Mechanics Science that studies the motion and balance of bodies. It is traditionally divided into three parts: kinematics , dynamics and static , which study motion, regardless of its causes, motion in relation to the causes that determine it and the balance of bodies, respectively.
- History of mechanics
1.1 The birth of mechanicsThe first writings in which, in classical antiquity, mechanical problems (of static and dynamics) are dealt with are Physics and The Sky of Aristotle and the Mechanical Questions always attributed to Aristotle. While Aristotelian statics rests on generally acceptable foundations, its dynamics is based on the principle, which has remained in force through the work of commentators until the time of Galileo , that there can be no movement in the absence of a driving force. For Archimedes are due to the early theoretical research on the centers of gravity, an equilibrium theory of the lever (in which one finds, in embryo, the modern principle of virtual work), the law of hydrostatic that bears his name.
The contributions of the Greek philosophers arrived in the West, through the Arabs and the commentators of Aristotle, almost unchanged, and underwent considerable reworking, especially by the Parisian school of the 14th century. Alongside the block (and almost unknown to his contemporaries), Leonardo’s work flourished, between the fifteenth and sixteenth centuries, a variety of research and studies, due to philosophers and mathematicians; however the birth of the new m. it is marked by the astronomical reform of N. Copernico , then carried out by J. Kepler , which establishes the laws of motion of the planets.
The construction of the foundations of the new science centers around the three great names of G. Galilei , C. Huygens , and I. Newton . Galilei was the first to clearly affirm the concept that the determining circumstances of motion identify acceleration and gives a precise notion of this; with the formulation of the laws of motion of the serious in the void, it essentially establishes the dynamics of the material point subject to constant force; enunciates the principle of relativity (now called Galilean ); formulates clearly, even if limited to a particular case, the law of inertia; discovers the law of isochronism of the pendulum etc. Galilei’s discoveries and Descartes’ scientific and philosophical workthey form the basis on which Huygens’ work develops, to which we owe, among other things, the notions of moment of inertia and centrifugal force, the dynamics of the composite pendulum, some precursor ideas on energy conservation. The conservation of the momentum of an isolated system was instead affirmed by Descartes. Newton’s ideas and results are summarized: he fixes the concept of mass; generalize that of strength; enunciates, in all their generality, the three fundamental laws (principle of inertia, relationship between force and acceleration, principle of action and reaction); formulates the law of universal gravitation, the cornerstone of m. heavenly.
Based on Newtonian principles, any mechanical problem can be mathematically set, but its effective resolution requires the use of appropriate mathematical tools: the concepts and methods of infinitesimal analysis had its creators in Newton himself and in GW Leibniz ; the m. basically ends up being experimental and becomes rational.
1.2 Mechanics from the 18th centuryIn this century, progress is largely linked, in each branch of the field, to the improvement of the analytical tool. Giacomo, Giovanni and Daniele Bernoulli establish in their generality the theorem of living forces, while D. Bernoulli and L. Euler found the dynamics of perfect fluids and give, independently of each other, the momentum theorem of the momentum that as a second cornerstone of system dynamics, it joins the momentum theorem already established by Newton. J.-B. d’Alembert enunciates the principle that goes today under his name and that with the principle of virtual works, modernly formulated by G. Bernoulli, constitutes the first foundation of the so-called analytical mechanics. This, which began with Euler, developed in the late eighteenth and early nineteenth centuries, mainly by GL Lagrange and WR Hamilton. Connected in many respects to the developments of m. analytic is the progress of celestial mechanics , which finds in P.-S. Laplace and then in J.-H. Poincaré insights and developments of fundamental importance.
Fruitful of results is the 19th century, in which individual branches and particular issues of m. are being studied: in particular in m. of continuous systems A.-L. Cauchy sets some general theorems intended to form the foundation of the theory; d’Alembert, Lagrange, H. Helmholtz , GR Kirchhoff , lord WT Kelvin bring new and essential contributions to m. perfect fluids; S.-D. Poisson , GG Stokes , JWS Rayleigh , O. Reynolds develop m. of viscous fluids born from the experiences of C.-A. Coulomb ; L.-M.-H. Navier , G. Lamé , A. de Saint-Venant,RF Clebsch lay the foundations of m. of elastic systems giving birth to the mathematical theory of elasticity. Always in the 19th century. a new branch of m., the so-called statistical mechanics , was created by JC Maxwell , JW Gibbs and L. Boltzmann .
Two truly revolutionary facts occur at the dawn of the 20th century: on the one hand the introduction of relativistic principles, as opposed to the classic ones of Galilei and Newton, on the other the advent of the discontinuous quantum conception, energy. The fundamental postulates of relativity, affirmed (1905) by A. Einstein , and initially accepted not without contrasts and perplexities, took root in the fertile ground prepared by the revision of the classical principles and concepts operated by H. Hertz , E. Mach and Poincaré, as well as the unsustainability of certain hypotheses (such as that of the cosmic ether): relativistic mechanicswhich was born does not supplant the m. classical but it contains it in the extreme case of the so-called slow phenomena , characterized by speeds much lower than the speed of light in a vacuum. The need to introduce in the m. Atomic an element of discontinuity, which does not appear in the laws governing ordinary-sized bodies, was recognized for the first time in 1900 by M. Planck . The address inaugurated by Planck has had developments of the greatest importance for the knowledge of the laws that govern the microcosm; these developments are now systematically organized in quantum mechanics , mainly by W. Heisenberg , E. Schrödinger and PAM Dirac. The m. quantum contains as limit case the m. classical (correspondence principle).
