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Physics seeks to understand how systems function by analyzing their components and interactions, formulating hypotheses and mathematical models that describe their behaviors.

To establish a hypothesis, one can use thought experiments that help visualize the phenomenon, while the equations making up the model are directly derived from these working hypotheses.

When applying these models, its essential to perform calculations efficiently, so we propose a framework to facilitate this process. This framework organizes a models equations and variables into an interconnected network. As calculations are automated, this network allows for the identification of a strategy to solve the problem and then uses that same network to efficiently calculate the target value.

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The goal of physics is to understand how things work. To achieve this, we need to understand the components of a physical system, how its different parts interact, and how they accomplish the observed outcome. René Descartes represented these ideas graphically, as shown here in his diagram explaining binocular vision:

Diagram of the functioning of binocular vision and the pineal gland [1].

However, to be confident that our verbal explanation is accurate, we must define measurable variables associated with the elements described, develop a mathematical representation of the explanation, calculate our predictions, and verify them empirically by comparing the measured values of the variables with the calculated values.

The descriptive part, which forms our hypothesis and outlines our model of reality, is crucial for starting the modeling process. The mathematical framework is essential for analyzing and evaluating whether the obtained values align with observed ones, thus validating our explanation according to the current state of knowledge. Moreover, these mathematical models become tools that ultimately enable the practical application of the model.

[1] "Traité de l'homme" (The Treatise of Man), René Descartes, 1664.

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The first challenge in developing a hypothesis is to establish a clear idea of how the physical system works. Often, this does not involve straightforward mechanical processes where one simply describes a mechanism; instead, these are often abstract ideas that are hard to identify and formulate. One way to arrive at a hypothesis is through thought experiments, where a physical system is defined and, using logic, different valid or true scenarios are explored to draw conclusions about a general behavior or law.

A classic example of this reasoning is how Galileo Galilei concluded that all objects fall in the Earth's gravitational field with the same acceleration, contradicting the popular belief that heavier objects fall faster than lighter ones.

Galileo Galileis Thought Experiment [2].

Galileo's thought experiment [1] imagined two objects of different masses falling, with the heavier one positioned above the lighter one. If the heavier object truly fell faster, it would catch up to the lighter one, and they would then move together. In this case, the new system's velocity would fall between the two original velocities, but as the combined mass would be greater than the heavier object alone, the speed should logically increase. This contradiction is only resolved if both objects fall at the same rate, suggesting that the fall of objects in a gravitational field is independent of their mass.

This reasoning led Galileo to conclude that gravitational acceleration acts equally on all objects, regardless of their mass. The famous story of Galileo dropping objects from the Leaning Tower of Pisa is likely a legend, as precise enough instruments were not available at the time to measure such subtle differences. Today, we understand why this idea seemed counterintuitive: people perceived that heavier objects fell faster because lighter ones are more affected by air resistance, which tends to slow them down. Later, Newton showed that although gravitational force depends on mass, so does inertia, balancing the effect and canceling out the mass dependency in free fall.

Another physicist who intensely used thought experiments was Albert Einstein, who developed much of the theory of special and general relativity based on numerous thought experiments. Like Galileo, Einstein also lacked instruments to measure extremely short times at the beginning of the 20th century, so many of the hypotheses on which he based his theory were only empirically confirmed years after he proposed them.

[1] "Discorsi e dimostrazioni matematiche intorno a due nuove scienze" (Mathematical Discourses and Demonstrations Relating to Two New Sciences), Galileo Galilei, 1638.

[2] MikeRun, CC BY-SA 4.0 , via Wikimedia Commons

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In general, a hypothesis is often a statement that can be directly linked to a mathematical expression. In some cases, this association is enough to develop the model and explore its implications. In other cases, existing mathematical tools may be limited, requiring the development of new methods to support the calculations needed for the model.

For example, when Galileo Galilei discovered that objects fall with an acceleration independent of their mass, he introduced the concept of acceleration $a$, which in this case is constant. The constant that we now call $g$, estimated at $9.81 m/s^2$, was first measured by Galileo, leading to the fundamental equation:

$a = g$

Although this expression may seem simple, it was crucial in beginning to understand how objects behave under Earth's gravitational pull.

