Principles Of Classical Mechanics Research Articles

Dissipation, fluctuations, and conservation laws

A large particle moves through a sea of small particles. On the microscale, all particle collisions are elastic. However, on the macroscale, where only the large particle is properly resolved, dissipative forces and fluctuating random forces are observed. These forces are connected by a fluctuation–dissipation theorem proved in two different ways, first via statistical mechanics, and second from fundamental classical mechanical principles of momentum and energy conservation. The novel classical mechanics proof elucidates the relation between micro- and macroscale behaviors, and offers new insights into the physics behind the fluctuation–dissipation result.

HAMILTON'S PRINCIPLE AND SCHRÖDINGER'S EQUATION DERIVED FROM GAUSS' PRINCIPLE OF LEAST SQUARES

It is shown that the Hamilton's principle in classical mechanics and the Schrodinger equation in quantum mechanics can both be derived from an application of Gauss' principle of least squares.

Laboratory study of acoustic emission and particle size distribution during rotary cutting

The coupling of crack growth rate with the crack trajectory is critical for adequate modeling and control of hydraulic fracture initiation and growth and has many important field applications. The identification of optimal breakdown rate for minimizing the near-wellbore fracture tortuosity and the evaluation of fracture penetration depth in previously undrained low-permeability formations during refracturing operations are two applications with significant economic impact. Both of these problems require an adequate account of the rate effect on fracture curvature during its quasi-static growth. A variety of criteria for the direction of crack growth have been proposed since the early sixties. Some of them are expressed as the maximum of certain stress components or as strain energy density in the vicinity of the crack tip. Others use fracture mechanics parameters, such as the maximum stress intensity factor KI, vanishing stress intensity factor, KII, or the maximum of the energy release rate, GI. Recent experimental examination of the above criteria revealed their inadequacy in cases when a process zone (PZ) is formed in front of the crack tip. The PZ in the form of strain localizations is usually created in the vicinity of the crack tip as a material response to stress concentration. It is commonly observed at various loading conditions in metals, polymers, overconsolidated clays, and rocks. Thus, the equations for crack trajectory and for growth rates along the trajectory are still open to debate. A new application of the least-action principle of classical mechanics to the crack growth problem is presented. It leads to the identification of the crack driving forces, including the conjugate to the curvature of the crack trajectory. Such consideration suggests that the crack trajectory depends on the crack growth rate during quasi-static fracture propagation. The experimental setup of laboratory tests designed and performed to examine the rate dependency of fracture curvature are presented, and the test results support this proposition.

Rate sensitivity of fracture trajectory

Abstract The coupling of crack growth rate with the crack trajectory is critical for adequate modeling and control of hydraulic fracture initiation and growth and has many important field applications. The identification of optimal breakdown rate for minimizing the near-wellbore fracture tortuosity and the evaluation of fracture penetration depth in previously undrained low-permeability formations during refracturing operations are two applications with significant economic impact. Both of these problems require an adequate account of the rate effect on fracture curvature during its quasi-static growth. A variety of criteria for the direction of crack growth have been proposed since the early sixties. Some of them are expressed as the maximum of certain stress components or as strain energy density in the vicinity of the crack tip. Others use fracture mechanics parameters, such as the maximum stress intensity factor K I , vanishing stress intensity factor, K II , or the maximum of the energy release rate, G I . Recent experimental examination of the above criteria revealed their inadequacy in cases when a process zone (PZ) is formed in front of the crack tip. The PZ in the form of strain localizations is usually created in the vicinity of the crack tip as a material response to stress concentration. It is commonly observed at various loading conditions in metals, polymers, overconsolidated clays, and rocks. Thus, the equations for crack trajectory and for growth rates along the trajectory are still open to debate. A new application of the least-action principle of classical mechanics to the crack growth problem is presented. It leads to the identification of the crack driving forces, including the conjugate to the curvature of the crack trajectory. Such consideration suggests that the crack trajectory depends on the crack growth rate during quasi-static fracture propagation. The experimental setup of laboratory tests designed and performed to examine the rate dependency of fracture curvature are presented, and the test results support this proposition.

Simplified hydrodynamic analysis of superfluid turbulence in He II: Internal dynamics of inhomogeneous vortex tangle.