1.3 The various branches of mechanicsCo Common foundation of the various industries in which it has developed the m. as science it is that body of doctrines traditionally known by the name of rational mechanics . The phrase may indicate any part of the m. that it develops with deductive mathematical procedure from some general principles. Moreover, the term is to be considered rather as a synonym of classical macroscopic mechanics , that is of that part of the m. which, on the basis of Newtonian postulates ( ➔ dynamics ), studies mechanical problems related to bodies of ordinary dimensions, in the various schemes adopted to represent them, regardless of any other phenomena concomitant with the actual mechanical fact (such as, for example, the phenomena chemical,electrical, thermal etc.).
- Analytical mechanics
The m. analytical , like m. rational, mainly has as its object the deduction, from some extremely valid principles, of the laws of motion under assigned stresses; the systems it takes into consideration are more general and complex than those examined by m. rational proper, and the analytical methods it uses are thus more general and complex. Among these systems, holonomic systems are particularly important, whose configurations can, at any time, be placed in one-to-one correspondence with a certain finite number, n , of arbitrary parameters, independent of each other (degrees of freedom or Lagrangian coordinates) and whose possible constraints are all holonomic (that is, they imply, directly, only limitations to the positions and not to the act of motion).
The m. analytic is based on d’Alembert’s principle and on the principle of virtual works. As regards the first, imagine that you have a system consisting of N points and let m s be the mass of the generic point P s , F s the active force applied to it, δP s any virtual displacement of P s instantly in exam; due to the constraints the acceleration to s of the point is not given by the ratio F s / m s , i.e. the product m s a s is different from F s , and (Fs – m s a s ) represents the ‘lost force’: d’Alembert’s principle expresses the fact that at every moment during the movement of the system the ‘lost forces’ and the ‘constraining forces’ balance each other. The virtual work principle then states that the virtual work of constraint forces offered by frictionless constraints can never be negative ( ➔ work ). The intervention of this principle as the basis of the m. Analytic subordinates the essential developments of this theory to the hypothesis that sensitive frictions do not come into play. The combination of the two principles leads to establish the symbolic relationship of the dynamics or relationship of d’Alembert and Lagrange:
If the system in question is a holonomic system subject to all bilateral constraints, so that the sign of equality holds in , the  itself gives rise to Lagrange’s equations. The system has n degrees of freedom and both q 1 , q 2 , …, q n a n- tuple of suitably chosen Lagrangian coordinates. If the system is in motion, the q are naturally to be considered as a priori unknown time functions; but the fact that they are suitable for identifying the configurations of the system is equivalent to saying that the position of the generic point of this can be expressed a priori as a function of the q same (and time if the constraints are mobile):
P s = P s ( q 1 , …, q n ; t ).
In consequence of this they are immediately expressible in function, in general, both of the q both of their derivatives q with respect to time, both of t the same, so the kinetic energy T of the system, which is a quadratic function of q , as the so-called Lagrangian components,
of the active solicitation. These quantities, as demonstrated from , are linked by relationships
which are precisely the Lagrange equations or, with more precise denomination, the ‘second form of the Lagrange equations’. It is a system of n second order differential equations in the n unknown functions q 1 ( t ), q 2 ( t ), …, q n ( t ), therefore a system able to completely determine the motion when the initial values of q and q̇ are also assigned . Its structure is simple and extremely synthetic: the active stress is collected in the Lagrangian components Q h, the material properties of the system in the expression T of the kinetic energy. It is noteworthy that the motion of systems, even apparently very different, but with the same expression for T and for Q h , is governed by the same equations as Lagrange: this is the case of the so-called dynamically equivalent systems. In the particularly important case in which the active stress is conservative, where it is denoted by U = U ( q 1 , …, q n ) the potential, it results Q h = ∂ U / ∂q h , and indicating with Lthe sum of the kinetic energy and the potential, L = T + U , the  can be placed in the form
 formula .
The function L ( q , q̇ , t ) is called the Lagrange or Lagrangian function; for a system subject to conservative stress it summarizes all the properties of the system and the stress. The first-member quantities of  are remembered as Lagrangian binomials; the quantities p h = ∂ L / ∂ q̇ h as kinetic moments . Generally speaking, any differential system that has the structure of the system , whatever the meaning of L , is still called the Lagrangian system.
