Stefan Rauch-Wojciechowski
Publications:
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Rauch-Wojciechowski S., Przybylska M.
When Knowledge of a Single Integral of Motion is Sufficient for Integration of Newton Equations $\ddot{\boldsymbol q}=\boldsymbol{M}(\boldsymbol{q})$
2025, vol. 30, no. 5, pp. 866-885
Abstract
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.
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