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Main Question or Discussion Point
Is there a particularly good reason that angular momentum was defined as the cross product of position and linear momentum vectors?
I'm not sure about this, but I think it's just to do with the development of the rotational equivalents of the translation dynamics equations. Torque is defined as [tex]r\times[/tex][tex]F[/tex]. We need the r here because physically there is a dependence on the distance of the force from the rotation axis. And we want the rotational equivalent of [tex]\sum[/tex] F = [tex]dP/dt[/tex] which is [tex]\sum\tau = dL/dt[/tex] and so we define angular momentuml as [tex]r\times[/tex][tex]p[/tex]Is there a particularly good reason that angular momentum was defined as the cross product of position and linear momentum vectors?
note: small typo above (should be angular momentum not torque)In particular, if we cross-product it with another vector, it's still valid.
r is a vector, and the cross-product with r is [itex]\sum\tau = r\times p[/itex] .
i want to know where this angular momentum or torque equaton comes from and why we use the cross product.In a lot of sites they only gave the definiton of angular momentum and torque and said that theses definitions are rotational equivelants of 2nd law.But how do they get there, i mean from pure 2nd law which doesn't say anytihng about rotation as far as i can see, how can we derive these torque equations?What was your question?
It is not possible to derive the conservation of angular momentum from Newton's laws. It a simple thing to postulate a counter example that satisfies the Newton's laws, but violates conservation of angular momentum. So you cannot derive the torque equations either, from Newton's laws alone.i want to know where this angular momentum or torque equaton comes from and why we use the cross product.In a lot of sites they only gave the definiton of angular momentum and torque and said that theses definitions are rotational equivelants of 2nd law.But how do they get there, i mean from pure 2nd law which doesn't say anytihng about rotation as far as i can see, how can we derive these torque equations?
violates conservation of angular momentum,what do you mean?if we can not derive them from newton's eqs, than they didn' completely explain the motion and angular momentum is a whole different thing?It is not possible to derive the conservation of angular momentum from Newton's laws. It a simple thing to postulate a counter example that satisfies the Newton's laws, but violates conservation of angular momentum. So you cannot derive the torque equations either, from Newton's laws alone.
I'll use coordinates for two dimension in the example: The particle A is in position (-1,0), the particle B is in position (1,0), the particle A exerts a force F_(AB)=(0,1) to the particle B, and particle B exerts a force F_(BA)=(0,-1) to the particle A. There force vectors satisfy the Newton's third law, because they are equal in magnitude, and in opposite direction. However, the angular momentum is not conserving, because the two particles will start rotating counter clockwise on their own.violates conservation of angular momentum,what do you mean?
In order to have a complete explanation of the motion, we need to of course specify what kind of forces particles use to interact. This is something different than the Newton's laws. Newton's laws are only a set of rules which the forces must obey, but there can be more rules for the forces too. On possible restriction to the forces is that the force vector must point in the same direction as the spatial separation vector between the particles. In this case the angular momentum will be conserved.if we can not derive them from newton's eqs, than they didn' completely explain the motion and angular momentum is a whole different thing?
so you're saying that the static equilibrium conditions: sum of all forces must be zero come from newtons law and the sum of the moment must be zero comes from somewhere else.Then where is this moment conditon come from?I'll use coordinates for two dimension in the example: The particle A is in position (-1,0), the particle B is in position (1,0), the particle A exerts a force F_(AB)=(0,1) to the particle B, and particle B exerts a force F_(BA)=(0,-1) to the particle A. There force vectors satisfy the Newton's third law, because they are equal in magnitude, and in opposite direction. However, the angular momentum is not conserving, because the two particles will start rotating counter clockwise on their own.
This example proves, that if you merely assume that in some system Newton's laws are being satisfied, it is not yet possible to prove that under zero external torque, the angular momentum would be conserving.
In order to have a complete explanation of the motion, we need to of course specify what kind of forces particles use to interact. This is something different than the Newton's laws. Newton's laws are only a set of rules which the forces must obey, but there can be more rules for the forces too. On possible restriction to the forces is that the force vector must point in the same direction as the spatial separation vector between the particles. In this case the angular momentum will be conserved.
The Newton's approach is that we set a postulate, that for all forces there are corresponding counter forces. In similar spirit, we can simply set a postulate, that for all torques there are corresponding counter torques.so you're saying that the static equilibrium conditions: sum of all forces must be zero come from newtons law and the sum of the moment must be zero comes from somewhere else.Then where is this moment conditon come from?
Good luck on your searchdear jostpuur you're saying that it's a postulate and deal with it:), but i still believe that
it's a cosequence of the seceond rule ,and somewhere there must be a relation between 2nd rule and rotational stuff.
Jostpuur, Can you point me to a reference (preferably online) that discusses this in more detail. My mechanics texts do not really discuss this in a way that I find satisfactory. My assumption was that Newton's laws were more fundamental (in the v << c limit) than angular momentum conservation, and that angular momentum was only conserved under specific circumstances (like constant and non-relativistic mass).It is not possible to derive the conservation of angular momentum from Newton's laws.
Not really... I don't remember very well where I have learned this myself. I don't think I've ever read that claim from anywhere. I've only thought about it myself.Jostpuur, Can you point me to a reference (preferably online) that discusses this in more detail.It is not possible to derive the conservation of angular momentum from Newton's laws.
I looked, but can't find a derivation in my notes that does not use Geometric Algebra (I'm sure I did one, but it must be on paper somewhere). I've attached what I do have typed up... you can probably get an idea how to directly approach the same radial acceleration decomposition using traditional cross products. Like I said, the key is exploiting a cross product expansion of \rcap', and taking derivitives of r = rcap |r|. If you can't derive the rcap' expression yourself just use what I posted above (my old Salus/Hille Calculus text did that rcap' derivation but it was a bit involved and indirect).Peeter could you put the derivation or send it to me?I tried but my maths is weak.
Hi jostpuur!I'll use coordinates for two dimension in the example: The particle A is in position (-1,0), the particle B is in position (1,0), the particle A exerts a force F_(AB)=(0,1) to the particle B, and particle B exerts a force F_(BA)=(0,-1) to the particle A. There force vectors satisfy the Newton's third law, because they are equal in magnitude, and in opposite direction. However, the angular momentum is not conserving, because the two particles will start rotating counter clockwise on their own.
This example proves, that if you merely assume that in some system Newton's laws are being satisfied, it is not yet possible to prove that under zero external torque, the angular momentum would be conserving.
Yes … obviously, we could define forces in a way that violates Newton's second and third law, which your example does.In order to have a complete explanation of the motion, we need to of course specify what kind of forces particles use to interact. This is something different than the Newton's laws. Newton's laws are only a set of rules which the forces must obey, but there can be more rules for the forces too. On possible restriction to the forces is that the force vector must point in the same direction as the spatial separation vector between the particles. In this case the angular momentum will be conserved
The Newton's laws were not being violated in my example.Yes … obviously, we could define forces in a way that violates Newton's second and third law, which your example does.