Mechanics

Mechanics include the study of machines (simply fixed pole).

Mechanics (in Greek, Μηχανική and in Latin, mēchanica) or the art of building a machine is the branch of physics that studies and analyzes the movement and rest of bodies, and their evolution in time, under the action of forces. Modern mechanics includes the evolution of more general physical systems than mass bodies. In this approach, mechanics also studies the temporal evolution equations of physical systems such as electromagnetic fields or quantum systems where it is not correct to speak of physical bodies.

The set of disciplines covered by conventional mechanics is very broad and can be grouped into four main blocks:

Mechanics is a science belonging to physics, since the phenomena it studies are physical, which is why it is related to mathematics. However, it can also be related to engineering, in a less rigorous way. Both points of view are partially justified since, although mechanics is the basis for most of the classical engineering sciences, it is not as empirical in nature as these and, instead, due to its rigor and deductive reasoning, it resembles more to math.

This branch of physics traces its origins to Ancient Greece with the writings of Aristotle and Archimedes. During the early modern period, scientists such as Galileo, Kepler, and Newton laid the foundations for what is now known as mechanics classical. It is a branch of classical physics that deals with particles that are at rest or that move with velocities significantly less than the speed of light. It can also be defined as a branch of science that deals with motion and forces on bodies that are not in the quantum realm. Today, the field is less well known in terms of quantum theory.

Classical mechanics

Classical mechanics is made up of areas of study ranging from the mechanics of rigid bodies and other mechanical systems with a finite number of degrees of freedom, to systems such as the mechanics of continuums (systems with infinite degrees of freedom). There are two different formulations, which differ in the degree of formalization for systems with a finite number of degrees of freedom:

• Newtonian mechanic. It gave origin to the other disciplines and is divided into several of them: the kinematics, study of the movement itself, without addressing the causes that originate it; the static, which studies the balance between forces and the dynamic that is the study of the movement according to its origins, the forces.
• Analytical mechanics, a very powerful mathematical formulation of the Newtonian mechanics based on Hamilton's principle, which employs the formalism of differentiable varieties, specifically the configuration space and the phase space.

Applied to three-dimensional Euclidean space and inertial reference frames, the two formulations are basically equivalent.

The basic assumption that characterizes classical mechanics is predictability: theoretically infinite, mathematically if in a certain moment the positions and speeds of a finite system of N particles can theoretically be known future positions and speeds, since in principle there are vector functions {r→ → i=r→ → i(t;r→ → i(0),v→ → i(0))!i=1N{displaystyle displaystyle {{vec {r}}_{i}={vec {r}}_{i}{i}(t;{vec {r}}_{i}}}{{(0)}},{vec {v}}_{i}{(0)}}_{i=1}{n}{n} which provide the positions of the particles at any time. These functions are obtained from general equations called equations of movement that manifest in a differential way by relationing quantities and their derivatives. Functions {ri→ → (t)!i=1N{displaystyle displaystyle {vec {r_{i}}}(t)}{i=1}^{n}}}} are obtained by integration, once known the physical nature of the problem and the initial conditions.

There are other areas of mechanics that cover various fields although they are not global in nature. They do not form a strong nucleus to be considered as a discipline:

• Continuous media mechanics
• Statistical mechanics

Mechanics of continuums

The mechanics of continuums deals with extensive material bodies that are deformable and cannot be treated as systems with a finite number of degrees of freedom. This part of mechanics deals with:

• The mechanics of deformable solids, which considers the phenomena of elasticity, plasticity, viscoelasticity, etc.
• Fluid mechanics, which includes a set of partial theories such as hydraulic, hydrostatic or fluidstatic, and hydrodynamic or fluiddynamic. Within the study of the flows is distinguished between compressible flow and incompressible flow. If the fluids are treated according to their constituent equation, they have perfect fluids, Newtonian fluids and non-Newtonian fluids.
• Acoustic, classical ondulatory mechanics.

Usual continuum mechanics is a generalization branch of classical mechanics, although during the second half of the XX century relativistic formulations of the continuum were developed, although there is no equivalent quantum analogue since said theory interprets the continuum as particles.

There is also relativistic continuum mechanics, although there are some open problems regarding relativistic generalizations of classical media mechanics. On the other hand, there are no quantum generalizations that are the quantum analogue of continuum mechanics.

Statistical Mechanics

Statistical mechanics deals with systems with many particles and therefore have a large number of degrees of freedom, to the point that it is not possible to write all the equations of motion involved and, failing that,, tries to solve partial aspects of the system by statistical methods that give useful information about the global behavior of the system without specifying what happens with each particle of the system. The results obtained coincide with the results of thermodynamics. It uses both formulations from Hamiltonian mechanics and formulations from probability theory. There are studies of statistical mechanics based on both classical mechanics and quantum mechanics.