An Lagrangian system  can be transformed, in infinite ways, into a system of 2 n first order differential equations in 2 n unknown functions of time. One of the most remarkable of these transformations, called the Hamilton transformation, is the following. Next to the n Lagrangian coordinates q , the n kinetic moments (conjugate variables of the q ) linked to the q̇ by the relations p h = ∂ L / ∂ q̇ h which, resolved with respect to the q̇ , give q̇ h = u h( p / q / t ). Then introduced the function (Hamiltonian or Hamiltonian function):
 formula ,
where each of the q̇ is understood to be expressed, as mentioned, as a function of the p , q and t , the system  is transformed into the Hamiltonian or canonical system
 formula ,
consisting precisely of 2 n first order differential equations, called canonical, in the 2 n unknown functions p h , q h (canonical variables). The physical significance of the Hamiltonian in the mechanical case is particularly remarkable if the constraints do not depend on time; in this case the potential U (or the potential energy – U ) depends only on the q h , i.e. U = U ( q h ), so the kinetic moments p h are also equal to ∂ T / ∂ q̇ h(as L = T + U ). The first term of the second member of , for the fact that T is a quadratic homogeneous function of q̇ h , is equal to 2 T ; consequently it results H = T – U : that is, H , sum of the kinetic and potential energy, represents the total energy of the system. Again for fixed constraints, if H does not explicitly depend on time, the integral of energy (or of living forces) exists: H ( p , q) = const. The same integral exists even if the constraints are not fixed but the hypothesis that H does not depend explicitly on time remains valid : the relation H ( p , q ) = cost. it then has the name of generalized integral of energy. The same facts occur if, instead of H , it is L independent of time.
- Applied mechanics
Complex of disciplines in which the principles and methods of m. rational. It is possible to distinguish the m. applied in m. applied to buildings, m. applied to fluids and m. applied to machines. The mechanics applied to buildings have as their object the practical applications of the principles of m. and the theory of elasticity to the static and dynamic study of structures and is much more commonly known as construction science . The mechanics applied to fluids , called fluid dynamics most frequently, studies the motion of fluids and bodies immersed in them, and is in turn distinguished in hydrodynamics, aerodynamics, gas dynamics, magneto-fluid dynamics. The m. applied to buildings and m applied to fluids they can, as a setting, refer to m. of continuous systems. The mechanics applied to machines, on the other hand, concern the elements of the machines, both with regard to their ability to transmit motion and to that of transmitting forces. If the study is limited to identifying the motion allowed to the different machine parts as a consequence of the constraints, both external and internal, this study is essentially kinematic.
- Celestial mechanics
Part of the theoretical astrom that considers the movements of celestial bodies in relation to the forces that produce them and the laws that govern them. Generally, the m. celestial in its proper sense only deals with the bodies of the solar system , that is, planets, satellites, planets and comets, and this is the classic part, created by mathematicians and astronomers of the 17th, 18th and 19th centuries. after the discovery of Newton’s law. The subsequent extension of astronomical knowledge to star systemsand to the whole sidereal universe has made to extend and complete, in the 20th century, the searches of the m. classic light blue with the introduction of new methods and new calculation aids. But this new part soon detached itself from the old, rather taking the names of stellar dynamics and stellar statistics, while the name of m. celeste remained at m. of the planetary system.
The m. celestial is based on the fundamental equation of dynamics, F = m a, where m is the mass, at the acceleration of a celestial body and F the force acting on it, deriving from the law of universal gravitation. In the hypothesis that each planet is subjected only to the action of the Sun, the solution of the equation of motion allows to determine the orbit of the planet, for which Kepler’s laws are deduced ( ➔ Kepler, Johannes), which, originally obtained by pure observation, thus find themselves to be theoretically justified. But the action of the Sun, although predominant, is not the only force that urges the bodies of the planetary system; they also suffer their mutual attractions. These second forces, called planetary perturbations, are very small but they are nevertheless sufficient to explain the irregularities which alter the motions of the planets and which prevent them from strictly complying with the laws of Kepler.
The merit of having first calculated these irregularities of planetary motions dates back to Euler . At the end of the 18th century and at the beginning of the 19th century the fundamental work of Laplace and Lagrange substantially marks the definitive arrangement of the m. heavenly. The Mécanique céleste Laplace has deeply influenced all the great number of his followers, including Poisson, U.-J.-J. Le Verrier , C.-E. Delaunay , J.-C.-R. Radau, F.-F. Tisserand , P.-J.-O. Callandreau, Poincaré. None of these distinguished mathematicians and astronomers has ever dealt with celestial observations: when Le Verrier wanted to confirm the discovery of Neptune in heaven, he resorted to JG Galle , who, toBerlin managed to observe the star. Poincaré’s work, in particular, marked a turning point in m. celeste, with the introduction of new methods and with the achievement of new results, to which all subsequent research has been based up to the present day.
