The numerical simulation of fluid dynamics problems plays an important role in the studies of laser inertial confinement fusion (ICF) and magnetic confinement fusion (MCF). Due to the complexity of physical processes and the large deformations of flow field, the numerical simulation of these problems has considerable difficulty. Taking the advantages of the discontinuous Galerkin (DG) method and the Lagrangian scheme, a second-order Lagrangian type scheme for solving the compressible Euler equations is developed on unstructured triangular meshes and implemented by the Runge-Kutta (RK) DG method in this paper. The solver of node velocity in the scheme has good adaptability for many problems. Without considering the material derivatives of basis functions and the Jacobian matrix associated with the map between Lagrangian space and Eulerian space, our scheme is relatively succinct. A HWENO reconstruction is used to eliminate the false oscillations whose stencils involve only the von Neumann neighborhood so that the scheme keeps compact within the DG method. Finally, some numerical examples are presented to illustrate the accuracy, resolution, and robustness of our scheme.