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Publication# Friction force: from mechanics to thermodynamics

Résumé

We study some mechanical problems in which a friction force is acting on a system. Using the fundamental concepts of state, time evolution and energy conservation, we explain how to extend Newtonian mechanics to thermodynamics. We arrive at the two laws of thermodynamics and then apply them to investigate the time evolution and heat transfer of some significant examples.

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Concepts associés

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Publications associées

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Concepts associés (13)

Publications associées (4)

Frottement

En physique, le frottement (ou friction) est une interaction qui s'oppose au mouvement relatif entre deux systèmes en contact. Le frottement peut être étudié au même titre que les autres types de forc

Opérateur d'évolution

En mécanique quantique, l'opérateur d'évolution est l'opérateur qui transforme l'état
quantique au temps t_0 en l'état quantique au temps t résultant
de l'évolution du systèm

Mécanique (science)

vignette|Gyroscope. Le gyroscope tient en équilibre sur la pointe fixe par le jeu des forces mécaniques (en particulier le couple de rappel) engendrées par la rotation rapide du disque au centre.
La m

Chargement

Chargement

Chargement

This is an overview of a program of stochastic deformation of the mathematical tools of classical mechanics, in the Lagrangian and Hamiltonian approaches. It can also be regarded as a stochastic version of Geometric Mechanics.The main idea is to construct well defined probability measures strongly inspired by Feynman Path integral method in Quantum Mechanics. In contrast with other approaches, this deformation preserves the invariance under time reversal of the underlying classical (conservative) dynamical systems.

2012The balance of pseudomomentum is discussed and applied to simple elasticity, ideal fluids, and the mechanics of inextensible rods and sheets. A general framework is presented in which the simultaneous variation of an action with respect to position, time, and material labels yields bulk balance laws and jump conditions for momentum, energy, and pseudomomentum. The example of simple elasticity of space-filling solids is treated at length. The pseudomomentum balance in ideal fluids is shown to imply conservation of vorticity, circulation, and helicity, and a mathematical similarity is noted between the evaluation of circulation along a material loop and the J-integral of fracture mechanics. Integration of the pseudomomentum balance, making use of a prescription for singular sources derived by analogy with the continuous form of the balance, directly provides the propulsive force driving passive reconfiguration or locomotion of confined, inhomogeneous elastic rods. The conserved angular momentum and pseudomomentum are identified in the classification of conical sheets with rotational inertia or bending energy.

2021The Monge problem [23], [27], as reformulated by Kantorovich [19], [20] is that of the transportation, at a minimum "cost", of a given mass distribu- tion from an initial to a
final position during a given time interval. It is an optimal transport problem [28, sects. 1, 2]. Following the fluid mechanical solution provided by Benamou and Brenier for quadratic cost functions [4] ,[28, sects. 5.4, 8.1] and, by analogy with the fi
xed end problem in Analytical Mechanics, Lagrangian formulations are needed to solve this boundary value problem in time. They are also needed to determine the Actions as time in- tegral of Lagrangians, that are measures of the "cost"of the transportations [4, proposition 1.1]. Four versions of explicit constructions of Lagrangians are proposed in section 3. They are associated to the Hamiltonians of perfect and self-interacting systems presented in section 2. These Hamiltonians are ex- pressed in function of pairs of the well known canonically conjugated Clebsch variables, namely mass densities and velocity potentials [14], [15]. The fi
rst version consists in the elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations de- rived from given Hamiltonians. The second version consists in the elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector valued applications such that their divergences give the mass densities. It turns out that, up to a sign factor, these vector
elds are canonically conjugated to Euler velocity
elds. The third version is a generalization in nD of Gelfand mass coordinate, a constant of the motion in 1 D [17], by the introduction of n-dimensional vector valued applications that enable to determine the mass densities as the determinant of their Jaco- bian matrices. Comparison of this set of mass coordinates with other sets of constants of the motion familiar in Fluid Dynamics is made in sub-section 3.3. Note that version two and three are identical for one-dimensional problems. The fourth version is based on the introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. As illustrations, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, respectively, with spherically symmetric boundary densities are given in section 4. However, and up to one exception given in the sub-section 3.3, calculations of the actions associated to these illustrations are not reported in this paper, nor the important analysis of the convexity-concavity properties of our Lagrangians. Lastly, and for the same models as those evoked above, a survey of past work concerning weak solutions of the Cauchy problem obeying the Hopf-Lax variational principle extended to negative time and having cor- related initial conditions is given in the Introduction as well as the derivation of the continuum fluid limit from many particle Hamiltonians.