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

I
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. What is the process of deriving a generic equation from specific equations?

The process of deriving a generic equation from specific equations typically involves identifying common patterns or principles in the specific cases. This can include recognizing symmetries, applying mathematical transformations, and using dimensional analysis. The goal is to generalize the specific instances into a broader, more universal form that can apply to a wider range of situations.

2. How do you ensure the derived generic equation is accurate and applicable?

To ensure the accuracy and applicability of a derived generic equation, it is essential to validate it against known specific cases and experimental data. This involves checking that the generic equation reduces to the specific equations under the appropriate conditions and that it accurately predicts outcomes in various scenarios. Peer review and replication of results by other researchers also help in confirming the validity of the derived equation.

3. What mathematical tools are commonly used in deriving generic equations?

Common mathematical tools used in deriving generic equations include calculus (differentiation and integration), linear algebra (matrices and eigenvalues), differential equations, and statistical methods. Additionally, techniques such as non-dimensionalization, perturbation theory, and symmetry analysis are often employed to simplify and generalize specific equations.

4. Can you provide an example of a generic equation derived from specific laws?

An example of a generic equation derived from specific laws is the ideal gas law. The specific laws, such as Boyle's Law (P1V1 = P2V2 at constant temperature) and Charles's Law (V1/T1 = V2/T2 at constant pressure), can be combined and generalized into the ideal gas law: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. This generic equation applies to ideal gases under a wide range of conditions.

5. What are the challenges in deriving generic equations from specific ones?

Challenges in deriving generic equations include dealing with complex systems where specific equations may not easily generalize, handling non-linearities, and ensuring that the derived equation remains accurate across different scales and conditions. Additionally, simplifying assumptions made during the derivation process can sometimes limit the applicability of the generic equation. Careful consideration and validation are required to overcome these challenges.