Questions about deriving generic equations/laws from specific equations/laws

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Thread starter Hallucinogen
Start date Oct 14, 2023

Oct 14, 2023

Hallucinogen

I'd like to ask what the most well-known case is of a formula for a physical law being derived from another (or set of others)?
For example, is there a law for electromagnetism that describes a law of electricity and a law of magnetism, which was derived from combining the two? Or any similar derivation? Does Ampere's law have this kind of relationship with Maxwell's equations?

Another related question is, what are the most well-known "generic" laws of physics which apply everywhere, and which are the most well-known highly specific laws of physics? I'd guess the most generic laws are those of general relativity and thermodynamics, since no physical process is allowed to conflict with them? And I'd guess that laws in materials science are highly specific, as they have multiple conditionalities, for example Newton's law of viscosity. Or laws about forces being applied to specific shapes?

Many thanks

1. How can you derive a generic equation from a specific case?

To derive a generic equation from a specific case, you typically start by identifying the fundamental principles or laws that govern the specific case. Next, generalize these principles by removing any constraints or specific conditions that limit their applicability. This might involve introducing more general variables or parameters, using algebraic manipulation, and considering broader assumptions. The goal is to formulate an equation that holds under a wider set of conditions or for a broader class of systems.

2. What is the importance of dimension analysis in deriving generic equations?

Dimensional analysis is crucial in deriving generic equations because it helps ensure that the equations are physically meaningful and dimensionally consistent. By analyzing the dimensions of the quantities involved, you can verify that both sides of an equation have the same dimensions, which is a fundamental requirement for a valid physical law. Dimensional analysis can also provide insights into the form of the equation and help identify scaling laws and nondimensional parameters.

3. Can you provide an example of deriving a generic law from a specific equation?

A classic example is deriving the ideal gas law from experimental observations of specific gases. Initially, experiments with gases like hydrogen or oxygen at various temperatures and pressures might lead to specific equations describing their behavior. By recognizing the commonality among these gases and their behaviors, scientists were able to generalize these observations into the ideal gas law, PV=nRT, which applies to any ideal gas regardless of its specific type.

4. What role does symmetry play in the derivation of generic equations?

Symmetry plays a significant role in the derivation of generic equations because it allows for the simplification of physical laws under certain conditions. Symmetries in a physical system, such as rotational, translational, or mirror symmetry, can lead to conservation laws (e.g., conservation of momentum, energy, or angular momentum) through Noether's theorem. These conservation laws can then be used to derive more generalized equations that apply across different systems exhibiting the same type of symmetry.

5. How do assumptions and approximations affect the derivation of generic equations?

Assumptions and approximations are often necessary to derive generic equations, especially when dealing with complex systems. By simplifying the conditions under which the system operates, these assumptions make the problem more tractable. However, they can also limit the applicability of the derived equation to situations where the assumptions hold true. It's crucial to clearly state these assumptions and understand their implications on the accuracy and applicability of the generic equation.