- Nonlinear mechanics
Part of mathematical physics that studies oscillatory phenomena (of a mechanical, electrical nature, etc.), the theory of which can take place only through nonlinear differential equations. Recall, as a simple example of a mechanical system that performs free oscillations, a point P , movable on a straight line, subject to an elastic force proportional and opposite to the abscissa x of P and to a resistant force −2 pẋ proportional and opposite to the speed ẋ of P . Point P, on the other hand, performs forced oscillations, if another force is added to the previous ones, e.g. of the type F cos ( Ωt + α), i.e. sinusoidal function of time t . The quantities p , F , Ω , α are constant. By choosing the unit measure so that they are unitary mass of P and the elastic coefficient, the x ( t ) verifies the linear differential equation
 formula ,
which can be attributed to numerous other mechanical or electrical systems, by replacing the constant 2 p with a function εf ( x ) of x (and, in some cases, also of ẋ ):
where ε is a constant greater than zero; this is Liénard’s differential equation, nonlinear for the presence of the term f ( x ) ẋ , with F = 0 in the case of free oscillations, with F ≠ 0 in the case of forced oscillations. Other problems can be traced back to the nonlinear differential equation
obtained from  by replacing the elastic force with a force – ϕ ( x ) generic function of x . La , possibly generalized by placing f ( x , ẋ ) or also f ( x , ẋ , t ) in place of f ( x ) ẋ , and  are the most important differential equations of m. nonlinear relative to systems with only one degree of freedom. These systems are said to be autonomous if, in the equations that govern them, time does not explicitly appear: for example, if in , with f ( x )ẋ eventually replaced by f ( x , ẋ ), is F = 0; not autonomous in the opposite case: e.g. if in  is F ≠ 0, or a term f ( x , ẋ , t ) appears . The quantitative study of the ,  and their generalized can be done with various methods; of approximates (Poincaré, B. Van der Pol, N. Krylov and N. Bogoljubov , N. Minorsky etc.), valid for weak nonlinearity, i.e. for ε ≪1, and when ϕ ( x ) differs slightly enough from x. The qualitative study (existence of periodic solutions, uniqueness, stability etc.) is usually done, instead, by referring the ,  and their generals to first order differential systems and then making use of topological considerations: as, for example, in Poincaré’s theory for systems of differential equations of the first order, very useful in the particular case of autonomous systems.
The case in which, in , f ( x ) is negative in a neighborhood of the origin (i.e. for small values of ∣ x ∣), positive for x outside this neighborhood, is of considerable importance : such is, for example ., the case f ( x ) = x 2 −1, in which the  reduces to the classic Van der Pol equation. Obviously, when f ( x ) is negative, the work of the force – εf ( x ) ẋ is positive ( εbeing> 0), that is, energy is introduced from the outside into the system: it is therefore clear how it can perform free oscillations without damping. Making some other qualitative hypotheses on f ( x ), we reach, when it is F = 0, the following results: a ) the equilibrium position of the system (solution x ≡ 0 of ) is unstable and therefore is not assumed, in practice, by the system; b ) the other solutions of  are always oscillatory around the origin (unlike the linear case , in which they are such only if it is p<1) and tend asymptotically to a single stable periodic solution. In other words, the system governed by  practically performs periodic oscillations after a transitory time interval. However, it should be noted that, unlike the linear case, these oscillations do not depend on the initial conditions. In the plane ( x , ẋ ) said periodic solution is represented by a closed curve C (stable limit cycle, according to Poincaré), which surrounds the origin and to which all the other curves representing solutions of  with F = 0 tend . If ε ≪1, the periodic solution is significantly sinusoidal, with a period close to 2 π ; the amplitude, in the case of Van der Pol, is 2. If it is insteadε ≫1, as recognized by Van der Pol himself, the periodic solution of the equation (of type ) has a period of 1.614 ε and is identified with the abscissa x of a point P that moves along two segments MN , M ′ N ′, symmetrical with respect to the origin O ( OM > ON , O outside MN ), as follows: starting from M , the point P travels through MN ; arrived in N ‘jumps’ abruptly in M ′; then travels M′ N ′, jump in M ; and so on. Oscillations of this type, which are, in essence, a periodic succession of aperiodic phenomena, are said to be ‘relaxation’ and are observed in many physical phenomena: such is, for example, the trend of the current absorbed by a conveniently powered neon (➔ oscillator ).
- Quantum mechanics
6.1 Quantum mechanics and wave mechanicsIt can be defined as the m. systems (electrons, nuclei, atoms, molecules, etc.) of extremely small dimensions with respect to the dimensions of ordinary bodies, which do not obey the laws of m. and electromagnetism valid for ordinary bodies and systems. The m. quantum arose almost simultaneously in two apparently very different forms: the theory of matrices, also called, from the beginning, quantum mechanics ( Quantenmechanik ) by W. Heisenberg (1925), and the wave mechanics , of which the first idea is due to L. de Broglie (1924) while the subsequent systematic developments are mainly due to E. Schrödinger(1926). Schrödinger himself has shown that the two theories are equivalent, that is, they are ultimately only two different mathematical forms of the same theory. Avoiding artificial hypotheses that previous quantum theories could not avoid, the m. quantum allowed a more satisfactory interpretation of various experimental results; among these, the phenomenon of electron diffraction must be remembered (Davisson and Germer, 1927) which, inexplicable in the context of the old quantum theories, is instead foreseen and perfectly framed, also in terms of quantity, by m. wave. Subsequently the m. quantum has taken on a new and more general form, in which the theory of matrices and m are included as special cases. wave, with the so-called transformation theory, due to PAM Dirac and P. Jordan and based on convenient uses of the mathematical method of the operators; while other powerful investigation tools haveprovided the systematic introduction, by H. Weil , of the mathematical methods of group theory.
