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What are the variables?
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To formulate models based on hypotheses, it is essential to define the variables that describe the evolution of a system, assigning them clear symbols and specifying their units, such as the MKS system (meter-kilogram-second). Some variables must be observable, meaning they can be measured empirically to validate the models experimentally. However, there are variables that cannot be directly measured and need to be calculated from other observable variables. Additionally, some variables could theoretically be measured directly, but due to limited access to the necessary technology, they must be derived from calculations.
Isaac Newton introduced the concept of variables for describing physical systems in Philosophiae Naturalis Principia Mathematica, while Pierre-Simon Laplace expanded on this by suggesting that knowing all the variables of a system would allow one to predict its future and reconstruct its past, promoting a deterministic view of the universe.
Variables are classified into independent variables, such as time ($t$) and initial position ($s_0$), and dependent variables, such as position ($s$) and velocity ($v$). In the context of uniform, rectilinear motion, dependent variables describe how the system evolves based on the independent variables.
Heisenberg challenged the classical view with his uncertainty principle, demonstrating that there is a theoretical limit to the precision with which certain pairs of variables, such as position and momentum, can be measured simultaneously. This principle revealed that measurement affects the state of the system, known as the wave function collapse, and introduced probability into physics, altering the classical idea of observables as exact, predictable values and transforming the understanding of measurement and determinism.
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Variables and observables
Description
To formulate the model associated with the proposed hypotheses, it is essential to define the variables that describe the system's evolution. Each variable should have a clear description and be assigned a symbol, such as a letter, a set of letters, or numbers, to facilitate reference in both texts and equations. Additionally, the units of measurement for each variable must be specified, using a universal system such as MKS (meter-kilogram-second). For clarity, units are represented within square brackets, for example, time in seconds [s] and position in meters [m].
Since hypotheses and models must be experimentally validated, it is necessary for at least some of the variables to be observables, meaning variables that can be empirically measured. However, there are variables that cannot be directly measured and are therefore considered non-observable. These variables need to be calculated from other observable variables. Furthermore, there are variables for which direct measurement methods exist but may require difficult-to-access technology, making it necessary to obtain them through calculations based on other measurable data.
Consequently, the formulation of models should consider how each variable is determined, ensuring that the model is feasible and testable. A common issue is designing a model where the number of non-observable variables exceeds the number of available equations needed to determine them, preventing the complete resolution of the system.
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Definition of variables
Description
The initial concept of a variable dates back to Isaac Newton, who documented it in his work Philosophiae Naturalis Principia Mathematica [1]. Beyond providing the foundation for establishing the laws that describe the motion of bodies, Newton introduced the idea that variables are fundamental elements for describing the state of a physical system.
Pierre-Simon Laplace expanded on this concept by asserting that there is a set of variables that completely describes the state of a physical system. He also introduced the idea of a hypothetical intelligence - known today as "Laplace's demon" - which, if it knew all these variables, could calculate the entire future evolution of the system and even reconstruct its past up to the present moment [2]. This led to the deterministic view that both the past and the future are defined by natural laws, challenging the notion of free will, that is, the human ability to make autonomous decisions.
Pierre-Simon, Marquis de Laplace (1749-1827)
To facilitate working with variables, it is necessary to define not only their descriptions but also assign letters, symbols, and sometimes numbers for easy reference in both text and equations. Additionally, it is important to specify the units in which the variables are measured, which can be defined using a universal system such as MKS (meter-kilogram-second). For clarity, units will be enclosed in square brackets, such as [s] for time in seconds and [m] for position in meters.
A useful way to define the variables that describe a physical system is to classify them into two main types: independent variables and dependent variables.
Independent variables are those that can be freely defined and are not influenced by other variables in the system. In the case of uniform and rectilinear motion, these variables include time $t$, initial time $t_0$, and initial position $s_0$.
Dependent variables, on the other hand, are those that depend on the independent variables and describe how the system evolves. In the example of uniform and rectilinear motion, the main dependent variable is the position $s$, as it depends on time $t$. Although velocity $v$ is constant in this type of motion, it is classified as a dependent variable because it is determined by the initial conditions of the system and is not freely chosen.
