The development of analytical mechanics involves many aspects of theory and application. This paper summarizes the recent progress of analytical mechanics in three problems on the interdisciplinarity between analytical mechanics and mathematics. The first is to study the integration problem of equations of analytical dynamics by using some results of Lie groups and Lie algebras. The second is to apply the classical and modern integration methods of analytical mechanics to the integration problem of general differential equations. The third is to transform the equations of analytical dynamics into the equations of gradient system under certain conditions and then discuss the dynamical behaviors of the mechanical system by using the properties of gradient system.