6.2 The origins of wave mechanicsStarting point to understand the origins of the m. the consideration of certain contradictory aspects was undulatorysome luminous phenomena. In fact, while some of these phenomena (such as interference) are well explained by considering light as a wave phenomenon and seem incompatible with any corpuscular theory, others instead (such as the photoelectronic effect) are explained if the corpuscular nature of light is admitted and they are inexplicable on the assumption of the wave nature of it. The apparent contradiction is resolved considering that ideas and operations, perhaps only conceptually possible on the ordinary scale, can become, even conceptually, impossible within the microcosm. In particular, the phenomena with which we reveal a radiation (of visible or non-visible frequency) always consist of modifications undergone by electrons, atoms, or molecules (of the retina of the eye, of the photographic plate,
Energy is transmitted to those who, in particular photons, are not corpuscles in the ordinary sense of the term and do not behave as such in all circumstances. Photons and quanta in general, but also the so-called elementary particles(electrons, protons, neutrons, etc.), give rise to dual-aspect, corpuscular and wave-like phenomena, which only at first glance can appear in contradiction since they never occur simultaneously but appear as two ‘complementary’ aspects that integrate with affair (principle of complementarity). A rigorous discussion of the phenomena with which these particles are revealed has shown that, in the same way as for quanta, it is impossible to attribute to them a certain position and at the same time a certain speed because the experiences, even ideal, to determine the one are incompatible with those to determine the other ( ➔ uncertainty, principle of ).
An important link between classical theory and m. quantum is given by the so-called correspondence principle of N. Bohr which, in its more general formulation, asserts that the laws of m. quantum take the same form of the laws of classical physics when transitions between states characterized by quantum numbers having very high and little different values come into play.
6.3 Analogy between the laws of geometric optics and those of mechanicsSchrödinger’s starting point was the consideration: between the laws of geometric optics and those of m. of the punThere is an analogy, already indicated in a mathematically precise way by Hamilton, which can also be pushed to the quantitative aspect of the laws. The laws of geometric optics cease to be valid when screens, cracks etc. come into play. relatively small in size: in this case, the various diffraction phenomena arise, which prove that the exact laws of optics are wave-like and those of geometric optics represent only a valid approximation for sufficiently large physical systems. Since experience has shown that the laws of m ordinary no longer apply to systems of very small dimensions, it was led to think, guided by the analogy with optics, that even mechanical laws are, strictly speaking, of a wave nature, with wavelengths of the order of the atomic dimensions, so that in all systems of ordinary dimensions the approximation corresponding to the geometric optics can be applied, that is the m. classical, while new atomic dimensions, similar to those of wave optics, must be applied to atomic-sized systems. This idea, suggested by de Broglie’s previous brilliant intuitions, was then clarified and developed by Schrödinger.
Similarly to what optics do for photons, the m. wave of particles proposes to search for the probability that the considered particle is at a given instant in an element of volume dτ , and for the calculation of this probability it introduces a function, generally complex, ψ ( x , y , z , t ), called wave function or amplitude of probability, the square of whose module multiplies; to by dτ , ∣ ψ ( x , y , z , t ) ∣ 2 dτ, gives the probability that in one measure the particle under examination is at the instant t in a neighborhood of volume dτ of the point of coordinates x , y , z . Due to the fact that the total probability of presence in the whole space must be 1, the ψ remains subject to the condition, called normalization,
 formula ,
where the integral means extended to all the space. If E is the total energy of the particle, ψ can be placed in the form
with ν = E / h ; and, like the purely spatial function u , it must satisfy Schrödinger’s equation:
where U is the potential energy of the force field acting on the particle.  analytically represents the propagation of waves (de Broglie waves), with phase velocity
and with group speed
If ψ is null everywhere except in a region of space small enough to be considered point-like, so as to have a wave packet, this region, which in this case represents the approximate position of the particle, moves according to the laws of ordinary m. of the point. So the m. wave is connected with continuity to m. ordinary. The Schrödinger equation is analogous to the electromagnetic wave equation of frequency ν = E / h in a medium with a refractive index proportional to
for de Broglie’s waves the refractive index is therefore represented by a function of the potential and also depends on ν , as occurs for light in dispersive media. If the particle is not subject to forces, it is U = 0, and for the phase velocity we have then
from this expression and from the relation λν = V , valid for each wave propagation, we obtain
remembering then the expression of the group speed, which is now reduced to
we reach the wavelength of de Broglie
 formula ,
where p indicates the momentum of the particle. Each particle is thus associated with a wavelength inversely proportional to the momentum. The ψ of the form  has a unique frequency, and therefore represents waves analogous to monochromatic light: however, we can also have a ψ which is the sum of several terms of the form  with different ν , and this will represent, if the ψ total satisfies , a particle whose energy is not physically determined. The ψ , in this more general case, satisfies, instead of , the following equation:
[11 ′] formula .
