Is linear momentum conserved in an isolated system always?

Consider a isolated system of particles. If no external force acts on this system, the momentum of the system should remain constant.

If the isolated system of particles represents living creature. And if we assume you (the reader) to represent that system. If you even shake your hand, you are changing the positions of the particle of your hand.

I hope, we can move the particles of hand at any speed (till the limit allowed by laws), at least to cross the initial momentum of the system. Is this not possible? If it is possible, aren't we violating the principle of conservation of linear momentum?

• As long as the Lagrangian of the entire system is invariant up to a total derivative under $x \to x + x'$, for all $x' \in \mathbb{R}$, then the linear momentum will be conserved by Noether's theorem. – JamalS Nov 10 '14 at 16:12

If the (entire) system does not change if you displace it in space, then the total momentum (of all particles) will be conserved. Momentum is the generator of translation, which is a specific case of Noether's theorem.

To relate it to your example of living creatures: imagine cat on very slippery ice and neglect air resistance and friction. If you slide the cat with some initial momentum, no matter how the cat moves and rotates, it will not be able to affect its center of mass as a function of time, i.e. its momentum is conserved.

• Another good example to consider is a person in a space station which is moving inertially in space; if the station is rotating to create artificial gravity, the person can walk around on the inside surface of the station just like on earth, but each step they take, or even each motion of a body part (like a hand) when they are standing in place, will impart an equal and opposite momentum to the station, such that there is no net change in the momentum of the station + person. But the more massive the station, the smaller the change in its velocity for a given momentum imparted to it. – Hypnosifl Nov 10 '14 at 19:37
• @ulf: +1 Thank you for the answer. Are you saying that center of mass doesn't vary as a function of time, for any movement (and thus momentum is conserved) of the creature? – Immortal Player Nov 11 '14 at 1:41
• I'm not saying it doesn't vary, but that the cat cannot influence it by anything it could possibly do. The conservation is guaranteed by the translational symmetry. Just to add as a note: For rotations this is not true. Imagine you were to throw the cat out of a window (please don't though). Now you could ask a similar question for angular momentum and rotation. The cat's angular momentum is preserved during the fall, but the cat can very well influence the angle (equivalent to the center of mass) at which it lands. – ulf Nov 11 '14 at 15:43

Conservation of linear momentum, for a physical system whose particles are initially at rest in a given inertial reference frame is equivalent to the fact that the center of mass of the system remains fixed at its initial position. You see that this constraint is quite week for a system made of a large number of particles as a human body. Starting form a completely static situation, you can pass to move your hands or your legs or other parties of your body, very quickly preserving the position of the center of mass.

• +1 Thank you for the answer. Sorry, I feel that we can shift the position of center of mass. If I shift more of my mass to a particular side of my body (in the situation described in the question), by any of the movement, I can shift the position of center of mass, isn't it? – Immortal Player Nov 11 '14 at 1:40
• No, we cannot, otherwise conservation of total momentum would be violated. If you try to shift a part of your body in a direction, the other part moves along the opposit direction in order to leave fixed the position of the center of mass of the total body. – Valter Moretti Nov 11 '14 at 5:01

Linear momentum is always conserved. If you swing your hand there is always a change such that linear momentum is conserved. If you swung your hand in space there would be some motion. At least until the electrostatic forces in your body brought you to rest. On Earth, the friction between your body and the ground and the air drag wouldn't allow the resultant changes to be very visible.

• +1 Thank you for the answer. "If you swing your hand there is always a change such that linear momentum is conserved." What if you move the hand with greater speed such that to make the product of mass and final velocity greater than that of, product of mass and initial velocity? – Immortal Player Nov 11 '14 at 1:46
• The rest of your body will move in the opposite direction. This appears confusing at first, that the different parts of your body move in opposite directions. But eventually the electrostatic forces which hold your body together will pull you back. The human body is not a very good example because of the large number of parts involved. – jerry Nov 11 '14 at 14:43