Relativistic Mechanics

Relativistic mechanics or theory of relativity includes:

• The Theory of special relativity, which properly describes the classic behavior of the bodies that move at great speeds in a flat (non-curved) space-time.
• The General Theory of Relativity, which generalizes the previous one describing the movement in curved spaces-time, in addition to encompassing a relativistic theory of gravitation that generalizes the theory of Newton's gravitation.

There are several interesting properties of relativistic dynamics, including:

• Force and acceleration are generally not parallel vectors on a curved path, as the relationship between the acceleration and the tangential force is different than that between normal acceleration and force. Nor is the reason between the strength module and the acceleration module constant, since in it appears the reverse of the Lorentz factor, which is decreasing with speed, becoming null at speeds close to the speed of light.
• The time interval measured by different observers in relative motion does not coincide, so there is no absolute time, and there cannot be established a present common to all observers, although there are strict chance relationships.
• Another interesting fact of relativistic mechanics is that it eliminates the action at a distance. The forces that experience a particle in the gravitational or electromagnetic field caused by other particles depend on the position of the particles in an earlier moment, being the "retras" in the influence of some particles on others of the order of distance divided between the speed of light:

Δ Δ t≈ ≈ dc{displaystyle Delta tapprox {frac {d}{c}}}}

However, despite all these differences, relativistic mechanics is much more similar to classical mechanics from a formal point of view, than for example quantum mechanics. Relativistic mechanics remains a strictly deterministic theory.

Quantum mechanics

Quantum mechanics deals with mechanical systems of small scale or with very small energy (and occasionally macroscopic systems that exhibit quantization of some physical magnitude). In such cases the assumptions of classical mechanics are not adequate. In particular, the principle of determination by which the future state of the system depends entirely on the current state does not seem to be valid, so that systems can evolve at certain moments in a non-deterministic way (see postulate IV and collapse of the wave function), since the equations for the quantum mechanical wave function do not allow us to predict the state of the system after a specific measurement, an issue known as the measurement problem. However, determinism is also present because between two filter measurements the system evolves deterministically according to the Schrödinger equation.

Non-deterministic evolution and measurements on a system are governed by a probabilistic approach. In quantum mechanics, this probabilistic approach leads, for example, in the most common approach, to renounce the concept of a particle's trajectory. Worse still, the concept, the Copenhagen interpretation, completely renounces the idea that particles occupy a specific and determined place in space-time. The internal structure of some interesting physical systems such as atoms or molecules can only be explained by quantum treatment, since classical mechanics makes predictions about such systems that contradict physical evidence. In this sense, quantum mechanics is considered a more exact or more fundamental theory than classical mechanics, which is currently only considered a convenient simplification of quantum mechanics for macroscopic bodies.

There is also a quantum statistical mechanics that incorporates quantum constraints in the treatment of particle aggregates.

Relativistic quantum mechanics

Relativistic quantum mechanics tries to join relativistic mechanics and quantum mechanics, although the development of this theory leads to the conclusion that in a relativistic quantum system the number of particles is not conserved and in fact one cannot speak of a mechanics of particles, but simply of a quantum field theory. This theory manages to combine quantum principles and the theory of special relativity (although it fails to incorporate the principles of general relativity). Within this theory, states of the particles are no longer considered, but of space-time. In fact, each of the possible quantum states of space-time is characterized by the number of particles of each type represented by quantum fields and the properties of said fields.

That is, a universe where N_i particles of the type i exist in the quantum states E₁, …, E_Ni represents a different quantum state from another state we observe in the same universe with a different number of particles. But both "states" or aspects of the universe are two of the possible physically realizable quantum states of space-time. In fact, the notion of quantum particle is abandoned in quantum field theory, and this notion is replaced by that of quantum field. A quantum field is an application that assigns a self-adjoint operator to a smooth function over a region of space-time. The smooth function represents the region where the field is measured, and the eigenvalues of the number operator associated with the field the number of observable particles when making a measurement of said field.

Interdisciplinary studies related to mechanics

• Electromechanical engineering, which applies concepts of electromagnetism, electronics, electrical and mechanical sciences.
• Biomechanics, which applies mechanical concepts within biology and medicine.
• Econophysics, which applies technique of statistical mechanics to the economy.
• The ecological economy, which criticizes the application of classical mechanics to the conventional economy.