  By studying scattering Lie groups and their associated Lie algebras, we
introduce a new method for the characterisation of collision invariants for
physical scattering families associated to smooth, convex hard particles in the
particular case that the collision invariant is of class $\mathscr{C}^{1}$.
This work extends that of Saint-Raymond and Wilkinson (Communications on Pure
and Applied Mathematics (2018), 71(8), pp. 1494-1534), in which the authors
characterise collision invariants only in the case of the so-called canonical
physical scattering family. Indeed, our method extends to the case of
non-canonical physical scattering, whose existence was reported in Wilkinson
(Archive for Rational Mechanics and Analysis (2020), 235(3), pp. 2055-2083).
Moreover, our new method improves upon the work in Saint-Raymond and Wilkinson
as we place no symmetry hypotheses on the underlying non-spherical particles
which make up the gas under consideration. The techniques established in this
paper also yield a new proof of the result of Boltzmann for collision
invariants of class $\mathscr{C}^{1}$ in the classical case of hard spheres.
