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The identification of fundamental principles involves recognizing profound elements that indicate underlying and constant conditions in a system, beyond mere observation. These principles include invariances or verifiable relationships within a system. Among the main principles are spatial invariance, temporal invariance, and the relativity of velocity.

Spatial invariance states that the laws of physics are independent of position in space, meaning that experiments conducted under identical conditions should yield the same results in different locations. This principle is crucial for formulating coherent physical theories and is related to the homogeneity of space.

On the other hand, temporal invariance implies that physical laws do not change over time, ensuring that experiments under the same initial conditions produce identical results at any given moment. However, this invariance is lost in systems that experience variable forces or irreversible processes, such as friction or heat conduction.

The relativity of velocity emphasizes that it is not possible to measure an absolute velocity of an inertial system without an external universal reference, which does not exist in practice. Only relative velocities with respect to other systems can be measured, and the laws of physics remain the same in all inertial systems. The principle of inertia states that an object will remain at rest or in uniform rectilinear motion unless acted upon by external forces.

Noether's theorem, a fundamental result in theoretical physics, connects the symmetries of a system with conservation laws. In the case of spatial symmetry, linear momentum is conserved, while temporal symmetry is associated with the conservation of energy. These principles are essential for understanding why certain quantities, such as energy and linear momentum, remain constant in physical processes and underpin the formulation of scientific models and theories.

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The identification of fundamental principles goes beyond simply describing what is observed; it involves recognizing deeper, more significant aspects that point to underlying conditions within a system.

Typically, fundamental principles refer to elements that remain constant, known as invariances, or to factors that exhibit a clear and verifiable dependency.

In the case under discussion, three principles stand out as fundamental:

1. Spatial invariance

2. Temporal invariance

3. Relativity of velocity

Each of these principles is explained in detail below.

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Spatial invariance is a fundamental principle in physics that states that the laws of physics are independent of the position in space where they are applied. This means that experiments conducted under identical conditions in different locations in space should yield the same results. This principle is linked to the homogeneity of space, which is the notion that space is uniform and does not have properties that vary from one point to another.

Spatial invariance is essential for formulating consistent physical theories and for describing natural phenomena throughout the universe. It ensures that the same physical laws apply everywhere in space, which is a key assumption in the development of mathematical and physical models. Furthermore, it is a central component for ensuring that physics does not depend on any particular reference frame concerning spatial translations.

The concept of spatial invariance was formalized by Isaac Newton in his work Philosophiae Naturalis Principia Mathematica [1], where he established that the laws of physics are the same at any location in space, independent of position. This idea is fundamental for understanding the homogeneity of space and the formulation of coherent physical theories.

[1] Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), Isaac Newton, Royal Society of London, 1687.

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Spatial invariance is not maintained in situations where the properties or conditions of space vary from one point to another, implying that the behavior of a physical system depends on its position in space. Some examples and scenarios where spatial invariance is broken include:

• Non-Homogeneous External Fields: When a system is influenced by a force field that varies spatially, such as a gravitational or electromagnetic field with changing intensity and direction at different locations, the governing laws of the system can depend on its position. For instance, on the surface of a planet with a non-uniform mass distribution, the gravitational field varies from point to point.

• Non-Homogeneous Materials: In a medium where physical properties change with position, such as a material with variable density or a structure exhibiting heterogeneities (e.g., a medium with regions of differing elasticity or thermal conductivity), the system's response will depend on its spatial location. This breaks spatial invariance since the laws of motion or the materials reactions differ across locations.

• Crystals and Periodic Structures: In crystalline materials, the arrangement of atoms follows a regular pattern, implying that physical properties vary periodically with position. While these materials have specific symmetries (such as translational symmetry in crystal lattices), they do not exhibit continuous spatial invariance, as properties change when moving from one point to another within the structure.

• Background Structures in Spacetime: In theoretical physics and cosmology, spatial invariance may be broken when considering a non-homogeneous background structure of spacetime. For example, in the early universe or in regions with non-uniform matter distribution, such as galaxy clusters or areas with intense gravitational fields, spatial conditions can vary from one location to another.

• Systems with Spatial Barriers or Potentials: In physical systems that include position-dependent barriers or potentials, such as a quantum potential well or a region with an electric potential difference, the laws governing the behavior of particles or fields in that space depend on their position.

• Engineered Environments with Property Gradients: In engineering and technology, some devices and materials are designed with property gradients, such as optical lenses with spatially varying refractive indices to focus light in specific ways. In these cases, interactions with light or other waves vary from point to point, breaking spatial invariance.

These examples illustrate how spatial invariance can be disrupted, leading to physical behavior that depends on location within a system or environment.

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Temporal invariance refers to the principle that the physical laws governing a system do not change over time. This means that the equations describing the evolution of a physical system remain consistent at any point in time, ensuring that experiments conducted under the same initial conditions will yield identical results, regardless of when they are performed.

Temporal invariance is lost when a system is non-autonomous or subjected to conditions and forces that change over time, which affects the consistency of its physical properties and its evolution.

The concept of temporal invariance was formalized by Isaac Newton in his work Philosophiae Naturalis Principia Mathematica [1], where he established that the laws of physics remain constant over time and do not depend on when they are applied. This principle is fundamental for understanding temporal uniformity and for the formulation of consistent physical models.

