# Introduction

Matter is at the same time something very mundane and incredibly complex. We deal with material objects all the time and jet when we are press to define what matter is, we end up with a unsatisfactory answer like “matter is that of which things are made of”. And then we can give examples of different materials (type of matter): steel, water, soil, rock, air,…

Our understanding of matter has changed a lot over the last century with new scientific insights, but much is still debated, many of the positions defended have a long history at their backs. This topic has been a central issue of philosophy for over 2000 years, clearly a lot of though has gone into the matter.

# Philosophical understanding of matter

Matter has been a topic of great discussion from the time of the Greeks, and still is central debate between religious people, who in some way or other believe on the existence of non-material, let’s say spiritual, things and atheist or materialists who state that matter is all that exist.

## The discussion of the essence of matter in ancient Greeks

The ancient Greek thought matter was formed by mixing 4 elements: water, air, fire and earth. Different materials from wood to iron and gold were thought to be mixtures on different percentages of the four basic elements. Thales of Miletus thought the first principle was water, this came from the observation that there is moisture everywhere so water is par of everything. His pupil Anaximander said that water couldn’t be the first, principia, since water couldn’t produce fire, the same happened with the other elements, non were able to create their opposite, thus there was no principal element as such. He nonetheless proposed a perfect, unlimited, eternal and indefinite substance, the Aperion, from which all was created. Anaximenes, Anaximandres pupil, returned to the elemental principia, theory but proposed air as the original element from which everything else is created. He said that though rarification air produced fire and through compression water and subsequently earth.

Pythagoras of Samos said that numbers not matter were the origin of everything. Heraclitus said that no matter was possible since all in life is flux and continuous change. On the other hand Parmenides believed that the universe was static and the hold the only truth, our senses however were changing and unreliable, rendering knowledge of truth impossible. Leucippus and Democritus held that matter was composed of indivisible constituents, atoms. As one can see there was a long debate of what matter was even as far back as the 6th century before Christ.

John Dalton discovered the atom in 1803, giving evidence towards the views held by Leucippus and Democrates.

## Modern philosophy: idealism and materialism

The Idealism movement is a group of thinkers which held that reality is principally mental, all we know of the world is what our mind interprets of the world in itself which is out of our reach. The most well known of these thinkers are: Immanuel Kant, G. W. F. Hegel and A. Schopenhauer. In their theories matter is relegated to a secondary place since the world is basically immaterial, or at least our knowledge of the universe is strongly conditioned by our mental process meaning the original essence of matter is not critical since all we ever have a chance to know is mental.

On the opposite side the materialism is a form of philosophical monism which holds that matter is the fundamental substance in nature, and that all phenomena, including mental phenomena and consciousness, are results of material interactions.

There are other philosophical theories that fall in dualistic or pluralistic reality, meaning the world is not purely mental, spiritual or material but is a composition of 2 or more aspects. Descartes is probably one the best known representative of the dualistic nature of reality.

# Scientific understanding of matter

From a scientific perspective in classical mechanics a material object is characterized with a position in space and time and some physical properties like volume and mass. All of these properties of matter have a very specific definition and meaning but the only definition of matter that will extract from mechanics is: “matter is that which lays on a position in space and time and occupies a volume and has mass” .

If we go further into quantum and relativistic mechanics we find that ordinary matter is structured, formed by atoms which themselves are composed of particles: electrons , neutrons and protons. neutrons and protons are also composite particles each consisting of 3 quarks. Other more exotic forms of matter exist, formed by all sorts of particles: muons, tauons, neutrinos (3 flavous), mesons (formed by a quark anti quark pair)… In addition we know from relativity that mass and energy are really part of the same thing so massless particles like photons also qualify as matter. From this perspective matter is formed from a sea of particles which in turn are just things that occupy a position in spacetime and have some physical properties like mass, electric charge and spin.

