# Geometric approach to Hamiltonian dynamics and statistical mechanics

@article{Casetti2000GeometricAT, title={Geometric approach to Hamiltonian dynamics and statistical mechanics}, author={Lapo Casetti and Marco Pettini and E. G. D. Cohen Infm and Dipartimento di Fisica and P. D. Torino and Italy. and Osservatorio Astrofisico di Arcetri and Firenze and The Rockefeller University and New York and Usa}, journal={Physics Reports}, year={2000}, volume={337}, pages={237-341} }

## 185 Citations

Riemannian geometry of Hamiltonian chaos: hints for a general theory.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2008

We aim at assessing the validity limits of some simplifying hypotheses that, within a Riemmannian geometric framework, have provided an explanation of the origin of Hamiltonian chaos and have made it…

Introduction to a Geometrical Theory of Fluid Flows and Dynamical Systems

- Mathematics
- 2001

Dynamical systems can be formulated on the basis of the Riemannian geometry and Lie algebra, provided that a dynamical system has a group symmtery, namely it is invariant under a group…

Coherent Riemannian-geometric description of Hamiltonian order and chaos with Jacobi metric.

- PhysicsChaos
- 2019

It is shown that there is no inconsistency in the description of stability/instability properties of Newtonian dynamics and that the observed instabilities in the case of integrable systems using the Jacobi metric are artifacts.

The Hamiltonian Mean Field Model: From Dynamics to Statistical Mechanics and Back

- Physics
- 2002

The thermodynamics and the dynamics of particle systems with infiniterange coupling display several unusual and new features with respect to systems with short-range interactions. The Hamiltonian…

Geometric approach to chaos in the classical dynamics of Abelian lattice gauge theory

- Physics
- 1998

A Riemannian geometrization of dynamics is used to study chaoticity in the classical Hamiltonian dynamics of a U(1) lattice gauge theory. This approach allows one to obtain analytical estimates of…

The Geometric Theory of Phase Transitions

- Mathematics
- 2022

. We develop a geometric theory of phase transitions (PTs) for Hamiltonian systems in the microcanonical ensemble. This theory allows to reformulate Bachmann’s classiﬁcation of PTs for ﬁnite-size…

The Possibility of Introducing of Metric Structure in Vortex Hydrodynamic Systems

- Physics
- 2018

Geometrization of the description of vortex hydrodynamic systems can be made on the basis of the introduction of the Monge –Clebsch potentials, which leads to the Hamiltonian form of the original…

Controlling effect of geometrically defined local structural changes on chaotic Hamiltonian systems.

- Physics, MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

It is shown here that the criterion for instability is local in coordinate space and can be constructively used to modify locally the potential of a chaotic Hamiltonian model in such a way that stable motion is achieved.

## References

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The relationship between dynamic instability and curvature properties of the configuration space manifold is the main concern of this paper and the possibility of misleading conclusions that might be drawn from the geometric approach is warned.

Geometric approach to chaos in the classical dynamics of Abelian lattice gauge theory

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A Riemannian geometrization of dynamics is used to study chaoticity in the classical Hamiltonian dynamics of a U(1) lattice gauge theory. This approach allows one to obtain analytical estimates of…

Geometry of dynamics and phase transitions in classical lattice phi^4 theories

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We perform a microcanonical study of classical lattice phi^4 field models in 3 dimensions with O(n) symmetries. The Hamiltonian flows associated to these systems that undergo a second order phase…

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Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the…