American Institute of Mathematical Sciences

Advanced
December  2010, 28(4): 1469-1504. doi: 10.3934/dcds.2010.28.1469

New insights into the classical mechanics of particle systems

Eliot Fried 1,

1. 

Department of Mechanical Engineering, McGill University, Montréal, Québec H3A 2K6, Canada

Received  October 2009 Revised  February 2010 Published  June 2010

The classical mechanics of particle systems is developed on the basis of the precept that, kinematics and the notion of force aside, power expenditures are of the foremost importance. The essential properties of forces between particles follow from requiring that the net power expended within any subsystem of particles be frame-indifferent. Furthermore, requiring that the net power expended on any subsystem of particles by external agencies be frame-indifferent yields force, moment, and power balances. These balances account for inertia but hold relative to any frame-of-reference, inertial or noninertial. Assuming that each particle possesses an interaction energy that embodies the extent to which it is attracted or repelled by other particles leads to the proposition of an interaction-energy inequality that serves as a purely mechanical statement of the second law of thermodynamics. In combination with the power balance, this inequality provides an avenue to ensure that constitutive equations do not permit a violation of thermodynamics. This inequality is used to develop the simplest class of constitutive equations that account for both conservative and dissipative particle-particle interactions.
Keywords: frame-indifference, Particle mechanics, balance laws, power, constitutive equations..
Mathematics Subject Classification: Primary: 70A05, 70F25; Secondary: 37J6.
Citation: Eliot Fried. New insights into the classical mechanics of particle systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469
[1]

Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271
[2]

Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477
[3]

Rinaldo M. Colombo, Graziano Guerra. Hyperbolic balance laws with a dissipative non local source. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1077-1090. doi: 10.3934/cpaa.2008.7.1077
[4]

Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure & Applied Analysis, 2017, 16 (2) : 629-644. doi: 10.3934/cpaa.2017031
[5]

Claude Vallée, Camelia Lerintiu, Danielle Fortuné, Kossi Atchonouglo, Jamal Chaoufi. Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1641-1649. doi: 10.3934/dcdss.2013.6.1641
[6]

B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems & Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405
[7]

Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018
[8]

Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
[9]

Stephan Gerster, Michael Herty. Discretized feedback control for systems of linearized hyperbolic balance laws. Mathematical Control & Related Fields, 2019, 9 (3) : 517-539. doi: 10.3934/mcrf.2019024
[10]

Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035
[11]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
[12]

Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147
[13]

Nicolas Fournier. Particle approximation of some Landau equations. Kinetic & Related Models, 2009, 2 (3) : 451-464. doi: 10.3934/krm.2009.2.451
[14]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[15]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357
[16]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933
[17]

Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001
[18]

Ning Lu, Ying Liu. Application of support vector machine model in wind power prediction based on particle swarm optimization. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1267-1276. doi: 10.3934/dcdss.2015.8.1267
[19]

Michael Böhm, Martin Höpker. A note on modelling with measures: Two-features balance equations. Mathematical Biosciences & Engineering, 2015, 12 (2) : 279-290. doi: 10.3934/mbe.2015.12.279
[20]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences