The VariationalCalculus Package
The new VariationalCalculus package in Maple 8 provides routines for solving problems in the calculus of variations, which studies nature's most "efficient" curves and surfaces. Examples include: find the shortest path between two points on a 3-D surface, shape a ramp between two heights such that a ball rolling down it reaches the bottom in minimum time, and find the shape of a soap film having minimum surface area spanning a given wire frame.
Such problems can often be solved with the Euler-Lagrange equation, which generalizes the Lagrange Multiplier Theorem. The Euler-Lagrange equation is easy to write down in general but notoriously difficult to write down and solve for most practical problems. The VariationalCalculus package automates the construction and analysis of the Euler-Lagrange equation.
