A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory
Katarzyna Grabowska, Janusz Grabowski
Abstract
Abstract We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> -principal bundle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>τ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>M</mml:mi> </mml:math> , which is equipped with a homogeneous symplectic form ω . In other words, our understanding of contact geometry is that it is not an ‘odd-dimensional cousin’ of symplectic geometry but rather a part of the latter, namely ‘homogeneous symplectic geometry’. This understanding of contact structures is much simpler than the traditional one and very effective in applications, reducing the contact Hamiltonian formalism to the standard symplectic picture. We develop in this language contact Hamiltonian mechanics in the autonomous, as well as the time-dependent case, and the corresponding Hamilton–Jacobi theory. Fundamental examples are based on canonical contact structures on the first jet bundles <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="sans-serif">J</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mi>L</mml:mi> </mml:math> of sections of line bundles L , which play in contact geometry a fundamental rôle, similar to that played by cotangent bundles in symplectic geometry.