Consider a mass m moving in two dimensions with potential energy u (x, y) = 2 1 kr 2 , wherer 2 = x 2 + y 2 . write down the lagrangian, using coordinates x and y, and find the two lagrangeequations of motion. describe their solutions. [this is the potential energy of an ion in an "iontrap", which can be used to study the properties of individual atomic ions.]

Simon is writing a story about an astronaut whose spacecraft has been boarded by space pirates. the astronaut has her lucky penny in her hand behind her back as the space pirates break into the control room. she has just locked the controls so that the ship is accelerating in the direction of the control room’s ceiling. simon wants the astronaut to use the penny to hit a button on the control panel to turn off the lights and escape. the button is located a short distance behind and below the astronaut’s hands. how should simon use the theory of relativity to describe what the astronaut must do in order to hit the button?