scholarly journals On internal constraints in continuum mechanics

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
K. R. Rajagopal ◽  
Giuseppe Saccomandi
Keyword(s):  
Continuum Mechanics ◽  
Classical Mechanics ◽  
Dissipative Systems ◽  
Classical Particle ◽  
Response Functions ◽  
Internal Constraints ◽  
Material Response ◽  
Particle Mechanics

In classical particle mechanics, it is well understood that while working with nonholonomic and nonideal constraints, one cannot expect that the constraint be workless. It is curious that in continuum mechanics, however, the implications of the result in classical mechanics have not been clearly understood. In this paper, we show that in dealing with the response of dissipative systems, one cannot require that constraints do no work or ignore the fact that the material response functions depend on the constraint reaction. An example of this is the viscosity of a fluid depending on the pressure.

scholarly journals Axiomatic Foundations of Classical Particle Mechanics

1953 ◽  
Vol 2 (2) ◽  
pp. 253-272 ◽  
Author(s):  
J. McKinsey ◽  
A. Sugar ◽  
Patrick Suppes
Keyword(s):  
Classical Particle ◽  
Particle Mechanics

Hamiltonians in Relativistic Classical Particle Mechanics

1968 ◽  
Vol 166 (5) ◽  
pp. 1308-1316 ◽  
Author(s):  
Thomas F. Jordan
Keyword(s):  
Classical Particle ◽  
Particle Mechanics

scholarly journals The adiabatic invariance of the quantum integrals

1925 ◽  
Vol 107 (744) ◽  
pp. 725-734 ◽  
Keyword(s):  
Dynamical Systems ◽  
External Field ◽  
Classical Mechanics ◽  
Work Conditions ◽  
Stationary States ◽  
Adiabatic Invariance ◽  
Internal Constraints ◽  
Quantum Numbers ◽  
Adiabatic Motion ◽  
Pass Through

The postulate of the existence of stationary states in multiply periodic dynamical systems requires that if the condition of such a system, when quantised, is changed in any way by the application of an external field or by the alteration of one of the internal constraints, the new state of the system must also be correctly quantised. It follows that the laws of classical mechanics cannot in general be true, even approximately, during the transition. There is one kind, of change, however, during which one may expect the classical laws to hold, namely, the so-called adiabatic change, which takes place infinitely slowly and regularly, so that the system practically remains multiply periodic all the time. In this case the quantum numbers cannot change, and it should be possible to deduce from the classical laws that the quantum integrals remain invariant. This was attempted by Burgers, who showed that they are invariant provided there are no linear relations of the type Σ r m r ω r = 0 (1) between the frequencies ω r , of the system, where the m r are integers. In general, however, the frequencies will alter during the adiabatic change, and in so doing will pass through an infinity of values for which relations such as (1) hold. A closer investigation is therefore necessary, as was pointed out by Burgers himself. In the following work, conditions which are rigorously sufficient to ensure the invariance of the quantum integrals, are obtained in such a form that it is possible for one to see whether they are satisfied or not without having to integrate the equations of adiabatic motion.

Continuum mechanics ability to predict the material response at atomic scale

2010 ◽  
Vol 48 (2) ◽  
pp. 372-380 ◽  
Author(s):  
Seyed Ehsan Habibi ◽  
M. Farid ◽  
M.H. Kadivar
Keyword(s):  
Continuum Mechanics ◽  
Atomic Scale ◽  
Material Response

Classical Covariance

2021 ◽  
pp. 78-115
Author(s):  
Moataz H. Emam
Keyword(s):  
Continuum Mechanics ◽  
Gravitational Potential ◽  
Classical Mechanics ◽  
Geodesic Equation ◽  
Relativity Theory ◽  
Fourth Dimension ◽  
Point Particles ◽  
Rigid Objects ◽  
Classical Setting ◽  
Galilean Covariance

Classical mechanics, from point particles through rigid objects and continuum mechanics is reviewed based on the notions of tensors, transformations, and the metric, as developed in the first two chapters. The geodesic equation on flat and curved spaces is introduced and solved in a classical setting. Motion in a potential, particularly a gravitational potential, is discussed. Galilean covariance and transformations are introduced. Time as a fourth dimension is shown to arise even in a classical setting, even if not as rigorous as it would be in relativity theory.

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