# Infinitesimal Legendre symmetry in the Geometrothermodynamics programme

## Abstract

The work within the Geometrothermodynamics programme rests upon the metric structure for the thermodynamic phase-space. Such structure exhibits discrete Legendre symmetry. In this work, we study the class of metrics which are invariant along the infinitesimal generators of Legendre transformations. We solve the Legendre-Killing equation for a K-contact general metric. We consider the case with two thermodynamic degrees of freedom, i.e., when the dimension of the thermodynamic phase-space is five. For the generic form of contact metrics, the solution of the Legendre-Killing system is unique, with the sole restriction that the only independent metric function – Ω – should be dragged along the orbits of the Legendre generator. We revisit the ideal gas in the light of this class of metrics. Imposing the vanishing of the scalar curvature for this system results in a further differential equation for the metric function Ω which is not compatible with the Legendre invariance constraint. This result does not allow us to use Quevedo's interpretation of the curvature scalar as a measure of thermodynamic interaction for this particular class.

- Authors:

- Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.P. 70-543, 04510 México D.F., México (Mexico)
- (Mexico)

- Publication Date:

- OSTI Identifier:
- 22306101

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DEGREES OF FREEDOM; DIFFERENTIAL EQUATIONS; MATHEMATICAL SOLUTIONS; METRICS; PHASE SPACE; SYMMETRY; THERMODYNAMICS

### Citation Formats

```
García-Peláez, D., E-mail: dgarciap@up.edu.mx, Universidad Panamericana, Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac, 01050 México D.F., México, and López-Monsalvo, C. S., E-mail: cesar.slm@correo.nucleares.unam.mx.
```*Infinitesimal Legendre symmetry in the Geometrothermodynamics programme*. United States: N. p., 2014.
Web. doi:10.1063/1.4891921.

```
García-Peláez, D., E-mail: dgarciap@up.edu.mx, Universidad Panamericana, Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac, 01050 México D.F., México, & López-Monsalvo, C. S., E-mail: cesar.slm@correo.nucleares.unam.mx.
```*Infinitesimal Legendre symmetry in the Geometrothermodynamics programme*. United States. doi:10.1063/1.4891921.

```
García-Peláez, D., E-mail: dgarciap@up.edu.mx, Universidad Panamericana, Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac, 01050 México D.F., México, and López-Monsalvo, C. S., E-mail: cesar.slm@correo.nucleares.unam.mx. Fri .
"Infinitesimal Legendre symmetry in the Geometrothermodynamics programme". United States.
doi:10.1063/1.4891921.
```

```
@article{osti_22306101,
```

title = {Infinitesimal Legendre symmetry in the Geometrothermodynamics programme},

author = {García-Peláez, D., E-mail: dgarciap@up.edu.mx and Universidad Panamericana, Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac, 01050 México D.F., México and López-Monsalvo, C. S., E-mail: cesar.slm@correo.nucleares.unam.mx},

abstractNote = {The work within the Geometrothermodynamics programme rests upon the metric structure for the thermodynamic phase-space. Such structure exhibits discrete Legendre symmetry. In this work, we study the class of metrics which are invariant along the infinitesimal generators of Legendre transformations. We solve the Legendre-Killing equation for a K-contact general metric. We consider the case with two thermodynamic degrees of freedom, i.e., when the dimension of the thermodynamic phase-space is five. For the generic form of contact metrics, the solution of the Legendre-Killing system is unique, with the sole restriction that the only independent metric function – Ω – should be dragged along the orbits of the Legendre generator. We revisit the ideal gas in the light of this class of metrics. Imposing the vanishing of the scalar curvature for this system results in a further differential equation for the metric function Ω which is not compatible with the Legendre invariance constraint. This result does not allow us to use Quevedo's interpretation of the curvature scalar as a measure of thermodynamic interaction for this particular class.},

doi = {10.1063/1.4891921},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 15 00:00:00 EDT 2014},

month = {Fri Aug 15 00:00:00 EDT 2014}

}