The indefinite metric of R. Mrugala and the geometry of the thermodynamical phase space

Details The indefinite metric of R. Mrugala and the geometry of the thermodynamical phase space The indefinite metric of R. Mrugala and the geometry of the thermodynamical phase space

2005-09-12

arxiv.org/abs/math/0509267v1

Mathematics

Differential Geometry

Scientific paper

We study the indefinite metric $G$ in the contact phase space $(P,\theta)$ of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of $P$ - constitutive surfaces of different homogeneous thermodynamical systems. We established an isomorphism of the space $(P,\theta ,G)$ with the Heisenberg Lie group $H_{n}$ endowed with the right invariant contact structure and the right invariant indefinite metric. The lift $\tG$ of the metric $G$ to the symplectization $\tP$ of contact space $(P,\theta)$, its curvature properties, and its Killing vector fields are studied. Finally we introduce the "hyperbolic projectivization" of the space $(\tP,{\tilde \theta}, \tG)$ that can be considered as the natural {\bf compactification} of the thermodynamical phase space $(P,\theta, G).$

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