where T and U are the total kinetic energy and total potential energy of the system, respectively; q
is a generalised position (such as x, y, or r), and; p is a generalised momentum. Using this notation, Hamilton’s equations of motion are
Notice that q with a ‘dot’ represents a generalised velocity and p with a ‘dot’ represents a generalised force.
For those of us who are Earthbound, it’s easy to think of liquids and gases as being the most common fluids. But plasma–the fourth state of matter–is a fluid as well. Plasmas are essentially ionized gases, which, thanks to their freely flowing electrons, are electrically conductive and sensitive to magnetic fields. Their motions are described by a combination of the Navier-Stokes equations–the usual equations of motion for a fluid–and Maxwell’s equations–the equations governing electricity and magnetism. Studies of plasma motion often fall under the subject of magnetohydrodynamics and can include topics like planetary auroras, sunspots, and solar flares. (Video credit: SciShow)