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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2005
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0509267 |
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| _version_ | 1866916172446826496 |
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| author | Preston, Serge Vargo, James |
| author_facet | Preston, Serge Vargo, James |
| contents | We study the indefinite metric $G$ in the contact phase space $(P,θ)$ 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 $(P,θ,G)$ 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 $(P,θ)$, its curvature properties, and its Killing vector fields are studied. Finally we introduce the "hyperbolic projectivization" of the space $(\tP,{\tilde θ}, \tG)$ that can be considered as the natural {\bf compactification} of the thermodynamical phase space $(P,θ, G).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0509267 |
| institution | arXiv |
| publishDate | 2005 |
| record_format | arxiv |
| spellingShingle | The indefinite metric of R. Mrugala and the geometry of the thermodynamical phase space Preston, Serge Vargo, James Differential Geometry 53C50 (Primary), 53D10, 74A15 (Secondary) We study the indefinite metric $G$ in the contact phase space $(P,θ)$ 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 $(P,θ,G)$ 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 $(P,θ)$, its curvature properties, and its Killing vector fields are studied. Finally we introduce the "hyperbolic projectivization" of the space $(\tP,{\tilde θ}, \tG)$ that can be considered as the natural {\bf compactification} of the thermodynamical phase space $(P,θ, G).$ |
| title | The indefinite metric of R. Mrugala and the geometry of the thermodynamical phase space |
| topic | Differential Geometry 53C50 (Primary), 53D10, 74A15 (Secondary) |
| url | https://arxiv.org/abs/math/0509267 |