In summary, for uniform and rectilinear motion, the variables are classified as follows:
Independent variables:
$t$: time [s]
$t_0$: initial time [s]
$s_0$: initial position [m]
Dependent variables:
$s$: position [m]
$v$: velocity [m/s]
With [m/s] representing the units of velocity in meters per second.
[1] Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), Isaac Newton, Royal Society of London, 1687.
[2] Essai philosophique sur les probabilités (Philosophical Essay on Probabilities), included in Théorie analytique des probabilités (Analytical Theory of Probabilities), Pierre-Simon Laplace, Courcier, Paris, 1814.
[3] James Posselwhite, public domain, via Wikimedia Commons.
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Definition of observables
Description
Observables are the magnitudes or properties of a system that can be directly measured or quantified during an experiment. On the other hand, there are variables that cannot be directly observed, such as the total energy of a system and similar properties. Additionally, there are cases where certain variables could be measured directly, but when the appropriate technology is not available, they must be calculated from other observables.
In the context of uniform and rectilinear motion, the observables include:
• Position $s$.
• Initial position $s_0$.
• Time $t$.
• Initial time $t_0$.
Velocity $v$ is a special case: although it can be measured directly with the appropriate technology, in many experiments it is calculated from other measurements, such as the distance traveled and the time elapsed. In this context, velocity becomes a variable derived from two observables:
• Distance traveled $\Delta s$ [m].
• Time elapsed $\Delta t$ [s].
Both the distance traveled and the time elapsed can be measured directly, making them observables. However, they can also be calculated from the previously mentioned observables, such as $s$, $s_0$, $t$, and $t_0$. This illustrates how certain variables can be observables under specific conditions and derived in others, depending on the context and the available measurement tools.
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Limitations of observables
Description
In the era of Isaac Newton, measurements were viewed as a process without inherent limits to accuracy, constrained only by the technological limitations of the time. In that same spirit, Laplaces demon suggested that, with sufficient precision in measurements, one could predict with certainty the future evolution of any physical system.
However, Werner Heisenberg's work in the past century challenged this view by demonstrating that measurements have fundamental limitations beyond technological constraints. Heisenberg showed that there are not only precision limitations but also theoretical boundaries that persist, even in classical physics, when extreme accuracy is sought.
Werner Karl Heisenberg (1901-1976) [1]
Heisenberg introduced concepts originating in quantum mechanics that are also becoming relevant in classical contexts. The original idea that any variable could be determined with the desired accuracy has shifted to the understanding that there are inherent limits to possible measurements.
These limitations include:
1. Limit of measurement precision: In classical physics, it is assumed that any observable, such as position or velocity, can be measured with arbitrary precision, and that measurement errors are purely technical issues that can be reduced with better instruments. However, Heisenberg's uncertainty principle introduced a fundamental limit to the precision with which certain pairs of observables, like position and momentum, can be measured simultaneously. This changed the classical view by showing that there is a theoretical, not just technical, limit to the precision of measurements.
2. Non-commutativity of measurements: In classical physics, any set of observables can be measured in any order without affecting the results. However, in quantum mechanics, Heisenberg demonstrated that certain observables, such as position and momentum, do not commute, meaning the order in which they are measured affects the outcome. This implies that the classical concept of simultaneous measurement of certain observables must be reconsidered.
3. Impact of the act of measurement: In classical physics, it is assumed that the act of measurement does not significantly alter the system being observed. Heisenberg, however, introduced the idea that measuring an observable in a quantum system alters the systems state, a phenomenon known as wave function collapse. This means that observation itself changes the properties of the system, challenging the classical notion of passive and independent measurement.
4. Probability and determinism: The classical view of observables is deterministic; it assumes that if all initial conditions of a system are known, future states can be predicted with certainty. However, through his uncertainty principle, Heisenberg revealed that in the quantum world, only probabilities of outcomes can be predicted, not absolute results. This introduces an inherently probabilistic nature to quantum measurements, shifting the classical idea of observables as precise, determined values.
[1] Bundesarchiv, Bild 183-R57262 / Unknown author / CC-BY-SA 3.0
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