As we can see science is good at telling us the structure of matter, which are its constructive blocks, but can’t really answer what matter is. This is a consequence of the scientific method, through which hypothesis are falsified and theoretical predictions are verified. This process ensures that the lasting theory has endured and all the predictions based on it have been verified, this however doesn’t mean that some future prediction may fail meaning a new refinement of the theory is required. As a consequence what science can definitely say is how the real world is not. It can’t be anything that produces falsifiable results, since these results are a definite prove that the world is not how we propose. Thats why science will never be able to answer what anything is, only what it is not.

In this sense matter is not a continuous media, since it is made of discrete pieces (subatomic particles). Matter is not static since these particles are in continuous motion. Matter is definitely not mass since mass is only a measure of a body’s resistance to change of motion, inertia, which is a property of most matter (all except mass-less particles, like photons). Different type of matter interact with each other in different ways by four forces: gravitation, electromagnetism, weak force and strong force.

# Discusion

Today, in our scientific worldview most people have a materialistic perspective on nature and life. All matter is made of atoms and all that is or ever has been is made of the particles that constitute the standard model and possible some other particles not jet discovered. This begs the question of whether abstract concepts exist, and by “abstract concepts” here I include such things as chairs and tables, not only truly abstract ones as goodness and happiness. These concepts exist in our minds but not in nature, in a materialistic explanation these concepts don’t really exist they are generated from chemical reactions in our brains the same way awareness arises. And here is were the opposites touch, in a materialistic perspective abstract concepts don’t exist, because the are non-material, from the point of view of idealism or dualism they do exist but in a different “realm of ideas”. Both agree that these concepts are non-material and, in the sense that we use them every day, they undoubtedly must exist in some non-material way. The only difference is whether we disregard this non-material existence as non-existence. So in an ontological way the difference is really not that great as initially expected.

## El Cáos a través del péndulo doble

El péndulo doble es uno de los sistemas más sencillos, cualquiera puede construirse uno en su casa con dos masas sujetadas de dos barras, sin embargo demuestra la complejidad de la mecánica en la naturaleza. La imagen en la portada (fuente original) muestra lo compleja que resulta la trayectoria de este segundo péndulo; el primer péndulo recorre arcos de circunferencia, trayectoria roja sin embargo al segundo muestra toda clase de quiebros inesperados, linea amarilla.

Se trata de un sistema donde el comportamiento caótico se presenta de una forma aparente hasta para ángulos de desplazamiento iniciales no muy grandes.

Lo primero que haremos es deducir las ecuaciones del movimiento del péndulo doble. Esto llevará una buena parte del post deduciremos las ecuaciones por dos procedimientos distintos, el Newtoniano y el Lagrangiano. Después exploraremos en que sentido es el movimiento caótico a través del análisis de resultados de la simulación de las ecuaciones deducidas para distintas condiciones iniciales.

## Método Newtoniano

Vamos a empezar por plantear el problema desde un punto de vista Newtoniano. Veremos después como el planteamiento se estandariza y facilita si se plantea desde un punto de vista Lagrangiano.

Presentamos en la Figura 1 un esquema de fuerzas y aceleraciones que actúan sobre el sistema.

Figura 1: a) Esquema de fuerzas b) Esquema de aceleraciones, en un péndulo doble.

Según la segunda ley de Newton $F=ma$ (fuerza igual a masa por aceleración) en este caso aplicamos esta ley a las fuerzas de cada una de las masas en la direción vertical (y) y horizontal (x). Tenemos:

$F_2\mathrm{sin}{\theta}_2-F_1\mathrm{sin}{\theta}_1=m_1a_{x1}$

$F_1\mathrm{cos}{\theta}_1-F_2\mathrm{cos}{\theta}_2-m_1g=m_1a_{y1}$

$-F_2\mathrm{sin}{\theta}_2=m_2a_{x2}$

$F_2\mathrm{cos}{\theta}_2-m_2g=m_2a_{y2}$

De la tercera de estas 4 ecuaciones sacamos:

$F_2=-\frac{m_2a_{x2}}{\mathrm{sin}{\theta}_2}$

De la primera sustituyendo el valor obtenido pata $F_2$ tenemos:

