The equivalence problem and canonical forms for quadratic Lagrangians

The general equivalence and canonical form problems for quadratic variational problems under arbitrary linear changes of variable are formulated, and the role of classical invariant theory in their general solution is made clear. A complete solution to both problems for planar, first order quadratic variational problems is provided, including a complete list of canonical forms for the Lagrangians and corresponding Euler-Lagrange equations. Algorithmic procedures for determining the equivalence class and the explicit canonical form of a given Lagrangian are provided. Applications to planar anisotropic elasticity are indicated.