6.4 Eigenvalues and eigenfunctionsThe problem of determining a function ψ which satisfies  and , and which is also finite and continuous throughout the space likely to be occupied by the particle, is mathematically identical to the acoustic problem of determining the standing waves which they can be established in said space with a given distribution of the density of the medium. It is shown that it admits a solution only if the parameter E shown in  assumes particular values which are said to be eigenvalues of the equation and define, through the relation E = hν, the frequencies of standing waves. These eigenvalues are always in infinite numbers and can constitute a discrete sequence or even, at least in part, fill a continuous interval which is called the continuous spectrum of eigenvalues. In the first case, Schrödinger’s equation leads to establish the existence in the atoms of discrete energy levels (which is an experimentally ascertained fact) not through a postulate as did Bohr’s theory, but through the same mathematical procedure with which in acoustics it is established that a string can vibrate only with certain frequencies. Each eigenvalue E n can correspond to only one solution ψ n(normalized) which is called eigenfunction, and then the eigenvalue is said to be simple; or there may correspond infinite eigenfunctions which can all be obtained as linear combinations of a certain number p of them, independent of each other: then the eigenvalue is said to be a multiple of order p (case of degeneration). Two eigenfunctions corresponding to different eigenvalues enjoy the so-called orthogonality property:
where the integral is extended to all the space and the asterisk denotes the conjugate complex.
6.5 Development and applicationsThe theory and the results to which it led, in particular in atomic physics , have allowed, among other things, to eliminate arbitrary hypotheses and uncertainties in the interpretation of previous results. Some changes later (1928) made by PAM Dirac in the same setting of the m. wave, to put it in accordance with the principle of restricted relativity , allowed to derive those properties of the electron that in the previous quantum theory had been introduced a priori by GE Uhlenbeck and SA Goudsmit with the so-called hypothesis of the rotating electron, and that is, the existence of a moment of momentum (the spin ) equal to ± h / 4π is of a magnetic moment equal to a Bohr’s magnetone of opposite orientation. A further development of the theory, due to Dirac himself, accounts for the properties of the positive electron, or positron, and in particular the possibility of ‘combination’ of a positive electron and a negative electron, with the cancellation of electric charges and masses and radiation emission ( annihilation ).
Among the most remarkable applications of m. quantum we will remember the theory of the emission of α particles by radioactive nuclei, due to G. Gamow (1928), which is the first example of application to nuclear physics; the theory of impact between electrons and atoms and between atoms (or ions) due to M. Born, NF Mott and others; the theory of electronic conduction and that of dia- and paramagnetism ( ➔ magnetism
On the hidden variables in m. quantum ➔ hidden, variable .
- Mechanics of continuous systems
Part of the m. in which the bodies are schematized in ‘continuous’, that is, in systems in which the matter is thought to be distributed continuously, according to the case in a volume, on a surface or along a line (continuous three-, two- or one-dimensional); the law of distribution is identified at the generic instant by the function that expresses how density varies from point to point. For this definition, the m. of continuous systems should strictly include both the m. of the deformable continuums and that of the rigid ones; in reality this is usually kept distinct from the first, since the criteria of setting and development are different, so the phrase is used as a synonym of mechanics of deformable systems. In the development of the theory at first the physical properties characteristic of the body under examination are not specified in any way, and it regards indifferently liquids, aeriforms, elastic bodies etc. In a subsequent phase, specifying these properties, the theory specializes in fluid mechanics, elastomechanics etc. The m. of continuous systems can be developed from two points of view: that is, the molecular point of view, or Lagrangian, which considers the various quantities in correspondence with the individual particles of the system, and the local point of view, or Eulerian, which refers them instead to the generic point of the space occupied by the system itself. The system is generally subjected to a certain set of forces, both superficial and mass: internal, unknown tensions arise, which intervene in determining the trend,
To determine the internal state of tension, one must determine the distribution of specific stresses that is created around each point at the generic moment. The specific effort, Φ ( n ) ( P , t ), which is exerted at the instant t on the surface element conducted for a point P and identified, as regards the position, from its normal oriented, n , without prejudice to P and t , varies with n according to the so-called Cauchy tetrahedron theorem. Precisely, if you choose a triad of orthogonal axes x 1 , x2 , x 3 , with origin in P , and considering three surface elements having the positive axes x 1 , x 2 , x 3 respectively oriented as normal , are indicated by Φ (1) , Φ (2) , Φ ( 3) the specific stresses relating to the three aforementioned elements, the specific stress relating to the surface element having normal n can be expressed in the form
Φ ( n ) = Φ (1) n 1 + Φ (2) n 2 + Φ (3) n 3 ,
