In details

Symmetry, Anti-Symmetry, and Symmetry Breaking V


Therefore, a vector can represent a nature phenomenon and operations between vectors continue to represent nature phenomena. It is inevitable, then, to ask: what other operations between vectors represent nature's phenomena? Which vector algebras are able to unravel the behavior of nature? How far can Homo sapiens sapiens go in this investigation of nature?

Symmetry, Anti-Symmetry, and Symmetry Breaking IV

The sine functions f(what) = The sen (wq + what0) are mathematical models for electromagnetic signals or waves, for example, where THE is the amplitude, w is the signal frequency and what0 It is the initial phase. These mathematical models are very suitable for describing the voltages provided by alternating current generators in steady state circuits.

Given a reasonably well behaved periodic signal, we can describe it as an infinite sum of sinusoidal functions (remember that cosine functions are also a type of sinusoidal) as discovered by Joseph Fourier (1768-1830), a French mathematician very close to Napoleon, and the first to undertake a systematic study of approximations of functions by trigonometric series. In 1822 published his famous work Theorie Analytique de la Chaleur. Daniel Bernoulli (1700-1782) had already studied this kind of approach to solving vibrant string problems (1747). In 1824 Fourier obtained the infinite sum that describes the movement of a heat wave through a body.

Fourier Analysis, an important branch of contemporary mathematics that developed as a result of Fourier's discovery, is not a trivial study. In particular, dealing with simple electrical circuits mathematically is a nontrivial task. However, engineers have found that complex numbers, even if considered by many to be “imaginary” numbers, could greatly simplify the mathematical treatment of electrical circuits.

An RLC circuit, ie with resistor, coil and capacitor, in the case where the electromotive force (f.e.m.) and(t) is a sinusoid (or a co-sinusoid), an important case of alternating current, admits an elegant and simple solution to the charge that circulates in it, by an algebraic treatment using the plane vectors subordinated to the complex multiplication. .

It is the treatment of the elements of an electrical circuit by phasors. However, let us remember before the Kirchhoff Laws.

The German physicist Gustav Robert Kirchhoff (1824-1887) enunciated in 1845 the Laws that allow equating currents, voltages and resistances of electrical circuits. That is, in the resistance to voltage variation rate dVR/dt is proportional to the current i = dQ/dt in the instant twhere we wrote from:

dVR/dt = R i = R dQ/dt;

on the capacitor, the rate of change of voltage dVÇ/dt is inversely proportional to the load Q(t) at the moment twhere we wrote from:

dVÇ/dt = Q/Ç;

and, in the coil, the voltage change rate dVL/dt is proportional to the rate of change of current di/dt in the instant twhere we wrote from:

dVL/dt = L di/dt = L d2Q/dt2.

We are using the famous notation dy/dx of the German philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716) for the instantaneous rate of change of a quantity y due to another x. We observed that the second instantaneous rate of change d2Q/dt2 of the electric charge Q is the first instantaneous rate of change di/dt of the electric current i relative to time t.

Consider the second order Linear Ordinary Differential Equation (EDOL) given by the idea that the sum of the voltage drops is equal to f.e.m. supplied to the circuit:

L d2Q1/dt2 + R dQ1/dt + Q1 = V cos (w t)

Where V is the maximum value of f.e.m. provided.

In order to use Leonhard Euler's famous and beautiful formula (andjx = waistband x + j sen x), Where j is the complex unit such that j2 = -1, we imagine a hidden symmetry for the above electrical equation, which is nothing more than the same circuit receiving a f.e.m. given by V sen (w t), which will therefore respond with a Q2 and a current i2.

We then complete the complex symmetry of this EDOL by adding to it its imaginary symmetrical complement, that is, the equation:

L d2Q2/dt2 + R dQ2/dt + Q2 = V sen (w t).

The imaginary charge Q2 symmetrically connected to the load Q1 forms a complex charge Q just so we can use Euler's formula. This formula shows that solving one of these equations automatically means solving its symmetric one. This is due to the flat vector notation and Euler's complex notation that allow the two equations to be merged into one. Using quotation marks for notation dy/dx from Leibniz, we have:

L(Q1´´ + jQ2´´) + R(Q1´ + jQ2´) + (Q1 + jQ2) = V cos (w t) + jV sin (w t),

that is, we complexify the electric charges and the f.e.m.´s. Thus we have:

LQ´´ + RQ´ + Ç-1Q = V ejwt.

with Q = Q1 + jQ2, Q´ = Q1´ + jQ2´, Q´´ = Q1´´ + jQ2and andjwt = V (waistband (w t) + j sen (w t)).

