Classical mechanics In Physics

In the fields of physics, classical mechanics is one of the two main sub-fields of study in the science of mechanics, which has to do with the set of governing physical laws and mathematics that describes the movements of bodies. and the aggregates of bodies geometrically distributed within a limit determined by the action of a system of forces. The other subfield is quantum mechanics.

Classical mechanics is used to describe the microscopic motion of objects, from projectiles to parts of machinery, as well as astronomical objects such as ships, planets, stars, and galaxies. It produces highly accurate results within these domains, and is one of the oldest and largest subjects in science, engineering, and technology. In addition to this, many related specialties exist that deal with gases, liquids and solids, and so on. Furthermore, classical mechanics is enhanced by special relativity for the high speed of objects approaching the speed of light. General relativity is used to control gravity at a deeper level, and finally quantum mechanics deals with the wave-particle duality of atoms and molecules.

The term classical mechanics was coined in the 20th century to describe the system of mathematical physics initiated by Isaac Newton and many 17th-century contemporaries as philosophers of nature, based on the earlier astronomical theories of Johannes Kepler, which in turn were based in the precise observations of Tycho Brahe and Galileo’s projectile motion studies of terrestrial ecosystems, but before the development of quantum physics and relativity. Therefore, some sources excluded the so-called »relativistic physicists« from that category. However, a number of modern sources include Einstein’s mechanics, which he believes represents classical mechanics in their most developed and most accurate form.

The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed and the mathematical methods invented by Newton himself, in parallel with Leibniz, among others. This is described in the following sections. Abstract and general of most methods including Lagrangian mechanics and Hamiltonian mechanics. Much of the content of classical mechanics was created in the 18th and 19th centuries and extends far beyond (particularly in the use of analytical mathematics) Newton’s work.

The classical mechanics (also known as mechanics of Newton , named in honor of Isaac Newton , who made fundamental contributions to the theory) is the part of physics that analyzes the forces acting on an object. Classical mechanics is subdivided into the branches of statics , which deals with objects in equilibrium (objects that are considered in a reference system in which they stand), and dynamics , which deals with objects that are not in equilibrium (objects in movement ). Classical Mechanics reduces its study to the domain ofdaily experience , that is, with events that we see or feel with our senses. It has several extensions: Relativistic mechanics goes beyond classical mechanics and deals with objects moving at high speeds (of a value relatively close to the speed of light ). The quantum mechanics deals with systems of reduced dimensions (on this scale atomic), and the theory of the field quantum treated with systems that exhibit both properties.

Even being an approximation , classical mechanics is very useful since it is much easier to understand (and mathematically much easier to compute), and therefore easier to apply, being sufficiently valid for the vast majority of practical cases in a large number of very diverse systems. The theory, for example, describes with great accuracy systems such as rockets, planets, organic molecules, spinning tops, trains, and also the trajectory of a soccer ball.

Classical mechanics is widely compatible with other classical theories such as electromagnetism , and thermodynamics , also “classical” (these theories also have their corresponding quantum).

Table of Contents

Index

[ hide ]

oneDescription of the Theory
twoMagnitudes of Position and Positions
3Forces
4Energy
5Other results
6Formalization
7References

Description of the Theory

Magnitudes of Position and Positions

We denote the position of an object with the vector Cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http : //api.formulasearchengine.com/v1/ »:): {\ displaystyle \ vec {r}} , with respect to a fixed point in space. Yes Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http://api.formulasearchengine.com / v1 / »:): {\ displaystyle \ vec {r}} is a function of time t denoted byCannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http://api.formulasearchengine.com/ v1 / »:): {\ displaystyle \ vec {r}} (t), the time t is taken from an arbitrary initial time. So we have to speed, Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http: // api.formulasearchengine.com/v1/ »:): {\ displaystyle \ vec {v}} (also a vector since it has magnitude and direction) is denoted by:

Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http://api.formulasearchengine.com/ v1 / »:): {\ displaystyle \ vec {v} = \ frac {d \ vec {r}} {dt}}

The acceleration, or the amount of change in velocity (the derivative of v ) is:

The position indicates the place of the object that we are analyzing. If this object changes places, the function Cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http : //api.formulasearchengine.com/v1/ »:): {\ displaystyle \ vec {r}} describes the new location of the object. These amounts cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http: //api.formulasearchengine. com / v1 / »:): {\ displaystyle \ vec {r}} ,Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http://api.formulasearchengine.com/ v1 / »:): {\ displaystyle \ vec {v}} , and Cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (” Math extension cannot connect to Restbase. ») From the server« http://api.formulasearchengine.com/v1/ »:): {\ displaystyle \ vec {a}} can be described approximately, that is without using differential calculation, but the results are only approximatesince all these functions and quantities are defined according to the calculation. However, these approximations will give us an easier understanding of the equations.

