| Ops.3vel {lorentz} | R Documentation |
Arithmetic Ops Group Methods for 3vel objects
Description
Arithmetic operations for three-velocities
Usage
## S3 method for class '3vel'
Ops(e1, e2)
## S3 method for class '4vel'
Ops(e1, e2)
massage3(u,v)
neg3(u)
prod3(u,v=u)
add3(u,v)
dot3(v,r)
Arguments
e1, e2, u, v |
Objects of class “ |
r |
Scalar value for circle-dot multiplication |
Details
The function Ops.3vel() passes unary and binary arithmetic
operators “+”, “-” and “*”
to the appropriate specialist function.
The most interesting operators are “+” and
“*”, which are passed to add3() and dot3()
respectively. These are defined, following Ungar, as:
\mathbf{u}+\mathbf{v} = \frac{1}{1+\mathbf{u}\cdot\mathbf{b}/c^2}
\left\{
\mathbf{u} +
\frac{1}{\gamma_\mathbf{u}}\mathbf{v} +
\frac{1}{c^2}\frac{\gamma_\mathbf{u}}{1+\gamma_\mathbf{u}}
\left(\mathbf{u}\cdot\mathbf{v}\right)\mathbf{u}
\right\}
and
r\odot\mathbf{v} =
c\tanh\left(
r\tanh^{-1}\frac{\left|\left|\mathbf{v}\right|\right|}{c}
\right)\frac{\mathbf{v}}{\left|\left|\mathbf{v}\right|\right|}
where \mathbf{u} and \mathbf{v} are three-vectors
and r a scalar. Function dot3() has special dispensation
for zero velocity and does not treat NA entries entirely
consistently.
Arithmetic operations, executed via Ops.4vel(), are not defined
on four-velocities.
The package is designed so that natural R idiom may be used for three velocity addition, see the examples section.
Value
Returns an object of class 3vel, except for prod3()
which returns a numeric vector.
Examples
u <- as.3vel(c(-0.7, 0.1,-0.1))
v <- as.3vel(c( 0.1, 0.2, 0.3))
w <- as.3vel(c( 0.5, 0.2,-0.3))
x <- r3vel(10) # random three velocities
y <- r3vel(10) # random three velocities
u+v # add3(u,v)
u-v # add3(u,neg3(v))
-v # neg3(v)
gyr(u,v,w)
## package is vectorized:
u+x
x+y
f <- gyrfun(u,v)
g <- gyrfun(v,u)
f(g(x)) - x # should be zero by eqn10
g(f(x)) - x
(u+v) - f(v+u) # zero by eqn 10
(u+(v+w)) - ((u+v)+f(w)) # zero by eqn 11
((u+v)+w) - (u+(v+g(w))) # zero by eqn 11
## NB, R idiom is unambiguous. But always always ALWAYS use brackets.
## Ice report in lat 42.n to 41.25n Long 49w to long 50.30w saw much
## heavy pack ice and great number large icebergs also field
## ice. Weather good clear
## -u+v == (-u) + v == neg3(u) + v == add3(neg3(u),v)
## u+v+w == (u+v)+w == add3(add3(u,v),w)