Angular momentum

“ If we have just the particles, and the magnet, close together, and everything is standing still, I say there is angular momentum in the field, a hidden form of angular momentum which does not appear as actual rotation. ”

“ So that no angular momentum enters at B, and therefore on the whole none leaves at A. The transfer of angular momentum through A is of two kinds—first, that due to the passage of rotating matter, and, secondly, that due to the couple with which matter to the right of A acts upon matter to the left of A. The mass of matter moving through A per second is piro 2 U, where a is the radius of the wire and is its density. If is the angle of twist, the angular velocity is . ”

“ In general, the increase of angular momentum is
[7]
by which the fact can be confirmed again, that the component of (parallel to ) doesn't matter.
Eventually, we imagine that the lever is in a shell, at which an axis of rotation is mounted, and from which – for example by strained springs – forces and are exerted. The energy current considered above, is continued in the axis after it left the lever at , then it enters the shell through the mounting locations of the axes, from there it enters the spring which affects the lever at , and thus it is closed. ”

“ The particles are all moving in various directions, and we take any point and calculate how much area is being swept out by the radius from this point to each of the particles. In this calculation the masses which are heavier count more strongly; if one particle is twice as heavy as another its area will count twice as much. So we count each of the areas swept out in proportion to the mass that is doing the sweeping, add them all together, and the resulting total is not changing in time. That total is called the angular momentum, and this is called the law of conservation of angular momentum. ”

“ The axis of angular momentum would have an invariable position in space, and would travel with respect to the earth round the axis of figure with a velocity where is the sidereal angular velocity of the earth. ”

“ If the velocity of the electrons is not small compared with the velocity of light, the constancy of the angular momentum no longer involves a constant ratio between the energy and the frequency of revolution. ”

“ This couple will tend to increase the angular momentum of the system, and, acting cumulatively, will soon cause an appreciable change of period, disagreeing with observation if the speed is at all comparable with that of light. ”

“ One plan is to assume that any body whatever may be treated as if it were composed of material particles, i.e. mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction. In the case of a rigid body we must suppose that those forces adjust themselves so as to preserve the mutual distances of the various particles unaltered. On this basis we can predicate the principles of linear and angular momentum, as in § 15. ”

“ Hence, whichever form of postulate we adopt, we are led to the principles of linear and angular momentum, which form in fact the basis of all our subsequent work. It is to be noticed that the preceding statements are not intended to be restricted to rigid bodies; they are assumed to hold for all material systems whatever. ”

“ Thus one could allow those non-electromagnetic forces (similarly to the molecular forces) to compensate the electromagnetic torque. So one would have given up the (pure electromagnetic) energy- and momentum conservations theorems, and would only maintain the center of gravity theorem. ”

“ This conception, that the momenta are capable of being produced by instantaneous impulses on the system at rest, is introduced only as a method of defining the magnitude of the momenta, for the momenta of the system depend only on the instantaneous state of motion of the system, and not on the process by which that state was produced.
In a connected system the momentum corresponding to any variable is in general a linear function of the velocities of all the variables, instead of being, as in the dynamics of a particle, simply proportional to the velocity. ”

“ Thus in the theory of relativity momentum appears as an invariant mass multiplied by a modified velocity . The physicist, however, prefers for practical purposes to keep to the old definition of momentum as mass multiplied by the velocity . We have , accordingly the momentum is separated into two factors, the velocity , and a mass , which is no longer an invariant for the particle but depends on its motion relative to the observer's space and time. ”

“ Angular velocity always falls as angular displacement grows (and vice-versa) , so each full period will look like an ellipse, and the loss of energy to friction will mean that each period will be represented by a slightly smaller ellipse as the system spirals toward its equilibrium position of zero displacement and zero velocity: straight up and down, and not moving. See Figure 5.1 for a rough plot of what the graph of this situation would look like in a state-space for the pendulum. ”

“ The totality doesn't require any torque, it rather represents (as in the model consisting of angle-lever and shell) a "completely static system" [10] , which obeys the dynamics of the mass-point for stationary and quasi-stationary translatory motions.
That the momentum of the angle-lever has not the direction of its velocity, is also given by another consequence, which appears to be paradox in Newton's dynamics. ”

“ Now it has been shown mathematically in several different ways that when small particles collect into a larger mass, the collisions tend to make the resultant orbit of the combination both more circular and more conformant to the general plane than its constituents. But we may see this more forthrightly by means of the general principle enunciated above. For in fact both results are direct outcomes of the conservation of moment of momentum. Given a certain moment of momentum for the system, the total energy of the bodies is least when they all move in one plane. ”

“ The older theory needs a force whose work covers a portion of the energy radiation to maintain uniform motion. According to the theory of relativity, however, it completely stems from the energy of the light source.
The given values for the energy and momentum radiation are valid, since they depend only on the acceleration of the moving electron, even if it doesn't oscillate together with a dipole, but is moving in any other way. ”

“ As to the energy we know already a number of different forms: the kinetic energy, gravitation, the elastic deformation energy, heat, chemical energy, and it is therefore not a principally new feature when to these different forms, the electromagnetic energy is attached as another form. Conversely, until now only a single form was known as momentum: mechanical momentum. ”

“ Indeed, as the denser parts of the planet necessarily lie in its interior, and have therefore neither the velocity nor the radius of the more superficial portions, it seems necessary to admit that the moments of momentum of the planets will be proportional to some lower power of the radius than the fifth. The total moment of momentum of the planets by rotation, when multiplied by a constant factor, and added to the revolutional moment of momentum, will remain absolutely constant. ”