Newton's Law of Motion in Modern Notation
The modern approach to obtaining the relativistic version of Newton's law of motion can be obtained by writing Maxwell's equations in covariant form and identifying a covariant form that is a generalization of Newton's law of motion.
Newton's law of motion can be written in modern covariant notation in terms of the field strength tensor as (cgs units):
where m is the particle mass, q is the charge, and
is the 4-velocity of the particle. Here, is c times the proper time of the particle and is the Minkowski metric tensor.
The field strength tensor is written in terms of fields as:
Alternatively, using the four vector:
related to the electric and magnetic fields by:
the field tensor becomes:
where:
The fields are transformed to a frame moving with constant relative velocity by:
where 
is a Lorentz transformation.
In the magnet/conductor problem this gives
which agrees with the traditional transformation when one takes into account the difference between SI and cgs units. Thus, the relativistic modification to Newton's law of motion using the traditional Lorentz force yields predictions for the motion of particles that are consistent in all frames of reference with Maxwell's equations.
Read more about this topic: Moving Magnet And Conductor Problem
Famous quotes containing the words newton, law, motion and/or modern:
“The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St Paul’s, like the editions of Balbec and Palmyra.”
—Horace Walpole (1717–1797)
“The Reverend Samuel Peters ... exaggerated the Blue Laws, but they did include “Capital Lawes” providing a death penalty for any child over sixteen who was found guilty of cursing or striking his natural parents; a death penalty for an incorrigible son; a law forbidding smoking except in a room in a private house; another law declaring smoking illegal except on a journey five miles away from home,...”
—Administration for the State of Con, U.S. public relief program (1935-1943)
“If we shall stand still
In fear our motion will be mocked or carped at,
We should take root here where we sit, or sit
State-statues only.”
—William Shakespeare (1564–1616)
“I may be old-fashioned. I don’t like this modern movement.... And yet, there are certain sorts of work a woman may well do; teaching, being governess, or any taking care of children.”
—Julia Dent Grant (1826–1902)