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Many times a precise definition of something in physics is not available but yet there exist some rough definitions that guide us through. I need the same rough (if not precise) definition of physical laws. The reason being that we would consider $\mathbf a= \frac {d \mathbf v}{dt}$ as a mere definition but $\mathbf F =m \mathbf a$ as a law (as it is called as Newton's second law of motion). But to me it seem to be more inclined towards being as definitions rather than laws. I mean contrasting Newton's laws of motion to the law of conservation of energy (momentum, etc) it seem to be an attempt to define forces ( though I might be wrong at this (but I don't know reason for such)).

  • So what is the definition of a physical law that can encapsulated it's entirety?
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    $\begingroup$ Related, if not a dupe of, physics.stackexchange.com/q/77465/25301 and the linked posts therein $\endgroup$ – Kyle Kanos Dec 5 '19 at 15:37
  • $\begingroup$ Physical laws reflect symmetries of nature. $\endgroup$ – safesphere Dec 5 '19 at 16:06
  • $\begingroup$ These words have simply not been applied consistently. There have been some valient attempts to produce a useful (and roughly historically consistent) taxonomy but they are all plagued by errata. $\endgroup$ – dmckee --- ex-moderator kitten Dec 5 '19 at 16:14
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    $\begingroup$ It's also worth looking in the "Linked" and "Related" sidebars of Kyle's question and we've been over this ground several times. $\endgroup$ – dmckee --- ex-moderator kitten Dec 5 '19 at 16:15
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    $\begingroup$ @Dale You are right that they aren't mutually exclusive. "Speed of light in vacuum is $c$ " is a law but the it comes from the definition of speed itself( pls correct me if I'm wrong). So I would say that laws are like the special cases of a particular set of definitions. $\endgroup$ – user238497 Dec 5 '19 at 17:27
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When one studies mathematics, one knows that a mathematical theory starts from axioms and then mathematics is used to prove theorems. When axioms change, the theory changes. A good example is Euclidean Geometry versus spherical geometry. Also in mathematics, a theorem may be used as an axiom, and the original axiom proven as a theorem.

Physics theories use mathematics as a tool to describe the natural world. But the mathematics of differential equations used extensively in physics, have an infinity of solutions and forms. It is necessary to use new "axioms" in order to pick up those solutions that are relevant to measurements and observations. These physics "axioms" are called "laws" "postulates" "principles" and are extracted from many observations and measurements so that the physics theory not only fits the data,but, very important, predicts new situations. (Just fitting data makes a mathematical map). At the same time there are at the level of axioms statements identifying the objects measured, mass, charge, (plus a lot of quantum numbers for particles)

Take $F=ma$ for example. It is a law, because when using it axiomatically ( together with the other two) Newtonian mechanics theory works beautifully, from predicting the planetary system, to trajectories of rockets , to stability of buildings etc.

The concept of force and mass existed in everyday language, because of the need to quantify products and the need to quantify effort, both everyday manifestations. The brilliant use though of the relation between acceleration , mass, and force brought the existence a complete and self consistent theoretical model for describing nature.

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As pointed out by Safesphere, the physical laws that rule the world are expressed at a higher and more general level of (mathematical) abstraction which underlie Newton's laws. We can then derive Newton's laws from them as needed to solve everyday problems in real-world dynamics.

These more general relationships are generically called symmetries; if you know what they are for a given system then you can in a sense solve the problem once by working with those symmetry equations and apply the general form of the solution to any of an entire class of related problems, making them easier to solve. This is a very powerful and general technique which is why physicists use it.

As pointed out by dmckee, this question has been asked and answered before but the reason I am answering it here is in doing so I get the chance to test my own understanding of the topic. If I get it wrong, then the experts here will point out my errors and I then learn something new.

Please write back if you are interested in examples of symmetries and I'll furnish you with some.

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  • $\begingroup$ I (not the OP) would appreciate some examples. $\endgroup$ – user45664 Dec 5 '19 at 17:54
  • $\begingroup$ Sorry I got late at asking that but would love to hear more about it (symmetries) from you. You can chat here. $\endgroup$ – user238497 Dec 5 '19 at 18:25
  • $\begingroup$ @user45664 , the fact that the equations describing dynamics are symmetric with respect to time means energy is conserved. the fact that they are symmetric with respect to position means linear momentum is conserved. the fact that they are symmetric with respect to direction means angular momentum is conserved. For wavefunctions, the fact that they are symmetric with respect to phase means electric charge is conserved. search on noether's theorem for a complete explanation, you will find it very fascinating. -NN $\endgroup$ – niels nielsen Dec 7 '19 at 7:55
  • $\begingroup$ Would transmit-receive reciprocity for an aperture or antenna beam pattern be a symmetry? Ie. the transmit beam pattern can be obtained from the receive beam pattern by reversing the the propagation direction (or reversing time). $\endgroup$ – user45664 Dec 7 '19 at 17:22
  • $\begingroup$ that I do not know. $\endgroup$ – niels nielsen Dec 7 '19 at 19:17