This symmetrization is equally valid for the famous and important EDOL known as the spring-mass system, old known to engineers, where the external force applied to the system, which plays the role of f.e.m. applied to the electrical circuit, is periodic in the manner V cos (w t) or V sin (w t).

EDOL's complex electrical solution is a phasor, that is, a complex electrical charge that provides a complex electrical current i(t) = K andjwt. Therefore, an analogous complex spring-mass system can also be solved by phasors if the external force applied to the system is periodic as follows. V cos (w t) or V sen (w t).

What is our practical gain with this complex vector imagination of a phasor, that is, of a complex electric current i(t) = K andjwt? Thanks to the property that the instantaneous rate of change of the exponential function is itself, the practical gain is enormous.

Suppose that the complex vector electrical EDOL solution is a phasor, that is, a complex vector electrical charge that can be represented by a complex expression. So:

(t) = K andjwt = i(t) Þ Q(t) = ò K andwtj dt = - jw-1 K andwtj.

Calculating the first instantaneous rate of change of the current we have:

i(t) = K andwtj Þ i´(t) = w j K andwtj.

Then, substituting these expressions into complex vector EDOL, we get:

L (w j K andwtj) + R K andwtj - Ç-1 (jw-1 K andwtj) = V ewtj.

Dividing the two members of the equation by andwtjcomes that:

L (w j K) + RK - Ç-1 jw-1 K = V.

That is, the complex constant K is given by:

K = V / R + (Lw - Ç-1w-1) j Û K = V R - (Lw - Ç-1w-1) j / R2 + (Lw - Ç-1w-1)2.

So we can write: i(t) = i1(t) + j i2(t) = K andwtj Þ

i(t) = i1(t) + j i2(t) = V R + (Ç-1w-1 - Lw) j waistband(w t) + j sen(w t) / R2 + (Lw - Ç-1w-1)2,

from which we simultaneously conclude that:

i1(t) = V R waistband (w t) - (Ç-1w-1 - Lw) sen (w t) / R2 + (Lw - Ç-1w-1)2,

i2(t) = V (Ç-1w-1 - Lw) waistband (w t) + R sen (w t)/R2 + (Lw - Ç-1w-1)2.

To obtain the actual loads demanded, we simply integrate the actual currents:

Q1(t) = Vw-1 R sen(w t) + (Ç-1w-1 - Lw) waistband(w t)/ R2 + (Lw - Ç-1w-1,

Q2(t) = Vw-1 (Ç-1w-1 - Lw) sen(w t) - R waistband(w t)/R2 + (Lw - Ç-1w-1.

Engineers have found that there are even more interesting gains from defining a complex impedance. We define the complex impedance of the RLC circuit to be the quotient

Z = V/I = V ejwt / K andjwt = V ejwt / {V andjwt / R + (Lw - Ç-1w-1) j} = R + (Lw - Ç-1w-1) j.

Therefore, the complex impedance Z = R + (Lw - Ç-1w-1) j contains in its real part the resistance R of the circuit and in its imaginary part the constant L of inductance of the coil, the constant Ç capacitor and frequency w from f.e.m. supplied to the circuit.

From there the engineers greatly simplified their treatment of electrical circuits and succeeded in even more comprehensive and effective mathematical developments in circuit and electrical network analysis.

We are interested here to reflect on the intriguing ability of flat vectors, or complex numbers, to describe nature's behavior clearly, simply and efficiently, for example, as we saw above, in the case of simple electrical circuits or certain spring-mass systems. . Therefore, we return to our observation that a vector can represent a phenomenon of nature and operations between vectors continue to represent phenomena of nature. Even, as we saw above, their rates of instantaneous variation may still represent natural phenomena. Thus, phasors can represent an electric voltage, an electric charge, an electric current, or an electrical impedance, and the algebraic operations between them continue to signify behaviors from Nature. It is inevitable, then, to keep asking: what other operations between vectors represent nature's phenomena? Which vector algebras are able to unravel the behavior of nature?

How far can Homo sapiens sapiens go in this line of nature research?

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