Classical Mechanics
10/22/2007
Stanford
class with Leonard Susskind.
In
what follows the meaning of the index is that it
indexes the three space coordinates.
The
kinetic energy:
Take
the derivative of the kinetic energy. Use the chain rule.
Substitute
to
get
Take
the time derivative of the potential.
Define
the total energy.
Take
the time derivative.
Define
the applied force to be the space derivative of the potential.
then
Therefore,
conservation of energy.
Change
the meaning of the index so that it now
indicates the ith particle.
Assume
that the force on a particle is the sum of the forces applied by the other
particles.
Assume
that the forces are “equal and opposite”.
Since
then
Therefore,
conservation of momentum.
The
problem is to minimize the function .
at
the minimum or stationary point
also
at
the minimum
F=ma
is local. However, the Principle of Least Action is global. Need endpoints
specified, and not initial conditions.
variable
speed of light?
The
total time is
Consider
the Action “A” and the Lagrangian “L”.
which implies
when minimized.
Sometimes the Action is labeled with “I” or “S”. In general
The Principle of Least Action is