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Pretty much everything we do appears to boil down to practice of Newton's Third law. Even this rug I'm seated upon relies upon reaction from the floor to keep me seated. Amongst others the same reaction helps me move, sleep/recline .. and so on.

What I wish to know - is it possible for a body to move without relying upon a reaction from the objects/whatever in the direction it wishes to move?

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The principle behind the third law is the conservation of momentum (see also the last paragraph in this section on the third law).

If you start with your stationary mass its momentum is zero. If it could start moving on its own the momentum would now be non-zero i.e. momentum wouldn't be conserved. Noether's theorem tells us that conservation of momentum is the same as saying that the laws of physics are unchanged by a displacement in space, so it's a pretty fundamental aspect of the universe. In this case it means your mass can only start moving if something else moves in a different direction to keep the total momentum at zero.

So the answer to your question is no!

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John Rennie answer is almost correct, but there is a small (thirty orders of magnitude-small to be precise) counterintuitive exception to this rule when the spacetime is curved over the dimensions of an extended body. It is called spacetime swimming, which is basically a way to get a net displacement by doing internal displacements in a noncommutative order.

However the effect is extremely tiny, even in strong gravitational fields, net displacements that can be achieved in a shape cycle are of the order of $10^{-30}$ meters

Here is a reference to the original paper: http://dspace.mit.edu/handle/1721.1/6706

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+1 good point! Still, I was right if you restrict yourself to flat space :-) – John Rennie Jul 17 '12 at 18:15
I remember this paper, and I never worked out how it was doing it or whether it was doing it. It is easy to fool yourself in GR with unwarranted assumptions about rigidity, and also by making spatial sections. The device should be either emitting gravitational radiation backwards in the process (having looked at the paper, this is not so--- it can be done slowly), or changing an angular momentum potential somehow and kicking the source of the field with backreaction. There is something incomplete or wrong about this--- the conservation laws work in GR with asymptotically flat boundaries. – Ron Maimon Jul 18 '12 at 2:45
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Oh wait, you are talking about displacement, not momentum acquisition--- this is certainly possible, you are just moving between equal action orbits by swimming--- this is allowed--- but does it count as motion? It doesn't give you net momentum at the end, and the displacement cycle for sure has to displace the source of the gravitational field in an infinitesimal way at the end, to keep the center of mass conservation (which also works in asymptotically flat GR). – Ron Maimon Jul 18 '12 at 2:47
@RonMaimon, yes, there is this slightly paradoxical situation where you obtain a net displacement but never changing net momentum. Does it count as motion? hell, if it takes me from A to B i won't complain about how it went about its business. The reality is that it will take you just from A to A plus infinitesimals – lurscher Jul 18 '12 at 3:06
@lurscher: But if you do it while in orbit around the Earth, it will let you swim forward in your orbit, but at the same time it will take the center of mass of the Earth from a certain point to a point infinitesimally moved over, so that the center of mass of you+Earth isn't moved. It still did it by third law, the backreaction to the swimming is motion of the source. I am trying to think of an EM analog, where you can swim in a static electrostatic field, pushing the source. – Ron Maimon Jul 18 '12 at 4:58
up vote -1 down vote

Yes, of course.

For motion, there's no need of opposite reaction at all. Due to inertia, an object will continue to move uniformly unless an external force is applied (Newton's 1st Law of Motion).

When it comes to accelerate an object from zero to some velocity, an external force is required which can't necessarily be opposite reaction. e.g. Gravitational field can accelerate an object which isn't opposite reaction of action of the object in question.

Update:
When it comes to object's WISHFUL acceleration, its NOT possible without relying on Newton's 3rd Law of Motion. Its because an object can apply force on itself only by Newton's 3rd Law ju-ju.

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The example in your second paragraph is wrong. If you consider some object falling in e.g. the Earth's gravitational field then the Earth moves towards the object at the same time as the object moves towards the Earth. It's just that the Earth doesn't move much because it's so heavy. – John Rennie Jul 17 '12 at 18:17
@John So what? Gravitational force isn't opposite reaction of object's action. My example is 100% valid.. – Sachin Shekhar Jul 17 '12 at 18:21
The Earth curves spacetime and that makes the object move. However the object also curves spacetime and that makes the Earth move. So the action of the Earth is matched by the reaction of the object. Newton's third law applies to the system comprised by the object and the Earth. – John Rennie Jul 17 '12 at 18:53
@John Limit the system to just object and field... – Sachin Shekhar Jul 17 '12 at 19:46
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You can't limit the system to just the object and the field. If you do that momentum won't be conserved. That's like looking at two colliding billiard balls and saying limit the system to a single ball. – John Rennie Jul 17 '12 at 19:50
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