Download Section 9

Energy and Momentum
LL2 Section 9
From mechanics, the momentum of a particle is given by
v2 = va va
For v << c,
or c ,
p  mv.
If the force on the particle is perpendicular to the
velocity, then the velocity changes only in direction,
and
If the force on the particle is in the same direction as the velocity,
the velocity changes only in magnitude, and
HW. (Note typos in new
printing of book.)
The ratio of force to acceleration, dp/dt : dv/dt, is different in the two cases.
In classical mechanics, F/a = m.
Energy
Energy of a free particle does not go to zero as v  0.
For v/c << 1
Classical expression
Relativistic e is completely definite and positive.
Classical e is defined only to within and arbitrary additive constant. It may be positive, zero,
or negative.
An atom.
Composite body at rest:
Rest energies of constituents + their kinetic energy + their interaction energies = M c2

Total energy of
composite particle
at rest, including
internal kinetic
energy and
interactions.
Sum of rest energies of
parts, which does not
include kinetic energy or
interactions
No conservation of
mass in relativistic
mechanics.
Only conservation of
energy (including rest
energies) is valid.
The relation between
momentum p and
energy E:
= e2/c2
Hamiltonian
For
Usual classical
expression
Relation between
This will be used a lot
If
Thus, if
the particle cannot move at speed c.
Particles with m = 0 can have velocity c. Then p = e/c.
For ultra relativistic particles with mass
HW
The four dimensional form of the equation of motion
The action for a free particle
Principle of least action:
Integrate
by parts
Variation is of trajectories for the same end points
A free particle has
constant 4-velocity
Equation of motion
for a free particle
To find the momentum, we need S as a function of xi.
We need the variation of S with respect to a
change in the coordinates.
We had
Now consider the actual trajectory, so
Take the initial point to be fixed
Allow the final point to vary
Momentum 4-vector
From Mechanics
4-gradient of S is
= (e/c, -p)
Momentum & energy are components of a single 4-vector.
Transformation formulas follow immediately (6.1)
Square of 4-momentum
(7.3)
Force 4-vector
For free particle, force = 0
We already had
4-acceleration
Relativistic Hamilton-Jacobi Equation of a free particle
From mechanics
Transition to classical mechanics