[1] Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), Isaac Newton, Royal Society of London, 1687.

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Temporal invariance is not maintained in situations where the laws or conditions of a system change over time. This means that the behavior of the system and the results of experiments can depend on the moment they are conducted. Some examples of scenarios where temporal invariance is broken include:

• Dissipative or Non-Conservative Systems: In systems where energy is lost due to friction, drag, or dissipation as heat, the total energy is not conserved and depends on time. For instance, a pendulum that slows down and stops due to air resistance or friction at its suspension point.

• Irreversible Processes in Thermodynamics: In most real-world thermodynamic processes, such as gas diffusion or heat conduction, the system's evolution depends on time, and these processes cannot be reversed without affecting the surroundings. The entropy of an isolated system increases over time, indicating a break in temporal symmetry.

• Systems Subject to Variable External Forces or Conditions: When a system is influenced by external forces that change over time, such as a magnetic or electric field that varies, the system's response may differ at different times, breaking temporal invariance.

• Cosmology and the Expansion of the Universe: On a large scale, the universe is not temporally invariant, as physical conditions have changed dramatically since the Big Bang and continue to change due to the expansion of the universe. The laws of physics may manifest differently in different cosmic eras.

• Systems with Internal Structural Changes: If a physical system undergoes an internal transformation that changes its properties over time, such as a chemical reaction altering the system's composition or a phase transition, temporal invariance is broken.

• Quantum Fields and Dynamic Background Effects: In certain contexts of quantum field theory or general relativity, a background field that changes over time, such as a time-dependent gravitational field, can break temporal invariance.

These examples illustrate how temporal invariance can be disrupted in various systems, resulting in behavior that depends on when the system is observed or measured.

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Imagine traveling at a constant velocity through the universe and observing everything from our frame of reference, which we call inertial to emphasize that it moves at a constant speed.

The problem arises when we try to determine our own velocity through an experiment conducted within our system. The reality is that there is no way to perform an experiment that allows us to determine our absolute velocity without referencing an external parameter related to a hypothetical universal system. However, there is nothing in the universe identified as such a universal reference system. Therefore, it is only possible to make measurements that provide the relative velocity with respect to other systems, without implying the existence of a universal velocity.

This leads to the conclusion that, although it is possible to measure velocities between different systems, there is no absolute reference system in the universe. According to the principle of relativity, the laws of physics are the same in all inertial systems, and the notion of a universal system is neither necessary nor valid.

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The fact that it is not possible to determine the velocity of an inertial system with respect to a hypothetical universal reference frame implies that the laws of physics are invariant in any inertial system, regardless of the velocity it may have relative to another reference frame.

This leads to the following conclusions:

• If an object is initially at rest, it will remain at rest indefinitely as long as no external forces act upon it.

• If an object is in uniform rectilinear motion, it will continue to move in a straight line at a constant velocity. It will not change its direction or speed unless an external force intervenes.

This behavior is a direct manifestation of the principle of inertia, which states that an object tends to maintain its state of motion (whether at rest or in uniform rectilinear motion) in the absence of external forces.

The principle of inertia was established by Galileo Galilei [1] as a correction to an earlier concept introduced by Aristotle, which was documented during the Middle Ages [2]. Aristotle observed that moving bodies tended to come to a stop due to friction, and without understanding the underlying mechanism, he formulated the principle that all bodies tend toward rest. Galileo challenged this view by introducing the concept of classical relativity, noting that an object might be at rest relative to the system in which it has stopped, but not necessarily for an observer in another system moving uniformly relative to the first. Thus, Galileo generalized the principle, stating that velocity is constant and its value depends on the reference system in which it is measured, potentially even appearing at rest in one of those systems.

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

[2] "De Caelo" (On the Heavens), Aldus Manutius, Aldine Press, 1495.

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Noether's theorem is a fundamental result in theoretical physics and mathematics that establishes a deep connection between the invariances or symmetries of a system and conservation laws.

Emmy Noether (1882-1935) [2]

In general, the theorem states that [1]:

For every continuous symmetry of the action of a physical system, there is a corresponding conservation law.

This implies that if a physical system exhibits a certain symmetry, a conserved quantity can be associated with that symmetry over time.

For the cases of discussed invariances or symmetries, the following conservations are obtained:

Spatial Invariance (translational symmetry in space):

If a system is invariant under translations in space, meaning that the laws describing it do not change when the entire system is moved to another position in space, Noether's theorem implies the conservation of linear momentum.

This means that the total momentum of a closed system remains constant as long as no external forces are acting on it.

Temporal Invariance (translational symmetry in time):

If a system is invariant under translations in time, meaning that the physical laws governing it do not change over time, Noether's theorem implies the conservation of energy.

This means that the total energy of an isolated system remains constant as long as there are no external interactions altering the system.

In summary, Noether's theorem provides a powerful tool for deriving conservation laws from the symmetries of a system and is fundamental for understanding why certain quantities, such as energy and linear momentum, are conserved in physical processes.

[1] "Invariante Variationsprobleme" (Problemas de variación invariante), Emmy Noether, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse. 1918: 235257.

[2] Emmy Noether (1882-1935), Public domain, via Wikimedia Commons

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