$F_1=-\frac{m_1a_{x1}+m_2a_{x2}}{\mathrm{sin}{\theta}_1}$

De las dos ecuaciones restantes, la segunda y la cuarta obtenemos las ecuaciones que rigen el sistema:

$m_1a_{x1}+m_1g-\frac{m_2a_{x2}}{\mathrm{tan}{\theta}_2}+\frac{m_1a_{x1}}{\mathrm{tan}{\theta}_1}$

$a_{y2}+\frac{a_{x2}}{\mathrm{tan}{\theta}_2}+g=0$

Ahora solo queda deducir las aceleraciones del sistema, para ello partimos de las posiciones de los péndulos, las derivamos con respecto al tiempo para obtener las velocidades y las volvemos a derivar para obtener aceleraciones.

Empezamos por las posiciones, de la Figura 1b, se observa (tened en cuenta que el eje y lo consideramos positivo en el sentido ascendente y el origen esta en el punto fijo del péndulo):

$x_1=L_1\mathrm{sin}{\theta}_1$

$y_1=-L_1\mathrm{cos}{\theta}_1$

$x_2=L_1\mathrm{sin}{\theta}_1+L_2\mathrm{sin}{\theta}_2$

$y_2=-L_1\mathrm{cos}{\theta}_1-L_2\mathrm{cos}{\theta}_2$

Ahora derivamos con respecto al tiempo y obtenemos las velocidades:

$v_{x1}=L_1\mathrm{cos}{\theta}_1\dot{{\theta}_1}$

$v_{y1}=L_1\mathrm{sin}{\theta}_1\dot{{\theta}_1}$

$v_{x2}=L_1\mathrm{cos}{\theta}_1\dot{{\theta}_1}+L_2\mathrm{cos}{\theta}_2\dot{{\theta}_2}$

$v_{y2}=L_1\mathrm{sin}{\theta}_1\dot{{\theta}_1}+L_2\mathrm{sin}{\theta}_2\dot{{\theta}_2}$

dónde usamos la notación  $\dot{z}$ para denotar la derivada temporal de $z$. Por último realizamos la segunda derivada:

$a_{x1}=L_1(\mathrm{cos}{\theta}_1\ddot{{\theta}_1}-\mathrm{sin}{\theta}_1\dot{{\theta}_1)}$

$a_{y1}=L_1(\mathrm{sin}{\theta}_1\ddot{{\theta}_1}+\mathrm{cos}{\theta}_1\dot{{\theta}_1})$

$a_{x2}=L_1(\mathrm{cos}{\theta}_1\ddot{{\theta}_1}-\mathrm{sin}{\theta}_1\dot{{\theta}_1})+L_2(\mathrm{cos}{\theta}_2\ddot{{\theta}_2}-\mathrm{sin}{\theta}_2\dot{{\theta}_2})$

$a_{y2}=L_1(\mathrm{sin}{\theta}_1\ddot{{\theta}_1}+\mathrm{cos}{\theta}_1\dot{{\theta}_1})+L_2(\mathrm{sin}{\theta}_2\ddot{{\theta}_2} +\mathrm{cos}{\theta}_2\dot{{\theta}_2})$

## Método Lagrangiano

La teoría de Euler-LaGrange dice que en mecánica se cumple la ecuación

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0$

Donde $L$ es el Lagrangiano, $x$ es una coordenada generalizada del sistema y $\dot{x}$ su derivada temporal. En nuestro problema coordenadas generalizadas son las posiciones angulares, ${\theta }_1$ y${\theta }_2$. La ley de Euler-LaGrange se demuestra desde los principios newtonianos y es equivalente a aquellos. Se trata de un planteamiento desde el punto de vista energético.

El Lagrangiano se define como:

$L=T-V$

Donde $T$ es la energía cinética y V la energía potencial del sistema.

Recordemos que la energía potencial gravitatoria es $V=-mgy$  donde $y$ es la altura (signo menos porque hemos elegido el eje y hacia abajo).

