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We study $\mathfrak{g}$-valued Yang-Mills fields on cylinders$Z(G/H)=\mathbb{R} \times G/H$, where G/H is a compact seven-dimensional cosetspace with $G_2$-structure, $\mathfrak{g}$ is the Lie algebra of G, and Z(G/H)inherits a Spin(7)-structure. After imposing a general G-invariance condition,Yang-Mills theory with torsion on Z(G/H) reduces to Newtonian mechanics of apoint particle moving in $\mathbb{R}^n$ under the influence of some quarticpotential and possibly additional constraints. The kinematics and dynamicsdepends on the chosen coset space. We consider in detail three coset spaceswith nearly parallel $G_2$-structure and four coset spaces withSU(3)-structure. For each case, we analyze the critical points of the potentialand present a range of finite-energy solutions. We also study ahigher-dimensional analog of the instanton equation. Its solutions yieldG-invariant Spin(7)-instanton configurations on Z(G/H), which are special casesof Yang-Mills configurations with torsion.

Yang-Mills solutions and Spin(7)-instantons on cylinders over coset spaces with $G_2$-structure