being n 1 , n 2 , n 3 the cosine directors of n . In other words, knowledge of the distribution of specific stresses around each point requires knowledge of the components of the three vectors Φ (1) , Φ (2) , Φ (3) , components that will be indicated here with the symbol Φ rs , being r , variable from 1 to 3, the index of the axis to which the component refers, and s , also variable from 1 to 3, the index of the axis to which the effort refers (thus, for example, Φ 23will indicate the component according to the x 2 axis of Φ (3) ). The Φ rs , which ultimately identify the stress tensor, are 9 scalar quantities, but, as can be shown, only 6 of them (voltage characteristics) are distinct from each other, using the so-called symmetry relations for which Φ rs = Φ sr ( r ≠ s). The distribution of specific efforts therefore introduces 6 unknowns at each point. To these must be added the density and 3 other scalar unknowns for the determination of the motion (for example, from the Eulerian point of view the 3 components of speed expressed as a function of place and time). Thus, in the more general case, we have 10 unknowns, point by point, whereas only 4 equations are available point by point. These are the so-called continuity equation, in which the principle of mass conservation is analytically translated, and the 3 scalar equations corresponding to the fundamental equation of m. of continuous systems. In Eulerian form the continuity equation, where indicated with x 1 , x 2, x 3 the coordinates of the place, with ρ ( x 1 , x 2 , x 3 , t ) and with v ( x 1 , x 2 , x 3 , t ) respectively the local expressions of density and speed, with t the time, you can write
since ∂ ρ / ∂ t is the partial derivative (or Eulerian) of the density with respect to time: it is essentially a first order partial derivative equation in the independent variables t , x 1 , x 2 , x 3 for the 3 components v 1 , v 2 , v 3 of the vector ve for the function ρ . Still in Eulerian form, the 3 fundamental scalar equations, keeping the previously mentioned meaning to the other symbols and indicating with a ( a 1 , a 2, a 3 ) the acceleration, with g ( g 1 , g 2 , g 3 ) the unit mass force, can be written in the form
with s = 1, 2, 3. The above 4 equations are said to be undefined since they must be verified at the generic point of the system. The framework of the fundamental relations is completed with the addition of the boundary conditions, summarized in the equality, vector, between the intensity f of the external surface force and the specific effort Φ ( n ) , evaluated with respect to the internal normal n at the surface of side dish:
f = Φ ( n ) .
The broad indeterminacy that occurs (4 equations against 10 unknowns) appears quite natural if it is reflected in the broad generality of the approach, that is, in the lack of any clarification regarding particular properties to be attributed to the system: an elastic body and a liquid, for example, both can be represented by means of a deformable continuous system, but it is clear that the behavior of one is not that of the other and that therefore the equations valid for the first cannot be the same in all that for the second . Case by case indeterminacy is therefore eliminated taking into account particular properties, which can give rise to a reduction of unknowns or to the addition of equations. If, for example, you are dealing with a non-viscous fluid, the specific stress is, as is shown, a pressure at each point, with the same value whatever the direction considered: in this case the voltage characteristics from 6 are reduced to one, that is to the unknown pressure, and there is only one unknown factor more than the number of equations. Furthermore, if the hypothesis that the non-viscous fluid is an incompressible liquid is added, the density, not being able to vary around each point, leaves the number of unknowns and equality between unknown and equations is reached without it being necessary add other relations to the 4 basic equations.
- Statistical mechanics
Part of the m. c h proposes to deduce the properties of macroscopic atomic systems hypothesis, according to which the matter is composed of aggregates of a large number of atoms or molecules that move following the laws of m. classical or m quantum; the corresponding theory is called classical statistical mechanics or quantum statistical mechanics. The mathematical complications that you would face if you wanted to study with the methods of m. ordinary (classical or quantum) the motion of a system consisting of a huge number of particles would be very large. On the other hand, a relatively small number of parameters is sufficient to characterize a macroscopic system, that is, those necessary to specify its thermodynamic state (such as, for example, temperature, pressure, magnetization, etc.); therefore a possible solution of the equations of motion, depending on the enormous number of parameters relating to the particles that make up the system, would be unnecessarily detailed. Task of the m. statistics is to provide the prescriptions and justifications to carry out the elimination of irrelevant microscopic parameters, and move on to a macroscopic description,i.e. thermodynamics.
8.1 Classical statistical mechanicsIn the M. classical microscopic states of a system consisting of N particles are described by specifying the value of the momentum and positions of each of the particles at a given instant; in the hypothesis that the determination of these values can take place with unlimited precision, this is equivalent to specifying the 3 N +3 Ncoordinates that identify a point in the phase space of the system in question. However, this assumption obviously accepts a fact that such is not from an experimental point of view, that is, that it makes sense, at least in principle, to measure exactly and simultaneously positions and speeds of a huge number of molecules. It is therefore advisable to proceed thinking about the phase space divided into very small cells of equal size, each of which determines the momentum and position of each particle of the system, with the maximum precision allowed. If p 1 , p 2 , p 3 and q 1 , q 2 , q 3are the coordinates of momentum and position of the first particle, p 4 , p 5 , p 6 and q 4 , q 5 , q 6 those of the second, etc., the microscopic states of the system can be represented by cells consisting of the points of coordinates:
