Q_\chi,\qquad \text{respectively}\qquad \frac{\partial R_2}{\partial \ddot q_\chi}=Q_\chi \qquad (\chi=1,2,\ldots,k), \tag{9} \]
analogous to Appell’s equations
\[ \frac{\partial S}{\partial \ddot q_\chi}=Q_\chi \qquad (\chi=1,2,\ldots,k), \tag{10} \]
(\(S\) is the energy of acceleration), valid both for holonomic and for nonholonomic mechanical systems with linear constraints.
With the aid of Appell’s substitution (7)
\[ K_1=R_1-\sum_{\chi=1}^{k} Q_\chi \dot q_\chi, \qquad \text{respectively}\qquad K_2=R_2-\sum_{\chi=1}^{k} Q_\chi \ddot q_\chi \tag{11} \]
we obtain
\[ \frac{\partial Q_\chi}{\partial \dot q_\chi} = \frac{\partial Q_\chi}{\partial \ddot q_\chi} = 0 \qquad (\chi=1,2,\ldots,k), \]
\[ \frac{\partial K_1}{\partial \dot q_\chi}=0, \qquad \text{respectively}\qquad \frac{\partial K_2}{\partial \ddot q_\chi}=0, \tag{12} \]
coinciding with the equations to which the problem of finding the stationary values of the functions \(K_1\) and \(K_2\) is reduced.
3. Turning to nonholonomic systems with linear constraints with respect to the generalized velocities, respectively the generalized accelerations, we consider the expressions
\[ \sum_{\chi=1}^{k} a'_{\rho\chi}\dot q_\chi+a'_{\rho}=0, \qquad \text{respectively}\qquad \sum_{\chi=1}^{k} a''_{\rho\chi}\ddot q_\chi+a''_{\rho}=0 \tag{13} \]
\[ (\rho=1,2,\ldots,r<k), \]
where \(a'_{\rho\chi}\) and \(a'_{\rho}\) are functions only of time \(t\) and of the generalized coordinates \(q_\chi\), while \(a''_{\rho\chi}\) and \(a''_{\rho}\) are also functions of the generalized velocities \(\dot q_\chi\). Solving (13) with respect to \(\dot q_\rho\) and \(\ddot q_\rho\), we obtain the expressions
\[ \dot q_\rho=\sum_{\lambda=1}^{l} a'_{\rho\lambda}\dot q_\lambda+a'_{\rho}, \qquad \text{respectively}\qquad \ddot q_\rho=\sum_{\lambda=1}^{l} a''_{\rho\lambda}\ddot q_\lambda+a''_{\rho} \tag{14} \]
\[ (\rho=l+1,l+2,\ldots,l+r=k), \]
which we substitute into the functions \(K_1=K_1(\dot q_\chi)\), respectively \(K_2=K_2(\ddot q_\chi)\). Thus we obtain the functions \(K_1^*\), respectively \(K_2^*\), depending only on \(\dot q_\lambda\), respectively on \(\ddot q_\lambda\). The required equations of motion now take the form
\[ \frac{\partial K_1^*}{\partial \dot q_\lambda}=0, \qquad \frac{\partial K_2^*}{\partial \ddot q_\lambda}=0, \qquad (\lambda=1,2,\ldots,l), \tag{15} \]
or, by virtue of (14), the form (8)
\[ \frac{\partial K_1}{\partial \dot q_\lambda} + \sum_{\rho=l+1}^{l+r} \frac{\partial K_1}{\partial q_\rho}a'_{\rho\lambda} = 0, \qquad \frac{\partial K_2}{\partial \ddot q_\lambda} + \sum_{\rho=l+1}^{l+r} \frac{\partial K_2}{\partial q_n}a''_{\rho\lambda} = 0 \tag{16} \]
\[ (\lambda=1,2,\ldots,l). \]
- Suppose that the constraints are nonlinear with respect to the generalized velocities, respectively to the generalized accelerations, and have the form
\[ \Phi'_{\rho}(t,\ldots,q_{\chi},\ldots,\dot q_{\chi},\ldots)=0,\qquad \Phi''_{\rho}(t,\ldots,q_{\chi},\ldots,\dot q_{\chi},\ldots,\ddot q_{\chi}\ldots)=0 \tag{17} \]
\[ (\chi=1,2,\ldots,k;\ \rho=1,2,\ldots,r<2). \]
In this case one may proceed in two ways. In the first case we repeat the transformations carried out in § 3, for which it is necessary to differentiate (17) with respect to \(t\), namely
