While teaching introductory physics, my professor explained that the conservation of linear momentum, conservation of energy and conservation of angular momentum are based on deeper principles in classical mechanics ,namely, the homogeneity of empty space, homogeneity of time and isotropy of space respectively.
Though I somewhat understand the individual meanings, what are the relations to these with the conservation laws?
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9$\begingroup$ The deep principle here is Noether's theorem. $\endgroup$– ACuriousMind ♦Commented Mar 9, 2016 at 20:17
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2$\begingroup$ Too many questions here, and most of them actually have answers elsewhere on the site (though not always in the form they are asked here). $\endgroup$– dmckee --- ex-moderator kittenCommented Mar 9, 2016 at 21:18
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$\begingroup$ A book by L.D.Landau and E.M.Lifshits, "Mechanics" (Course of Theoretical Physics, v.1) is a good source for this subject. Using the Hamiltonian formualtion of mechanics, on can derive the conservation laws. $\endgroup$– user2320292Commented Mar 10, 2016 at 3:22
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$\begingroup$ symmetry leads to conserved quantities. $\endgroup$– user196418Commented Jun 10, 2019 at 20:05
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After Newton an equivalent formulation of mechanics was found called Lagrangian mechanics. This features differential equations of the form $\frac{\partial L}{\partial \mathbf{q}}=\dot{\mathbf{p}}$ where we define $\mathbf{p}:=\frac{\partial L}{\partial \dot{\mathbf{q}}}$. I'll give a very basic example of how symmetries are related to conservation laws. An infinitesimal transformation of the form $\delta\mathbf{q}=\epsilon\mathbf{k}$ achieves $$\delta L = \epsilon \left[\dot{\mathbf{p}}\cdot \mathbf{k}+\mathbf{p}\cdot \dot{\mathbf{k}}\right]=\epsilon\partial_t \mathbf{p}\cdot\mathbf{k}.$$ Thus $\mathbf{p}\cdot\mathbf{k}$ is conserved iff $\delta L=0$, which is a symmetry condition.