with the condition δpδq = h , where h is to be understood as a constant, a priori arbitrary, which has the interpretation of limiting the precision with which measurements of a coordinate of momentum and the corresponding position coordinate can be performed. The coordinates p 0 α and q 0 α are used to identify the center of the generic cell and therefore the cell itself. The space of the microscopic states is therefore the set of cells Δ , of volume h 3 N, with which the phase space is thought to be divided. The situation in which it is possible to perform perfect momentum and position measurements simultaneously, corresponds to the limit h which tends to zero in the most general theory. To translate the principle of mechanism inherent in the atomic hypothesis, it is necessary to define a law S, which describes the dynamics of the system by transforming the cells of the phase space between them: if the system at time t is identified by the cell Δ , at time t + τ will be identified by the cell Δ ′ = S Δ ; τit is a small unit of time with respect to the macroscopic time intervals on which the system is observed, however accessible by direct measurement, at least in principle. The evolution law S must verify the laws of m. Newtonian; this means that each cell is associated with 3 fundamental quantities: the kinetic energy T ( Δ ) = T (p 0 ), the potential one Φ ( Δ ) = Φ (q 0 ), the total one H ( Δ ) = H (p 0 , q 0 ) = T (p 0 ) + Φ (q0 ), where p 0 i q 0 i are the momentum and position of the i- but particle ( i = 1, …, N ) in the state corresponding to the center (p 0 , q 0 ) of Δ . Given the solutions of Hamilton’s equations of motion:
if Δ ′ is the cell containing the point where the initial datum evolves (p 0 , q 0 ) over time τ , S is defined so that it is S Δ = S Δ ′. Very important is the question whether S Δ 1 = S Δ 2 implies Δ 1 = Δ 2 : this property, which has an intuitive meaning and a clear interest in its connection with the problem of reversibility of motion, is certainly true only in the case point cells ( h = 0). In the limit h → 0 it is possible to choose τso that, unless a negligible set of cell pairs, Δ 1 ≠ Δ 2 implies S Δ 1 ≠ S Δ 2 . If, on the other hand, h > 0, and a posteriori, we must think that h is the Planck constant, this condition is satisfied only for sufficiently high temperatures; otherwise, the representation of microscopic states in terms of cells is no longer consistent, but in this case the problem should be treated with m. Quantum. A mechanical system of Nidentical particles can therefore be described, for temperatures that are not too low, in terms of an energy function, defined on the phase space at 6 N dimensions divided into cells of equal volume; the temporal evolution, observed over multiple time intervals of a unit τ , must be thought of as a permutation of cells of given energy. One wonders what the qualitative behavior of this system is with macroscopically fixed energy, that is between E – DE and E , being DE a small quantity with respect to E but macroscopic. L. Boltzmannhe assumed that in interesting cases the ergodic hypothesis was valid, according to which the action of S is as simple as possible: S is a one-cycle permutation of the cells of the given energy. In other words: over time each cell evolves by subsequently visiting all the others of equal energy. The basis of this famous and contested hypothesis is its conceptual simplicity: it says that, in the system in question, all cells of equal energy are equivalent. There are cases where this hypothesis is manifestly false; for example, if the system is enclosed in a perfect spherical container, evolution preserves the angular momentum whereby cells with the same energy but different angular momentum cannot transform into one another.
Given an observable quantity defined on the phase space, f (p, q), an important quantity that we want to study, and often the only one needed, is the time average of f :
 formula ,
where f ( Δ ) = f (p, q) if (p, q) is a point that identifies Δ . If Δ 1 = Δ , Δ 2 , Δ 3 , …, Δ N is the cycle to which the cell Δ belongs, we have:
 formula ,
and in the ergodic case the cycle consists of all the cells of energy equal to that of Δ . If the system energy is determined less than the DE error , the energy cells between E – DE and E are divided into variable energy cycles, on each of which the f should have the same average value (unless variations negligible, if DE is negligible compared to E ). So, remembering that the cells all have the same volume and indicating with J the domain of the variables (p, q) in which E – DE ≤ H (p, q) ≤ E, for the supposed weak dependence of f̅ ( Δ ) on H ( Δ ) from  follows:
 formula ,
if h is so small that the sum on the cells can be replaced with an integral. This report, which Boltzmann conjectured to be true except in exceptional cases, reads: “the temporal average of an observable is equal to its average on the surface of constant energy”. It is the heuristic basis of the microcanonical model of classical thermodynamics. Note that, if  is true, the average value of an observable depends only on Eand not from the particular cell in which the system is initially located. The latter property is a necessary prerequisite for any program that intends to deduce the macroscopic properties of matter from the atomic hypothesis, as these properties cannot depend on the detailed microscopic properties of the configuration in which the system is located at a certain instant. In applications, it is very important to know how to evaluate the speed with which the limit f̅ is reached: for  to be useful, the limit in  must be reached in a long time t with respect to τ, but very short compared to the times relevant for the macroscopic observations that you want to perform on the system. In fact, only on time scales of the order of t or longer does the observable f appear constant and equal to its average value. It is perfectly conceivable a situation in which the system is ergodic, but the value f ( Δ ) oscillates so much along the trajectory that the average value of f is reached on time scales of the order of magnitude of the time necessary to visit the entire surface of constant energy (recurrence time), which is enormous. For example, for 1 cm 3 of hydrogen at 0 ° C this time is of the order of 10 1019s, while the age of the Universe is (only) ≈ 10 17 s!
8.2 Quantum statistical mechanicsL to m. Quantum statistics is very similar to the classical one from the point of view of the formal structure. The phase space no longer makes sense and instead we only think of the set of observable quantities: they are described by linear operators on a Hilbert space and the statistical sets are defined in terms of the operator that describes the energy, usually denoted by H and said Schrödinger operator.