$y_1=L_1{\mathrm{cos} {\theta }_1\ }$
$y_1=L_1{\mathrm{cos} {\theta }_1\ }+L_2{\mathrm{cos} {\theta }_2\ }$
$V=-m_1gy_1-m_2gy_2$
$V=-m_1gL_1{\mathrm{cos} {\theta }_1\ }-m_2g\left(L_1{\mathrm{cos} {\theta }_1\ }+L_2{\mathrm{cos} {\theta }_2\ }\right)$
$V=-g\left[\left(m_1+m_2\right)L_1{\mathrm{cos} {\theta }_1\ }+{m_2L}_2{\mathrm{cos} {\theta }_2\ }\right]$

La energía cinética es un poco más difícil de sacar. Recordemos que

$T=\frac{1}{2}mv^2$

La velocidad de la masa 1 es inmediata.

$v_1=L_1\dot{{\theta }_1}$

Sin embargo el de la masa 2 resulta de la suma vectorial de la velocidad 1 más la velocidad de la masa dos relativa al sistema no-inercial.

$\overrightarrow{v_2}=\overrightarrow{v_1}+\overrightarrow{v_r}$
$v_r=L_2\dot{{\theta }_2}$

Es fácil de ver que el ángulo que forman $\overrightarrow{v_1}$ y $\overrightarrow{v_r}$ es ${\theta }_2-{\theta }_1$ y por tanto por el teorema del coseno:

${v_2}^2={\left(L_1\dot{{\theta }_1}\right)}^2+{\left(L_2\dot{{\theta }_2}\right)}^2+2L_1L_2\dot{{\theta }_1}\dot{{\theta }_2}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }$

Por tanto la energía cinética es:

$T=\frac{1}{2}m_1{v_1}^2+\frac{1}{2}m_2{v_2}^2$
$T=\frac{1}{2}m_1{L_1}^2{\dot{{\theta }_1}}^2+\frac{1}{2}m_2\left[{\left(L_1\dot{{\theta }_1}\right)}^2+{\left(L_2\dot{{\theta }_2}\right)}^2+2L_1L_2\dot{{\theta }_1}\dot{{\theta }_2}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }\right]$

El Lagrangiano por tanto:

$L=\frac{1}{2}(m_1+m_2){L_1}^2{\dot{{\theta }_1}}^2+\frac{1}{2}m_2\left[{\left(L_2\dot{{\theta }_2}\right)}^2+2L_1L_2\dot{{\theta }_1}\dot{{\theta }_2}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }\right]+ g\left[\left(m_1+m_2\right)L_1{\mathrm{cos} {\theta }_1\ }+{m_2L}_2{\mathrm{cos} {\theta }_2\ }\right]$

El siguiente paso por tanto es calcular las derivadas parciales del Lagrangiano como función $L({\theta }_1,{\theta }_2,\ \dot{{\theta }_1}\dot{{,\theta }_2})$:

$\frac{\partial L}{\partial {\theta }_1}=m_2L_1L_2\dot{{\theta }_1}\dot{{\theta }_2}{\mathrm{sin} \left({\theta }_2-{\theta }_1\right)\ }-g\left(m_1+m_2\right)L_1{\mathrm{sin} {\theta }_1\ }$
$\frac{\partial L}{\partial {\theta }_2}=-m_2L_1L_2\dot{{\theta }_1}\dot{{\theta }_2}{\mathrm{sin} \left({\theta }_2-{\theta }_1\right)\ }-gm_2L_2{\mathrm{sin} {\theta }_1\ }$
$\frac{\partial L}{\partial \dot{{\theta }_1}}=\left(m_1+m_2\right){L_1}^2\dot{{\theta }_1}+m_2L_1L_2\dot{{\theta }_2}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }$
$\frac{\partial L}{\partial \dot{{\theta }_2}}=m_2{L_2}^2\dot{{\theta }_2}+m_2L_1L_2\dot{{\theta }_1}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }$