\[ \sum_{\chi=1}^{k}\frac{\partial\Phi'_{\rho}}{\partial\dot q_{\chi}}\ddot q_{\chi}+\cdots=0,\qquad \sum_{\chi=1}^{k}\frac{\partial\Phi''_{\rho}}{\partial\ddot q_{\chi}}\dddot q_{\chi}+\cdots=0 \quad(\rho=1,2,\ldots,r). \tag{18} \]
Equations (18) are already linear with respect to \(\ddot q_{\chi}\), respectively \(\dddot q_{\chi}\). Then one must again use Tzenov’s equations (5), respectively equations (3) for \(n=3\), i.e. \((9)\),
\[ \frac{1}{3}\left(\frac{\partial\ddot T}{\partial\ddot q_{\chi}}-4\frac{\partial\dot T}{\partial q_{\chi}}\right)=Q_{\chi} \quad(\chi=1,2,\ldots,k), \tag{19} \]
and the transformations
\[ R_3=\frac{1}{3}\left(\ddot T-4\ddot T_0\right),\qquad K_3=R_3-\sum_{\chi=1}^{k}Q_{\chi}\dddot q_{\chi} \quad(\partial Q_{\chi}/\partial\dddot q_{\chi}=0). \tag{20} \]
In the second case one must apply Tzenov’s method, but not only for second-kind constraints nonlinear (with respect to velocity), but also for first-kind constraints; in the latter case one has again to use not Tzenov’s equations, but Nielsen’s equations. For this, let us differentiate the constraints (17). Eliminating from
\[ \sum_{\chi=1}^{k}\frac{\partial\Phi'_{\rho}}{\partial\dot q_{\chi}}\,d\dot q_{\chi}+\cdots=0,\qquad \sum_{\chi=1}^{k}\frac{\partial\Phi''_{\rho}}{\partial\ddot q_{\chi}}\,d\ddot q_{\chi}+\cdots=0 \quad(\rho=1,2,\ldots,r), \tag{21} \]
\[ dK_1\equiv\sum_{\chi=1}^{k}\frac{\partial K_1}{\partial\dot q_{\chi}}\,d\dot q_{\chi}+\cdots=0,\qquad dK_2\equiv\sum_{\chi=1}^{k}\frac{\partial K_2}{\partial\ddot q_{\chi}}\,d\ddot q_{\chi}+\cdots=0 \tag{22} \]
respectively \(r\) differentials \(d\dot q_{\chi}\) and \(r\) differentials \(d\ddot q_{\chi}\), we finally obtain the equations
\[ \frac{\partial K_1^{*}}{\partial\dot q_{\lambda}}\,d\dot q_{\lambda} = \left( \frac{\partial K_1}{\partial\dot q_{\lambda}} + \sum_{\rho=l+1}^{l+r} \frac{\partial K_1}{\partial q_{\rho}}\gamma'_{\rho\lambda} \right)d\dot q_{\lambda}=0 \quad(\lambda=1,2,\ldots,l), \tag{23} \]
\[ \frac{\partial K_2^{*}}{\partial\ddot q_{\lambda}}\,d\ddot q_{\lambda} = \left( \frac{\partial K_2}{\partial\ddot q_{\lambda}} + \sum_{\rho=l+1}^{l+r} \frac{\partial K_2}{\partial q_{\rho}}\gamma''_{\rho\lambda} \right)d\ddot q_{\lambda}=0 \quad(\lambda=1,2,\ldots,l). \tag{24} \]
Consequently, for the constraints encountered in practice, linear and nonlinear with respect to the generalized velocities \(\dot q_{\chi}\), the equations of motion have the form of Nielsen’s equations of motion (4), derived from Jourdain’s principle, and for constraints linear and nonlinear with respect to the generalized accelerations \(\ddot q_{\chi}\), the form of Tzenov’s equations (5), derived from Gauss’s principle and (19).
Sofia, Bulgaria
Received
5 II 1966
CITED LITERATURE
\(^{1}\) Bl. Dolaptschiew, Zs. angew. Math. u. Phys., 17 (1966).
\(^{2}\) Bl. Dolaptschiew, C. R., 262 (1966).
\(^{3}\) D. Mangeron, S. Deleanu, C. R. Bulg. Acad. Sci. (1963).
\(^{4}\) J. Nielsen, Elementare Mechanik, 1935.
\(^{5}\) Bl. Dolaptschiew, Zs. angew. Math. u. Mech., 47 (1966).
\(^{6}\) I. Tzenov, DAN, 89, No. 1 (1953).
\(^{7}\) P. Appell, Mécanique rationelle, 1909.
\(^{8}\) I. Tzenov, Dokl. Bulgarian Acad. Sci. (1965).
\(^{9}\) Bl. Dolaptschiew, Izv. Bulgarian Academy of Sciences, in press.