For an autonomous dynamical system of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11819_2025_7277_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation> differential equations each integral of motion allows for reduction of the order of equations by 1 and knowledge of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11819_2025_7277_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\((n-1)\)</EquationSource> </InlineEquation> integrals is necessary for the system to be integrated by quadratures. The amenability of Hamiltonian systems to being integrated by quadratures is characterised by the Liouville theorem where in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11819_2025_7277_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(2n\)</EquationSource> </InlineEquation>-dimensional phase space only <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11819_2025_7277_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation> integrals are sufficient as equations are generated by 1 function — the Hamiltonian.There are, however, large families of Newton-type differential equations for which knowledge of 2 or 1 integral is sufficient for recovering separability and integration by quadratures. The purpose of this paper is to discuss a tradeoff between the number of integrals and the special structure of autonomous, velocity-independent 2nd order Newton equations <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11819_2025_7277_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ddot{\boldsymbol{q}}=\boldsymbol{M}(\boldsymbol{q})\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11819_2025_7277_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(\boldsymbol{q}\in\mathbb{R}^{n}\)</EquationSource> </InlineEquation>, which allows for integration by quadratures.In particular, we review little-known results on quasipotential and triangular Newton equations to explain how it is possible that 2 or 1 integral is sufficient. The theory of these Newton equations provides new types of separation webs consisting of quadratic (but not orthogonal) surfaces.

When Knowledge of a Single Integral of Motion is Sufficient for Integration of Newton Equations \(\ddot{\boldsymbol{q}}=\boldsymbol{M}(\boldsymbol{q})\)

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发表日期：

2025年10月10日

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Stefan Rauch-Wojciechowski,
Maria Przybylska

摘要

For an autonomous dynamical system of \(n\) differential equations each integral of motion allows for reduction of the order of equations by 1 and knowledge of \((n-1)\) integrals is necessary for the system to be integrated by quadratures. The amenability of Hamiltonian systems to being integrated by quadratures is characterised by the Liouville theorem where in \(2n\) -dimensional phase space only \(n\) integrals are sufficient as equations are generated by 1 function — the Hamiltonian.

There are, however, large families of Newton-type differential equations for which knowledge of 2 or 1 integral is sufficient for recovering separability and integration by quadratures. The purpose of this paper is to discuss a tradeoff between the number of integrals and the special structure of autonomous, velocity-independent 2nd order Newton equations \(\ddot{\boldsymbol{q}}=\boldsymbol{M}(\boldsymbol{q})\) , \(\boldsymbol{q}\in\mathbb{R}^{n}\) , which allows for integration by quadratures.

In particular, we review little-known results on quasipotential and triangular Newton equations to explain how it is possible that 2 or 1 integral is sufficient. The theory of these Newton equations provides new types of separation webs consisting of quadratic (but not orthogonal) surfaces.