Concepts from differential geometry appear in various areas of physics, most notably, in the general theory of relativity. However, even a discipline such as classical mechanics admits an intriguing geometric structure. As is well known, the Hamiltonian formulation of classical mechanics rests on the notion of the phase space with the dynamics being described by the phase trajectories for a given Hamiltonian. One should note, however, that there are several universal features. For example, a phase space as encountered in classical mechanics is endowed with a Poisson bracket which is independent of the choice of any specific Hamiltonian -- that is, it appears from the intrinsic structure of the phase space and is not related to the specific physical problem that one wishes to study. Another feature is the Liouville's theorem which advocates for the preservation of the phase-space volume along any Hamiltonian flow -- once again, true for any system with any Hamiltonian indicating that this feature is also a universal one. Such universal features that are present for any Hamiltonian system indicates that phases spaces could possibly be studied under a single umbrella in a system-independent manner. This universal structure is termed the symplectic structure and the Poisson bracket or the Liouville's theorem are just simple consequences of it. Moreover, given any smooth function H on the phase space, the symplectic structure allows one to associate with its differential dH, a unique vector field X whose integral curves are the familiar phase trajectories. Equipped now with this geometric notion, one can ask two questions -- (a) are there generalizations of the symplectic geometry? (b) where else in physics does symplectic geometry or its cousins play a role?
The answers to both the questions are in the affirmative. There are indeed non-trivial generalizations of symplectic geometry, (i) where the generalization is based on the symplectic structure, namely, geometries dubbed cosymplectic, contact, cocontanct, etc., (ii) where the generalization is based on the Poisson structure, namely, structures on smooth manifolds dubbed as Poisson, Jacobi, Nambu-Poisson, Nambu-Jacobi, etc. All such generalizations have found interesting implications in physical problems. For example, cosymplectic geometry describes time-dependent Hamiltonian mechanics, contact geometry describes dissipative systems as well as thermodynamic systems, cocontact geometry describes dissipative systems with explicitly-time-dependent Hamiltonians, Poisson structures can describe conservative dynamics on odd-dimensional (also even) phase spaces, Jacobi structures generalize Poisson structures and naturally describe the local Lie brackets on contact manifolds, Nambu-Poisson structures describe the dynamics of completely-integrable systems, Nambu-Jacobi structures generalize the elements of both Jacobi and Nambu-Poisson structures, and so on. More modern developments include the use of such dynamical frameworks in convex optimization and neural networks. Therefore, this rich area of mathematical physics deserves special attention in order to explore connections between these intriguing geometries on one hand, and on the other hand, finding new physical scenarios where they appear naturally.
Much of my work in this fascinating area has been focused on contact geometry. I had been involved in investigations regarding the appearance of contact geometry in thermodynamics, particularly the thermodynamics of black holes [1] as well as nanoscale (quantum) systems in nonequilibrium steady states [2]. Following this, a Hamilton-Jacobi framework for thermodynamic transformations was put forward in [3] and a symplectic approach (distinct from the contact approach) towards equilibrium thermodynamics was presented in [4].
From the mechanics point of view, contact geometry can describe simple dissipative systems for which not only is the Hamiltonian not a conserved quantity, but the phase-space volume as defined in a standard manner is also not conserved. For such systems, a generalized virial theorem was developed in [5] and for dissipative systems including those described by contact geometry, a generalized notion of Liouville's theorem was presented in [6].
My current work includes a recent study on the Hamiltonian aspects of three-dimensional non-conservative systems (including chaotic systems) [7]. At present, I am also seeking to explore the Hamiltonian aspects of geometric optics.
Collaborators:
1) Chandrasekhar Bhamidipati (IIT-BBS)
2) Anindya Ghose-Choudhury (Diamond Harbour Women's University)
3) Partha Guha (Khalifa University)
4) E. Harikumar (University of Hyderabad)
5) Malay Bandyopadhyay (IIT-BBS)
Related publications:
[1] A. Ghosh and C. Bhamidipati, Phys. Rev. D 100, 126020 (2019) [arXiv].
[2] A. Ghosh, M. Bandyopadhyay, and C. Bhamidipati, Physica A 585, 126402 (2021) [arXiv].
[3] A. Ghosh, Pramana 97, 49 (2023) [arXiv].
[4] A. Ghosh and E. Harikumar, arXiv:2410.04622.
[5] A. Ghosh, J. Phys. A 56, 235205 (2023) [arXiv].
[6] A. Ghosh, Phys. Scr. 100, 035220 (2025) [arXiv].
[7] A. Ghosh, A. Ghose-Choudhury, and P. Guha, arXiv:2504.10729.