Equations of Motion (Review)

In Newtonian mechanics it is axiomatic that no particle or distribution of "matter" can exert a force on itself. If, relative to the set of inertial frames, a particle’s momentum varies with time, then according to Newton an external force must be acting on the particle. (Newton defined the set of inertial frames in his first law. If the momentum of a particle that is subject to no net external force is observed to be constant in time, then the particle is being observed from an inertial frame.)

Newton stated the relationship between an externally applied force and the time rate of change of the particle’s momentum as an equality:

. (1)

This, his second law, has stood the test of time remarkably well, even holding when the more recently discovered dependence of inertial mass on speed is taken into account.

Perhaps the most mysterious of Newton’s three laws is the third, which states that for every "action" (or externally applied force) there is an equal and oppositely directed "reaction." If an agent applies the external force, F, to a particle, causing the particle’s momentum to vary in time, then the particle applies an equal and oppositely directed force on the agent. The change in momentum, imparted to the particle by F acting over a given period of time, is precisely countered by a change of momentum imparted by the particle to the external agent. Similarly, the "positive" work done by F acting through a given distance is exactly matched by the "negative" work done by the particle on the source of F. In brief, Newton’s third law is a statement of the conservation of momentum and energy.

Newton’s third law is mysterious in that it offers no explanation for the source or mechanism of the equal and oppositely directed reaction force. It is as if, whenever an agent applies a force to a particle, some unseen entity applies an equal but oppositely directed force back on the particle/driving agent. In this mechanism the particle becomes the point of contact between the driving agent and the unseen counterpart. Any momentum imparted by the driving agent to the unseen counterpart is exactly matched by an equal and opposite momentum imparted to the driving agent. The total momentum remains unchanged.

The first insight regarding the nature of the unseen counter-agent followed Maxwell’s statement of the electromagnetic field equations and Lorentz’s statement of the electromagnetic force experienced by a charge in an electromagnetic field. For it is readily demonstrated that a distribution of electric charge experiences an electric force when a (the particle’s acceleration) and/or da/dt are nonzero. This electric force is attributable to the charge’s own electric field. If the charge and its field are considered to be a single entity, then the Newtonian axiom, that no object can exert a force on itself, seems to be in doubt. An alternative is to view the charge as a point of contact between an external, driving agent and the electromagnetic field. The driving agent pours momentum (and energy) into the electromagnetic field by exerting a "contact" (or non-electromagnetic) force on a charged particle. And the field responds with an equal and oppositely directed electric force on the particle, said force "passed through" to the driving agent.

The discovery of atoms, and the classical model of a positively charged nucleus embedded in a "cloud" of negative charge, dispels some of the mystery surrounding Newton’s reaction force. For although the electromagnetic field of an atom is nominally zero outside of its boundaries, powerful fields exist within. And if, for example, an agent somehow accelerates the nucleus by exerting a force on it, momentum pours into the inter-atomic field. To the extent the sub-atomic constituent particles are themselves assemblies of charged "quarks," similar arguments apply to protons, neutrons, etc.

A convenient charge distribution (and one used in several articles on this site) is the spherical shell of charge. A caveat in the case of any static distribution of charge is that some non-electromagnetic mechanism must be assumed if the distribution does not disperse with time. Having said this, we can show that the momentum in a spherical shell’s electromagnetic field is proportional to its velocity:

. (2)

Here q is the charge and R is the shell’s radius. An obvious definition of the shell’s electromagnetic mass is thus

. (3)

If an external contact force is applied to the shell, then (a) the shell accelerates, and (b) momentum flows into the shell’s field precisely as Newton prescribes:

. (4)

Furthermore, the shell’s acceleration induces electric field components in the shell’s surface, resulting in an electric reaction force, and the net effect precisely counteracts the applied contact force:

. (5)

It is customary to refer to this force as the inertial reaction force. In effect Newton’s laws lie implicit in the electrodynamics of Maxwell and Lorentz (although the Newtonian laws were formulated before the underlying electric nature of "neutral" matter was understood).

But the theory of Maxwell/Lorentz transcends Newtonian theory in one important regard. In Newtonian theory (assuming mass is constant in time)

. (6)

But according to Maxwell/Lorentz a nonzero da/dt results in a second induced electric force. This second component of induced force is referred to as the radiation reaction force, FRadReact. It is independent of a given charge distribution’s particulars, and was determined to be

. (7)

The total reaction force for the spherical shell of charge can therefore be (non-relativistically) written as

(8)

.

If the entire reaction force acts back upon the driving agent (or upon the source of the external contact force, F), then the correct law for the spherical shell of charge is

. (9)

An important distinction between the two parts of F in Eq. 9 has to do with "recoverability." Any work (or impulse), expended by q2a/6peoRc2 to accelerate the charge, can be recovered when the charge is decelerated to its initial speed. This part of the external force thus obeys the Work-Energy Theorem (quite as the external force in Newton’s F=ma does). But any work expended by (-q2/6peoc3)(da/dt) is lost in the form of radiation. Indeed this is why (q2/6peoc3)(da/dt) is referred to as the radiation reaction force. For example, in the case of periodic motion, say

, (10)

the net work per cycle done by mema is zero:

. (11)

But the work per cycle done by (-q2/6peoc3)(da/dt) is always greater than zero. In fact the work per cycle done by this part of the total driving force precisely equals the field energy flux per cycle through an enclosing surface:

. (12)

Now in general it appears in the real world that charge is always linked to some quantity of "mechanical" mass. The total mass of a charged particle is the sum of this mechanical mass and the electromagnetic mass. Whereas the momentum associated with electromagnetic mass resides in the electromagnetic field, the momentum associated with mechanical mass appears to be local to the particle. In this sense the mechanical momentum is as "mysterious" as the momentum of Newton’s particles of neutral matter. Factoring mechanical mass into the equation of motion changes Eq. 9 to

. (13)

The total reaction force is then

. (14)

Like –mmecha, the electromagnetic inertial reaction force hypothetically always acts on the external agent (the source of F in Eq. 13). But the radiation reaction force may not invariably do so. It only acts on the driving agent when that agent counteracts all of the reaction forces. But in some situations the external agent may counteract only the inertial reaction forces. In such cases the radiation reaction force (if nonzero) may act in concert with the external agent force, and Eq. 13 must be replaced by

. (15)

Outwardly Eqs. 13 and 15 may appear to be the same. But there is a subtle difference. In the case of Eq. 13 the driving force, F, is such as to counteract all reaction forces and, in the case of periodic motion, it cannot be a function of the particle’s position. That is, part of its total power expenditure is always positive and equal to the rate at which radiant energy is emitted into infinite space. In the case of Eq. 15 F is typically position dependent. For example, it might be the force exerted on a charged particle by a spring. If the motion is that specified by Eq. 10, then the radiation reaction force always opposes F, and the particle’s acceleration is consequently less than it would be if q equaled zero.

In many articles featured on this site it has been assumed that the external force counteracts all of the reaction forces. In a more recent article consideration is given to a case where the radiation reaction force acts in concert with an external, position-dependent force, resulting in non-periodic motion. In effect the sum of the radiation reaction force and the position-dependent force constitutes the external force, and the counteraction to this force is provided by the negative rate at which the particle’s momentum varies in time.