If for example we did an experiment and we could measure time (t), and we could know the position of an object ( Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid answer (“Math extension cannot connect to Restbase.”) From the server “http://api.formulasearchengine.com/v1/” :): {\ displaystyle \ vec {r}} ) at that time (t), we could define the earlier amounts more easily. We first denote the initial time as t ₀ which is when we start the timer for our experiment, and we denote the final time simply as the _final tot . If we denote the initial position asCannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http://api.formulasearchengine.com/ v1 / »:): {\ displaystyle \ vec {r} _0} , so we designate the end position with the symbol Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): answer no valid (“Math extension cannot connect to Restbase.”) from server “http://api.formulasearchengine.com/v1/” :): {\ displaystyle \ vec {r}} orCannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http://api.formulasearchengine.com/ v1 / »:): {\ displaystyle \ vec {r} _ {final}} . Now, having defined the fundamental quantities, we can express the physical quantities in approximate terms as follows.

The velocity of the object is denoted by:

also with the expression:

Acceleration is denoted by

Forces

The Fundamental Principle of Dynamics (Newton’s Second Principle) relates the mass and velocity of a body to a vector quantity, force . If m is assumed to be the mass of a body and Cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) Of server “http://api.formulasearchengine.com/v1/” :): {\ displaystyle \ vec {F}} the vector resulting from adding all the forces applied to it (resulting or net force), then

where m is not necessarily independent of t. For example, a rocket expels gases decreasing the mass of fuel and therefore its total mass, which decreases as a function of time. To quantity Cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http: //api.formulasearchengine .com / v1 / »:): {\ displaystyle m \ vec {v}} is called momentum or momentum . When m is independent of t (as is frequent), the previous equation becomes:

The exact form of Cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http: // api. formulasearchengine.com/v1/ »:): {\ displaystyle \ vec {F}} is obtained from considerations about the particular circumstance of the object. Newton’s third law gives a particular indication about Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server « http://api.formulasearchengine.com/v1/ »:): {\ displaystyle \ vec {F}} : if a body A exerts a forceCannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http://api.formulasearchengine.com/ v1 / »:): {\ displaystyle \ vec {F}} on another body B, then B exerts a force (called reaction) of the same direction and opposite direction on A, It cannot be interpreted (MathML with SVG or PNG as alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) from server “http://api.formulasearchengine.com/v1/” :): {\ displaystyle – \ vec {F}} (Newton’s third principle or principle of action and reaction).

An example of a force is friction or friction , which for movement within gases is a function of the velocity of the particle (although this effect is neglected at small speeds). For example:

where k is a positive constant. If we have a relation for Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http: // api .formulasearchengine.com / v1 / »:): {\ displaystyle \ vec {F}} similar to the one above, can be substituted in Newton’s second law to obtain a differential equation, called equation of motion. If friction is the only force acting on the object, the equation of motion is:

What can be integrated to obtain:

where Cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) from server “http://api.formulasearchengine.com / v1 / »:): {\ displaystyle \ vec {v} _0} is the initial speed (a limit condition in the integration). This tells us that the velocity of this body decreases exponentially to zero. This expression can be re-integrated to get Cannot interpret (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http: //api.formulasearchengine.com/v1/ »:): {\ displaystyle \ vec {r}} .

The absence of forces, when applying Newton’s second principle, leads us to the conclusion that the acceleration is null (Newton’s first principle or Principle of inertia)

Important forces are the gravitational force (the force that results from the gravitational field ) or the Lorentz force (in the electromagnetic field ).

Energy

If a force cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): invalid response (“Math extension cannot connect to Restbase.”) From server “http: //api.formulasearchengine .com / v1 / »:): {\ displaystyle \ vec {F}} is applied to a body that scrolls Cannot be interpreted (MathML with SVG or PNG as an alternative (recommended for modern browsers and accessibility tools): Invalid response (“Math extension cannot connect to Restbase.”) from server “http://api.formulasearchengine.com/v1/” :): {\ displaystyle d \ vec {r}} , work done by force is a scalar magnitude of value:

If the mass of the body is assumed to be constant, and _total dW is the total work done on the body, obtained by adding the work done by each of the forces acting on it, then, applying Newton’s second law, can demonstrate that:

where T is the so-called kinetic energy . For a point particle, T is defined:

For large objects made up of many particles, the kinetic energy is the sum of the kinetic energies of the constituent particles.

A particular type of forces, known as conservative forces, can be expressed as the gradient of a scalar function, called potential energy , V:

If all conservative forces on a body are assumed, and V is the potential energy of the body (obtained by summing the potential energies of each point due to each force), then:

This result is known as the law of conservation of energy , indicating that the toral energy E = T + V is constant (not a function of time).

Other results

Newton’s second law allows various other results to be obtained, in turn considered as laws. See for example angular momentum .

Formalization

There are two important alternative formalizations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics . They are equivalent to Newton’s laws and their consequences, but they are more practical for solving complex problems than directly applying them.