The hydrodynamic theory of superfluid turbulence is presented in a new simplified form. It applies to flow situations frequently encountered in practice, in which the thermohydrodynamic environment of a superfluid turbulent tangle of quantized vortices may be considered, in a first order of approximation, as given. Flow quantities like the mass density, the entropy density, and the drift velocities of mass and elementary excitations act, accordingly, as external parameters with respect to the internal dynamics of the vortex tangle. The internal dynamics is completely specified by a kinematic equation governing the time evolution of the line-length density of the quantized vortices and a dynamic equation involving the impulse density of the vortex tangle. The derivation of these equations starts from a variational principle that is reminiscent of Hamilton's principle in classical mechanics and proceeds, in order to include dissipative effects, by using methods of the thermodynamics of irreversible processes. A new quantity called superfluid turbulent pressure is introduced which shows many properties that are familiar from the ordinary pressure in a classical fluid. Two important particular cases are considered in more detail, viz., homogeneous superfluid turbulent flow and flow situations in which the vortex tangle is in permanent internal equilibrium. When diffusion of the vortex-tangle impulse is taken into account and dispersive effects are disregarded, the dynamic equation of the vortex tangle assumes, in the case of internal equilibrium, the form of Burgers' equation with a nonlinear source term. This equation, which is new, may be considered as a natural generalization of Vinen's equation to inhomogeneous superfluid turbulence. Some exact solutions which represent uniformly propagating superfluid turbulence fronts are listed in the Appendix.

Balance equations for physical systems with corpuscular structure

The usual method in nonequilibrium statistical mechanics allows the derivation of the balance equations only for the collision invariants (mass, momentum, energy) and only by a complete knowledge of the microscopic structure. In this paper the existence of the balance equation for an arbitrary physical quantity is proved for any corpuscular system satisfying the local equilibrium assumption, if the microscopic components obey the classical mechanics principles and can be generated or destroyed as a result of some instantaneous processes. We discuss the fundamental equations of the continuum mechanics for mass and momentum.

Statistical origin of classical mechanics and quantum mechanics.

The classical action for interacting strings, obtained by generalizing the time-symmetric electrodynamics of Wheeler and Feynman, is exactly additive. The additivity of the string action suggests a connection between the area of the string world sheets and entropy. We find that the action principle of classical mechanics is the condition that the total entropy of the strings be at an extremum, and the path-integral representation of the quantum density matrix element is an approximation to the partition function of the string theory.

New Variational Approach in the Theory of Nonequilibrium Stationary Processes

The variation principle by Nakano in the quantum theory of transport processes is reformulated in the form of Hamilton's principle in classical mechanics, which provides us with a new possibility of studying nonequilibrium steady states if we can distinguish the phenomenological currents from the driving forces. From the form of the variation principle, Nakano's variation principle becomes evident by inspection.

Sui principi variazionali nella meccanica dei continui classici

In this paper, along with the well established lines of the analytical mechanics, we aim to show an autonomous formulation of the fundamental principle of Lagrange-d'Alembert and of the subsequent Hamilton principle in classical mechanics of deformable continua in the sense that the afore—mentioned principles do not turn out to be a reformulation of the well known undefined equations of motion.

Critical Ornstein-Uhlenbeck processes

The Ornstein-Uhlenbeck position process with the invariant measure is shown to satisfy a variational principle quite analogous to Hamilton's least action principle of classical mechanics. To prove this, a stochastic calculus of variations is developed for processes with differentiable sample paths, and which form a diffusion together with their derivative. The key tool in the derivation of stochastic Euler-Lagrange-type equations is a symmetric variant of Nelson's integration by parts formula for semimartingales simultaneously adapted to an increasing and a decreasing family ofσ-algebras. An energy conservation theorem is also proved.

A frame-independent description of the principles of classical mechanics

A frame-independent description of the principles of classical mechanics

Hamilton's law or Hamilton's principle: A response to Ulvi Yurtsever

The law of varying action and Hamilton's principle of classical mechanics are discussed. It is now clear that the law of varying action, introduced by Hamilton in his papers of 1834 and 1935, was never recognized by either the mathematicians or other scientists who followed him. Why this occurred is discussed in this paper.

An application of the equivalence principle in classical mechanics

It is shown that for the case of a classically interacting point particle system, Einstein's equivalence principle provides a direct means of writing down equations of motion in the centre-of-mass system without resort to a centre-of-mass coordinate transformation.

