"Points of Interest" (POI) are points in a coordinate space of a frame of reference. Note that POI is a point of interest, which may refer to a particle, but more generally simply refers to a point in space.
The notation used here is :
POI (POIo,POIp) = "Point Of Interest" is a specific point, and is typically where estimates of [B,E,F, etc] are required. Normally, the POI is STATIONARY in EITHER :
RFp - particle reference frame (eg POIp,POIo(t))
RFo- observer reference frame (eg POIo,POIp(t))
Note that a POI cannot be simultaneously be fixed in BOTH RFo and RFp at a specific time t, EXCEPT when BOTH vo & vp = zero.
POIo(t) = POI fixed in particle RFp, but seen in the observer Rfo (eg for EIocv(POIp(t),t),B,etc)
POIp(t) = particle location, as a POI, usually expressed in the RFo.
For a (POI) fixed in the particle reference frame (RFp) => (POIp) :
(POIp) applies to to basic measures and their derivatives for (POI)s fixed with respect to (RFp). This is often used for coordinate based measures such as :[rpcs,Rθ0pcs,RθPI2pcs,sin(Opc),cos(Opc)], etc)
CAUTION : The following two symbols are a "flip of the reference frame" of POIp, in the sense that they refer to points in the OBSERVER frame of reference RFo that describe the trajectory of POIp through RFo.
(POIp(t)) refers to the position over time, in the OBSERVER frame of reference RFo, of a point of interest that is fixed in the particle reference frame RFp. Effectively, this is a trajectory of POIp in RFo.
(POIp(tx)) refers to a point of interst in the OBSERVER reference frame (RFo) of a fixed point in the particle reference frame RFp (POIp), the trajectory of which, at the specific time tx, coincides with a fixed POIo, defining a common point FIXED in RFo.
Note that :
dp[dt : POIp(tx)] = dp[dt : POIo] = 0,
whereas
dp[dt : POIp(t)] = Vonv(particle)
applies to to basic measures and their derivatives for (POI)s that move with respect to (RFp). This is often used for (RFo) coordinate based measures with respect to the observer origin such as [rocs,Rθ0ocs,RθPI2ocs,sin(Ooc),cos(Ooc)], etc).
It is also used for derived measures and their derivatives for (POI)s fixed with respect to (RFp), but which are time dependent (no examples yet 03Feb2016).
It is also used for derived measures and their derivatives for (POI)s that move with respect to (RFp), and which are obviously time dependent (no examples yet 03Feb2016).
For a (POI) fixed in the OBSERVER reference frame (RFo) => (POIo) :
(POIo) applies to to basic measures and their derivatives for (POI)s fixed with respect to (RFo). This is often used for coordinate based measures such as [rocs,Rθ0ocs,RθPI2ocs,sin(Ooc),cos(Ooc)], etc)
CAUTION : The following two symbols are a "flip" of the meaning of POIo, in the sense that they refer to points in the PARTICLE frame of reference RFp that describe the trajectory of POIo through RFp.
(POIo(t)) refers to the position over time, in the PARTICLE frame of reference RFp, of a point of interest that is fixed in the particle reference frame RFo. Effectively, this is a trajectory of POIo in RFp.
(POIo(tx)) refers to a point of interst in the PARTICLE reference frame (RFp) of a fixed point in the observer reference frame RFo (POIo), the trajectory of which, at the specific time tx, coincides with a fixed POIp, defining a common point FIXED in RFp.
Note that :
dp[dt : POIo(tx)] = dp[dt : POIp] = 0,
whereas
dp[dt : POIo(t)] = -Vonv(particle)
applies to basic measures and their derivatives for (POI)s that move with respect to (RFo). This is often used for coordinate based measures with respect to the particle such as [rpcs,Rθ0pcs,RθPI2pcs,sin(Opc),cos(Opc)] etc).
It is also used for and their derivatives for (POI)s fixed with respect to (RFo), but which are time dependent such as [E0odv,BTodv,etc] and their derivatives..
It is also used for derived measures and their derivatives for (POI)s that move with respect to (RFo), and which are obviously time dependent([ E0odv(POIo), BTodv(POIo,t), etc).
Examples :
rocs(POIo) = rpcs(POIo,t) but which also = rocs(POIo,t) = rpcs(POIo)
For the electrostatic field :
E0odv(POIo,t) = E0pdv(POIo(tx),t) != E0pdv((POIo,t) = Eodv(POIo)
For an induced electroic field :
EIpdv((POIo) = EIodv(POIo,t) but which also != EIpdv((POIo,t) = EIodv(POIo)
??? need more explanation.
??nyet : My interpretation is that this relates to Bill Lucas' comments :
p71h0.15 "... In order to obtain the results observed in the laboratory rame, it is necessary to evaluate Ei(rvt,t) at t=0 when the moving frame coincides with the laboratory frame after all terms for a given iteration have been evaluated. In order to simplify the iteration f successive terms the r'= r -vt terms are left in place in order to keep track of the correct power of r' for the derivative in the iterative term. The vt(cos(theta')) terms are explicitly dropped. ..."
CAUTIONS :
???You should ?ALWAYS? see the same "o" or "p" qualifier (meaning observer or particle reference frame) for the variable symbol and the point of interest symbol (for example, both "o"s in Rocs(POIo)). ???
This redundancy provides a good check for some errors that may srise in formulae (mostly when not paying enough attention to the frames of reference).
Note that time lags SHOULD affect field formulae unless field movement across infinite distance is instantaneous, but this issue is not addressed in this document..
Vibrating neutral diploes - are also looked at in a previous Chapter of Lucas's book. My nomenclature specific to that situation have yet to be specified.
General comment – where a "measure" (eg [R,v,E,B, etc]) is sought in a reference frame, then a POI fixed (stationary) in the same reference frame is not a function of time (i.e. doesn't move), whereas a POI fixed (stationary) in another reference frame IS a function of time.
It is recommended that the POI symbol ALWAYS be written in terms of the frame of reference in which the POI is fixed (stationary). For example :
EIocv(POIp(t),t) → the POI is stationary in the particle reference frame (RFp), and therefore is a function of time in the observer reference frame (Rfo) as is EIocv for that POI. POIp(t) makes both aspects clear.
EIpcv(POIp,t) → the POI is stationary in the particle reference frame (RFp), and therefore is NOT a function of time in the particle reference frame (RFp). POIp makes both aspects clear. This is an intermediary expression, as EIpcv in this case is not a function of time! This expression becomes EIpcv(POIp) = 0 as there is NO induced [B,E] in this case!!! But "EIpcv(POIp,t)" is a proper starting expression, as the simplifications easily follow in subse2quent steps.
EIpcv(POIo(t),t) → the POI is stationary in the observer reference frame (RFo), and therefore is a function of time in the particle reference frame (Rfp) as is the case for EIpcv. POIp(t) makes both aspects clear.
While one could write alternate forms
EIocv(POIo(t),t)
EIpcv(POIo(t),t)
these are NOT recommended, as they tend to "confuse" the issue (for me anyways, if I am not consistent), as the expression "EIocv(POIo(t),t)" loses the general sense that the POI referred to is fixed in the "o" frame (observer), which is NOT the case.