| MadSci Network: Physics |
Jenna,
Sophia Kovalevskaya was a very talented mathematician; perhaps the best female mathematician to ever live. Her work was on the physics of rotating bodies.
There are three cases of rigid body rotational motion that are solvable in closed form. The three cases are the Euler, Lagrange, and Kovalevsky solutions.
Lagrange studied the motion of a rigid body with the center of mass of that body fixed on the axis of rotation of the body at the center of the body. For this system, it is possible to derive the equations of motion using the symmetry of the problem to your advantage. With the Lagrangian formalism, it is possible to directly integrate the energy relations and end up with a complete set of equations of motion for the system. This model essentially gives you the motion of a typical heavy, symmetric toy top.
Euler studied a more general problem, where the center of mass of the system was not at the center of the body, but still on the axis of rotation. Euler invented a technique for general solution of the rotating symmetric body problems. Using the Euler equations of motion, it is possible to use symmetry again to solve the problem through direct integration. The Euler approach is quite general for symmetric rotating bodies, but some cases are not integrable in closed form, and gives rise to elliptic integrals. These are also sometimes integrable, but not easily.
Kovalevskaya studied the non-symmetric rotating case. Kovalevskaya showed that for some classes of rigid rotating systems, the solution is attainable by direct integration, but again, not easily. Specifically, Kovalevskaya studied the case where the rotating object has the same moment of intertia I in two axes, but the third axis has a moment of I/2. This case turns out to be solvable directly, and the method is general to other similar cases. While the details are rather complex, this method of solution gives a stable model for Saturn's rings. For this work, she was given the Bordin medal in 1888.
For more information, check out the following links:
Good luck!
-Fred
Try the links in the MadSci Library for more information on Physics.