Galileo did not have access to the tools of differential calculus, which we now use to analyze this relationship, where acceleration is defined as the derivative of velocity $v$ with respect to time $t$:

$a = \displaystyle\frac{dv}{dt}$

This allows us to deduce that velocity increases linearly with time, represented by an equation in the form:

$v = v_0 + gt$

where $v_0$ is the initial velocity, assuming that the initial time is zero. Similarly, we can determine the position $s$ as a function of time, given that velocity is the time derivative of position:

$v = \displaystyle\frac{ds}{dt}$

which leads to the result that, with an initial position $s_0$, position over time follows a parabolic path:

$s = s_0 + v_0 t + \displaystyle\frac{1}{2}gt^2$

Galileo did not have these tools, as Newton and Leibniz, who developed differential calculus, were born shortly after his death. Nevertheless, Galileo managed to reach similar conclusions using the mathematics of his time, which relied primarily on geometry.

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No one expects us to make groundbreaking discoveries or develop new mathematics. The typical user of physics generally studies well-established hypotheses and models developed over the years and applies them to their own calculations. However, this is where the real challenges begin. Understanding a hypothesis and its model is often manageable, as it is rooted in logic and conceptual thinking. Sometimes, these ideas may be difficult to accept because they counter our intuition, but ultimately, with effort, one gains a solid understanding and feels ready to apply them.

For many, the real challenge begins when they must tackle the mathematics, which can feel like an impenetrable jungle. At first encounter, they face a series of definitions and conventions that often seem abstract and far removed from everyday experience. As they progress, variables are introduced that, with some practice, start to connect to concrete measurements. However, these variables soon become intertwined in a complex system of equationsa tangle that is difficult for beginners to unravel.

Although fundamental principles provide structure and a sense of order, when attempting to apply them to specific problems, the novice often feels overwhelmed. Even when it seems they have found an equation to solve the problem at hand, they quickly realize that the equation does not quite apply to their situation.

Thus, it is essential not to view equations solely within the general context of theory but rather as a set of interconnected tools that work in unison. This approach helps prevent getting lost in theory, allowing one to focus on understanding the model and its associated tools. Even so, there remains the inherent challenge of mathematics, which must be approached from a different perspective to be effectively managed.

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To overcome the initial barrier that mathematics can represent, we propose an alternative approach that transforms mathematics into an accessible and functional tool rather than an obstacle. We leverage the fact that todays advanced technological tools allow us to perform algebraic calculations and solve differential equations. This way, we focus on extracting the physical concepts and associated variables, prioritizing the understanding of the underlying principles rather than diving into the complete mathematical process.

Consider, for example, the equation for the energy $E$ of a particle of mass $m$ moving at a height $h$ with velocity $v$ in Earths gravitational field, where $g$ is the gravitational acceleration:

$E = \displaystyle\frac{1}{2}mv^2 + mgh$

This equation can be represented by a node (in this case, light blue) connected to nodes (white) for its variables: $E$, $m$, $v$, $h$, and $g$:

Representation of an equation as an element of a network of equations and variables

Each equation allows us to calculate one variable at a time. For example, if we want to isolate velocity $v$, we can highlight it in the node with an orange color:

Identification of a variable (v) to be calculated (orange)

To perform this calculation, we need the values of all other variables in the equation, which we highlight in light green to indicate that their values are known:

Identification of a variable required for the calculation (green)

At this stage, we have rearranged the equation to solve for $v$, and weve changed the color of the equation node from light blue to blue to indicate that it has been used in the calculation.

In this example, we assume that all known variables (green) are available. However, in a model with multiple interconnected equations, it is possible that one or more of these required variables come from prior calculations using a different equation.

In this way, we can represent each model as a network of interconnected equations and variables, allowing us to develop calculation strategies and focus on understanding the meaning of the variables, how they are measured, and how they are calculated. Additionally, we ensure that we understand the hypotheses and conceptual foundations of the model, reserving the equations for specific calculations and for behavior analysis where applicable.

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By representing models as networks of equations and their variables, the mathematical work is significantly simplified, as calculations can be performed automatically. This approach allows us to focus on understanding the variables, their physical meaning, and their valid ranges.

The use of equations is reduced to three possible situations:

• when performing calculations, which are executed by support systems;
• when equations restrict the validity range of a variable, such as in cases of singularities or the non-existence of solutions;
• when it is necessary to modify the model, and thus the supporting equations.

In everyday use, a model is conceived as a network of equations interconnected through their variables. For example, consider a simple model with two equations interconnected by a variable (in this case, $z$):

Model of two interconnected equations

If we want to calculate the variable $x_0$, we see that it can be done using equation $eq_1$, provided that the variables $x_1$ and $z$ are known. If $z$ is unknown, we would need to know the variables $y_0$, $y_1$, $y_2$, and $y_3$:

Variables needed to calculate the variable $x_0$

In this way, we can first calculate $z$ using equation $eq_2$ and then use $eq_1$ to find the value of $x_0$:

Calculation of $x_0$ using both equations

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