Derivation of an adiabatic time-dependent Hartree-Fock formalism from a variational principle

A derivation of the adiabatic time-dependent Hartree-Fock formalism is given, which is based on a variational principle analogous to Hamilton's principle in classical mechanics. The method leads to a Hamiltonian for collective motion which separates into a potential and a kinetic energy and gives mass and potential parameters in terms of the nucleon-nucleon interaction. The adiabatic approximation assumes slow motion but not small amplitudes and can therefore describe anharmonic effects. The RPA is a limiting case where both amplitudes and velocities are small. The variational approach provides a consistent way of extracting coordinates and momenta from the density matrix and of obtaining equations of motion when particular trial forms for this density matrix are chosen. One such choice leads to Thouless-Valatin formula. An other choice leads to irrotational hydrodynamics.

On the dynamics of a human body model

Equations of motion for a model of the human body are developed. Basically, the model consists of an elliptical cylinder representing the torso, together with a system of frustrums of elliptical cones representing the limbs. They are connected to the main body and each other by hinges and ball and socket joints. Vector, tensor, and matrix methods provide a systematic organization of the geometry. The equations of motion are developed from the principles of classical mechanics. The solution of these equations then provide the displacement and rotation of the main body when the external forces and relative limb motions are specified. Three simple example motions are studied to illustrate the method. The first is an analysis and comparison of simple lifting on the earth and the moon. The second is an elementary approach to underwater swimming, including both viscous and inertia effects. The third is an analysis of kicking motion and its effect upon a vertically suspended man such as a parachutist.

Consideration of Selected Agricultural Products as Viscoelastic Materials

SUMMARYA demand currently exists for information on the mechanical behavior of foods and agricultural products during handling, processing and quality evaluation. Many of the older subjective tests have now been replaced by various techniques from which empirical and partially objective measurements may he obtained; however, the majority of these measurements are not well‐defined in terms of accepted physical constants, thereby making their interpretation difficult. To provide consistency between various investigations, all mechanical properties may be evaluated in terms of common engineering parameters as the first approximation. From the point of view of mechanics, many products such as fruits, vegetables, and cereal grains in their natural state may be considered as convex bodies for which classical solutions can be applied for evaluation of compression, shear and tension properties.In this paper previous and current investigations in which the engineering approach to mechanical property evaluation has been used are reviewed. In all of these investigations, the common methods for product testing are hydrostatic and uniaxial compression. Instrumentation and techniques are described for obtaining and interpreting data from these tests in terms of physical constants.The mechanical behavior of most agricultural products is time dependent. Therefore, characterization of mechanical behavior requires the application of viscoelasticity principles in which both viscous and elastic responses are combined. The fundamental principles of viscoelasticity are presented briefly and analogies are developed for relating the observed behavior to well‐established mechanical systems.To illustrate the application of the engineering approach to mechanical property evaluation, a study on McIntosh apple fruits is described. This study demonstrates that many of the principles of classical mechanics are applicable to selected agricultural materials.

Principles of Classical Mechanics and Field Theory. (Vol. III, Part 1, of Encyclopedia of Physics, edited by S . Flügge.) Springer, 1960. 902 pp. DM. 198.

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Principles of Classical Mechanics and Field Theory. Vol 3, Part 1 of Encyclopedia of Physics

<i>Principles of Classical Mechanics and Field Theory</i>. Vol 3, Part 1 of Encyclopedia of Physics

Collision and Activated Complex Theories for Bimolecular Reactions

By using the principles of classical mechanics, the specific rates k′ of bimolecular reactions which proceed without activation energies were obtained by taking the average of πbc2g, where bc is the critical impact parameter and g is the relative molecular velocity. The result is k′=(β/μ)(kT)(s−4)/28C2/8.Here, C and s are the constants appearing in the attractive potential, C/rs(s&amp;gt;2), between two reacting molecules separated by a distance r, β is a dimensionless quantity involving s, μ is the reduced mass, and other symbols have their usual meaning. After substituting proper potential parameters into the above equation, we obtained the rates of the reactions for the systems, ion-molecule and radical-radical, in exact agreement with the rates in the literature obtained using the activated complex theory. The reason for the agreement was considered, and it was shown that under two conditions pointed out in the text the equations of k′ obtained from the activated complex theory transform to those derived from the classical collision theory.

A Principle in classical mechanics with a ‘relativistic’ path-element extending the principle of least action

1. A particle with mass m and coordinates x1x2, x3 relative to a set of rectangular axes fixed in Newtonian space is moving in a field of conservative forces with a potential energy V(x1, x2, x3) and a kinetic energyThe equations of motion, written(representing the three equations i = l, i = 2, i = 3 in a way to be used in this paper), constitute, as they stand, a sufficient condition in order to ensurein the sense that the Hamiltonian integral has a stationary value if the actual motion is compared with neighbouring motions with the same terminal positions and the same terminal values of the time as in the actual motion.