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Given a newtonian mechincal system with $n$ objects, we may think of it as living in $\mathbb{R}^{6n+1}$ ; one dimension is time, $3n$ dimensions for velocities, and $3n$ for positions.

We then have qualities that are kept under for any initial conditions; momentum, angular momentum. I want to make sure we found all of those, let's formalize this question;

I think we don't want the time to be involved, so I am asking for all continous $F : \mathbb{R}^{6n} \to \mathbb{R}$ so that for any initial conditions and the resulting path in $\mathbb{R}^{6n}$ (over time), $F$ is constant. Those continous $F$ form a vector space, and I want to understand it.

Examples -

$F$ is the mass of a fixed body.

$F$ is the sum of the momentums.

$F$ is the angular momentum.

Qmechanic
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Andy
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    Shouldn't that be $\mathbb{R}^{6n+1}$? After all, both position and velocity are three-dimensional quantities. – probably_someone Oct 11 '19 at 13:49
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    Also, without imposing any constraints on the type of system you have in mind, none of the examples of $F$ you provided are actually necessarily constant. There's nothing preventing the mass of particular systems from changing over time (for example, a system of trucks that have sand leaking from their beds), a system subject to an external force will have a non-constant sum of momenta, and a system subject to an external torque will have a non-constant angular momentum. You should narrow down what kind of system you're talking about. – probably_someone Oct 11 '19 at 14:01
  • @probably_someone Thanks for your comment. Can I assume a system with outside force whatsoever? I'm just imaginging balls colliding and maybe using force on one another (both momentum and angular momentum aren't affected by that). – Andy Oct 11 '19 at 14:07
  • We then have qualities that are kept under for any initial conditions; momentum, angular momentum. I want to make sure we found all of those... I don't understand really. There are an unlimited amount of things you can define. Can you be more specific? – BioPhysicist Oct 11 '19 at 14:20
  • @AaronStevens So I am considering all possible trajectories of any initial conditions and any symmetric force between any two objects (by being symmetric it has to be a vector in the direction of their difference), I think I mean I want the force (between two particles) to act well with rotations, translations, etc. Then there are infinitely many good functions, and I want to classify all of them (are they a finite dimensional vector space?) – Andy Oct 11 '19 at 14:25
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    But saying "I want to find all of those" or "I want to understand it" is pretty broad. – BioPhysicist Oct 11 '19 at 14:37
  • @AaronStevens When you mean broad you mean undefined yes? I agree I should be more specific in the forces I allow, let's say the forces acting from each pair acting on each other, and a pair of objects (with speed,position,mass) $(v_1,p_1,m_1)$, $(v_2,p_2,m_2)$ act on each either along their directions with magnitude (that may be negative or positive, implying pushing or pulling) via some $Force(p_1-p_2, m_1,m_2,v_1,v_2)$. Now certainly my question is well defined enough? – Andy Oct 11 '19 at 15:36

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