Is there a rigorous definition of a law of physics? Is there, in some physics or mathematics text, a rigorous definition of what a law of physics is? I have read many physics books and papers, and while they all talk about particular laws of physics (like light's speed in a vacuum being constant), they don't actually define what a law of physics is. I know that philosophers have tried to define laws of physics, but I am looking for a physics text or a mathematical physics text that tries to rigorously define what a law of physics is.
 A: It seems to me the word "Law" is typically used of individual equations or relationships, such as Newton's 2nd Law $F=ma$, Newton's Law of Universal Gravitation $F=Gm_1m_2/r^2$ (or collectively Newton's Laws of Motion), the First Law of Thermodynamics, etc. In contrast "theory" often refers to a comprehensive set of assumptions and postulates that contain many physical relationships, such as Newton's Theory of Gravitation, the Theory of General Relativity, Quantum Field Theory. All of these essentially rely on a model or picture of how light and matter or space and time behave and what they are composed of, and their predictions (which may include statements of Laws) are based upon these models of the underlying realities being true.
For example, Quantum Mechanical Theory predicts the relationship known as Wien's Displacement Law, which states that the temperature $T$ of a hot body is inversely proportional to the peak wavelength $\lambda$ of the light spectrum that it emits, or in other words $\lambda T = \mathrm{Constant}$.  This law is predicted by the framework of Quantum Theory, but the law was obverved to hold true from experimental data before QT was developed.  And if QT is superseded in the future by a different picture of reality, that new theory will have to agree with Wien's Law.
Thus I would summarize by saying a Law is typically a targeted statement about a certain phenomenon or physical relationship that can often be expressed in a single equation, and is almost always supported by observational data. It does not necessarily claim to explain why that physical relationship is true. A Theory is a comprehensive set of assumptions and postulates designed to explain the "why" and to make a broad variety of predictions.  Because of its broader scope and predictions that apply to essentially every physical situation, it cannot usually be rigorously called "true," because there may always be some unique observation down the line that disagrees with its predictions.  But to answer your main question, much of this is a matter of historical convention, and I don't think there are rigorous definitions of the terms – certainly not ones that all scientists would agree upon.
A: A scientific law is a prediction, statement, equation, etc. that is consistent with known observations (possibly within some restricted scope of validity). It falls short of a scientific theory, which does the same but also explains "why" the prediction/equation works from some notion of first principles. Newtonian gravity is a law ($F \propto r^{-2}$), while General Relativity is a theory ("space tells matter how to fall, matter tells space how to bend"). Physics is a science, so physical laws meet the same criteria as scientific laws.
A: In my opinion, a law of physics is a mathematical relationship developed to explain observed phenomena, that over the course of time has been found to accurately describe said phenomena.  Examples are: Newton's second law, Euler's equations for motion of a rigid body, Einstein's equivalence of mass and energy, etc.
