Interrelating Lagrangians, Hamiltonians, and Autonomous Systems of Differential Equations, using Computer Algebra

In order to facilitate education in upper-level physics courses a multi-purpose user-friendly program has been written in wxMaxima (software freely available) to incorporate the two functions: Lagrangians, quadratic or linear in the velocity components, and Hamiltonians. Here, as is often the case, neither function will depend explicitly on the time, t. The program also provides for the use of either independent or related (to the Hamiltonian) autonomous systems of differential equations. The user can input a Lagrangian, see the corresponding Euler-Lagrange equations, generalized momenta and (if it exists) the Hamiltonian which is the total energy. In addition, the user can obtain Hamilton's equations beginning with a Hamiltonian and generate solutions to those equations, in terms of algebraic expansions. For systems of autonomous differential equations arising independently of the Lagrangian or Hamiltonian formalisms, the user can find approximate (and in some cases, complete) computer-algebraic solutions near a specified value of, and as expansions to arbitrary order in t; and, moreover, in a form that can facilitate graphing. This paper has significant pedagogical value, for both instructors and students, in a number of upper-level courses in physics and is also useful for mathematics and engineering. Recent applications of the program are shown in the areas of classical mechanics and electromagnetism; and there is a description of its use in neural networks.