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{{\theta }_1}}\right)=\left(m_1+m_2\right){L_1}^2\ddot{{\theta }_1}+m_2L_1L_2\left[\ddot{{\theta }_2}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }-\dot{{\theta }_2}(\dot{{\theta }_2}-\dot{{\theta }_1}){\mathrm{sin} \left({\theta }_2-{\theta }_1\right)\ }\right]$

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{{\theta }_2}}\right)=m_2{L_2}^2\ddot{{\theta }_2}+m_2L_1L_2\left[\ddot{{\theta }_1}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }-\dot{{\theta }_1}(\dot{{\theta }_2}-\dot{{\theta }_1}){\mathrm{sin} \left({\theta }_2-{\theta }_1\right)\ }\right]$

Por tanto, aplicando $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{{\theta }_i}}\right)-\frac{\partial L}{\partial {\theta }_i}=0$, las dos ecuaciones que rigen el sistema (simplificando los términos que se cancelan) son:

$\left(m_1+m_2\right)L_1\ddot{{\theta }_1}+m_2L_2\left[\ddot{{\theta }_2}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }-{\dot{{\theta }_2}}^2{\mathrm{sin} \left({\theta }_2-{\theta }_1\right)\ }\right]+g\left(m_1+m_2\right){\mathrm{sin} {\theta }_1\ }=0$

$L_2\ddot{{\theta }_2}+L_1\left[\ddot{{\theta }_1}{\mathrm{cos} \left({\theta }_2-{\theta }_1\right)\ }-{\dot{{\theta }_1}}^2{\mathrm{sin} \left({\theta }_2-{\theta }_1\right)\ }\right]+g{\mathrm{sin} {\theta }_1\ }=0$

Para que estén escritas de la misma manera que en el apartado anterior solo habría que despejar $\ddot{{\theta }_1}$ y $\ddot{{\theta }_2}$, cosa que vamos a dejar para el lector.

La ventaja del método Lagrangiano es que no requiere calcular las fuerzas que soportan los péndulos que es un proceso engorroso. Además la manera de llegar hasta las ecuaciones es siempre igual, a cambio requiere realizar varias derivadas.

## Comportamiento Caótico del sistema

En Física se dice que un sistema es caótico cuando pequeñas diferencias en las condiciones iniciales del sistema conducen a situaciones muy distintas con el trascurso del tiempo. Para mostrar esto voy a simular dos casos en el primero los péndulos van a soltarse con unos ángulos: ${\theta}_1=30^{\circ} \quad (\frac{\pi}{6} \mathrm{rad})$ y ${\theta}_2=60^{\circ} \quad (\frac{\pi}{3} \mathrm{rad})$; el segundo se soltara desde:${\theta}_1=31^{\circ} \quad (\frac{31\pi}{180} \mathrm{rad})$ y ${\theta}_2=61^{\circ} \quad (\frac{61\pi}{3} \mathrm{rad})$. Una diferencia de tan solo 1º produce diferencias importantes en tan solo 3s de simulación. De la misma manera diferencias de 1″ llevarían a errores inaceptables al cabo de unos minutos. Los péndulos simulados tienen una longitud de 1 m y unas masas de 1 kg.

De la Figura 3 vemos que los dos sistemas se siguen bastante bien durante 1.5 s o así, a partir de ahí empiezan a separarse bastante bruscamente. La Figura 2 muestra la diferencia entre el caso cuyas condiciones iniciales son (30º, 60º) y aquel que tiene (31º,61º). Durante los primeros 1.75s o así las graficás de la Figura 2 estan bastante cerca de cero, indicando que la diferencia entre los dos sistemas es escasa. Sin embargo a los 2s diverge sustancialmente ya que llegan a mas de 2 rad (114º) de diferencia. Observar que la máxima diferencia posible es de 180º, significando que estan en la posición opuesta.

Notar que un error de 1º en la posición de un péndulo de 1m es bastante grande ya que equivale a unos 17 mm de desplazamiento, por esta razón los dos casos divergen a los pocos segundos.