Using the rigid magnetic vortex model, we develop a substantially modified Landau theory approach for analytically studying phase transitions between different spin arrangements in circular submicron magnetic dots subject to an in-plane externally-applied magnetic field. We introduce a novel order parameter: the inverse distance between the center of the circular dot and the vortex core. This order parameter is suitable for describing closed spin configurations such as curved or bent-spin structures and magnetic vortices. Depending on the radius and thickness of the dot as well as the exchange coupling, there are five different regimes for the magnetization reversal process when decreasing the in-plane magnetic field. The magnetization-reversal regimes obtained here cover practically all possible magnetization reversal processes. Moreover, we have derived the change of the dynamical response of the spins near the phase transitions and obtained a “critical slowing down” at the second order phase transition from the high-field parallel-spin state to the curved (C-shaped) spin phase. We predict a transition between the vortex and the parallel-spin state by quickly changing the magnetic field—providing the possibility to control the magnetic state of dots by changing either the value of the external magnetic field and/or its sweep rate. We study an illuminating mechanical analog (buckling instability) of the transition between the parallel-spin state and the curved spin state (i.e., a magnetic buckling transition). In analogy to the magnetic-disk case, we also develop a modified Landau theory for studying mechanical buckling instabilities of a compressed elastic rod embedded in an elastic medium. We show that the transition to a buckled state can be either first or second order depending on the ratio of the elasticity of the rod and the elasticity of the external medium. We derive the critical slowing down for the second-order mechanical buckling transition.