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VIDEO LIBRARY |
International youth conference "Geometry & Control"
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Differential Invariants of Feedback Transformations for Quasi-Harmonic Oscillation Equations Dmitry Gritsenkoa, Oleg Kiriukhinb a Lomonosov Moscow State University, Moscow, Russia b University of Chicago Booth School of Business, Chicago, Illinois, USA |
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Abstract: The classification problem for a control-parameter-dependent second-order differential equations is considered. The algebra of the differential invariants with respect to Lie pseudo-group of feedback transformations is calculated. The equivalence problem for a control-parameter-dependent quasi-harmonic oscillation equation is solved. Some canonical forms of this equation are constructed. Consider the problems of equivalence and classification for the differential equation: $$ \tag{1} \frac{d^{2}y}{d x^{2}}+f(y,u)=0, $$ with respect to the feedback transformations [1]: $$ \tag{2} \varphi\colon (x,y,u)\longmapsto (X(x,y),Y(x,y),U(u)), $$ where the function Definition. Operator $$ \tag{3} \nabla = M\frac{d}{dy} + N\frac{d}{du} $$ is called Theorem. Differential operators \begin{align} \tag{4} \nabla_1 = \frac{z}{z_{y}} \frac{d}{dy}, \\ \tag{5} \nabla_2 = \frac{z}{z_{u}} \frac{d}{du} \end{align} are Theorem. Functions $$ J_{21} =\frac{z_{yy}z}{z_y^2},\quad J_{22} =\frac{z_{yu}z}{z_{y}z_u} $$ form a complete set of the basic second-order differential invariants, i.e.any other second-order differential invariants are the functions of Theorem. Quasi-harmonic oscillation equation differential invariants algebra is generated by second-order differential invariants Let us call an equation $$ dJ_{21}(f)\wedge dJ_{22}(f)\neq 0. $$ Here Theorem. Suppose that the functions Language: English References
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