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I know that Lagrangian system is required to demonstrate the relationship between homogeneity(or isotropy) and conservation.

Is any expression or equation pertaining to the homogeneity of time and space and isotropy of space when we prove conservation of energy, momentum and angular momentum in Newton's system?

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All you need to prove conservation of energy, momentum and angular momentum in the framework of newtonian mechanics are the three newtonian axioms.These proofs are indeed very simple and of purely algebraic nature (give it a try!)

Isotropy and homogeneity of space and time are still important concepts in newtonian mechanics and implied by the axioms. Newtons axioms don't differentiate between direction and position in (free) space and are thus isotropic and homogeneous (prove it!). A shift in time wont't affect the axioms as well. Thus we conclude that classical mechanics is an isotropic and homogeneous theory.

So it's not necessary to state these symmetries explicitly as an extra axiom.

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I am not sure if this is what you are looking for, but the relation between symmetry and conservation law follows from Noether's theorem: https://en.wikipedia.org/wiki/Noether%27s_theorem

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  • $\begingroup$ thanks for answer but I wanna know how to prove homogeneity and isotropy by using vector form of second law instead of using lagrangian form $\endgroup$
    – peter.H
    Commented Apr 18, 2017 at 13:20
  • $\begingroup$ @peter.H I suggest to try to end your sentences with point... $\endgroup$
    – peterh
    Commented Apr 18, 2017 at 15:51
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What you call "Newton's system" is based on three simple axioms: the principle of relativity (the laws of mechanics are the same in every possible inertial reference frame --> the laws of mechanics are invariant under Galilean boosts, time translations and space rotations), Newton's 2nd law in differential form (dp/dt = vector sum of all Forces) and Newton's 3rd law (the principle of reciprocity of forces, F_12 = F_21 in vector form). Homogeneity of space and time are included in all 3 axioms, especially in the Galilei princple of relativity.

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