Summary: We present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects
\[ \begin{aligned} & u^{\Delta^2}(t)+A(\sigma(t))u(\sigma(t))+\nabla F(\sigma(t),u(\sigma(t)))=0, \quad \Delta \text{-a.e.} \;t\in[0,T]^\kappa_\mathbb T; \\ & u(0)-u(T)=u^\Delta(0)-u^\Delta(T)=0,\\ & (u^i)^\Delta(t^+_j)-(u^i)^\Delta(t^-_j)=I_{ij}(u^i(t_j)), \;i=1,2,\dots,N,\,j=1,2,\dots,p, \end{aligned} \]
where \(t_0=0<t_1<t_2<\dots<t_p<t_{p+1}=T\), \(t_j\in [0,T]_\mathbb T \;( j=0,1,2,\dots,p+1)\), \(u(t)=(u^1(t),u^2(t),\dots,u^N(t))\in \mathbb R^N\), \(A(t)=[d_{lm}(t)]\) is a symmetric \(N\times N\) matrix-valued function defined on \([0,T]_\mathbb T\) with \(d_{lm}\in L^\infty([0,T]_\mathbb T, \mathbb R)\) for all \(l,m=1,2,\dots, N\), \(I_{ij}:\mathbb R\to \mathbb R \;(i=1,2,\dots,N,\, j=1,2,\dots,p)\) are continuous and \(F:[0,T]_\mathbb T\times \mathbb R^N\to \mathbb R\). Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.