Projective Quadrics, Poles, Polars, and Legendre Transformations
In a study of the solution of certain differential equations, Legendre discovered a transformation with remarkable properties, which has since been found to have applications in analysis, mechanics, and thermodynamics. (See Courant & Hilbert (1962).) In classical mechanics, for example, the Legendre transformation (LT) is used to make the transition from the Lagrangian to the Hamiltonian formalism. In modern texts, this transition is usually given a differential geometric interpretation: Lagrangian mechanics takes place on the tangent bundle TM of the configuration space manifold M, Hamiltonian mechanics on the cotangent bundle T* M. However, there is another geometric context in which the LT may be viewed (regardless of the branch of physics in which it is applied): the LT is in fact a particular case of a construction in projective geometry. The symmetry, or “duality”, between the new and old variables in the transformation (reflected in the duality of TM and T* M in the differential geometric formalism) may be expressed as the projective duality between so-called “poles” and “polars” in the theory of projective quadrics.
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