The basis for Special Relativity is that the laws of physics are independent of which inertial coordinate system we write them in.
We wish to write equations in terms of scalars, 4-vectors, and tensors, with both sides of the equation transforming the
same way under rotations and boosts.
Such equations are said to be
covariant, because both sides of the equation transform in the same standard way.
The Lagrangian is based on the principle of least Action.
The Action as we know it so far is clearly not a vector or a tensor so our obvious choice is to make it a scalar.
Our definition of the Action so far is.
We should then have
be a scalar quantity.
But what scalar do we have?
For a particle, the
mass is the only scalar we can use.
Clearly this is not related to external potential applied to the particle.
agrees with our previous result
indicating that we have picked the right kinetic term for the Lagrangian.
The (velocity independent) potential can be subtracted.
This is an important relativistic basis for our use of
in nonrelativistic problems.
To have fully relativistic problems we need to have potentials, for example, that are relativistically correct.
We will take a look at the relativistic equations of electromagnetism, but this is really beyond the scope of this course.
In field theory, there will be scalar interaction terms in the Lagrangian,
like the interaction between the electric current due to particles with the vector potential,
a dot product between 4-vectors.
Hamiltonian can be computed from the Lagrangian.
It is the
It is quite interesting to look at this Action which is to be minimized in Minkowski space
and to try to formulate the principle of least Action in a covariant way.
What Action should we use for particles in Special Relativity?
The proper time is directly related to the space-time interval.
So we can write our action to be minimized in terms of the spacetime interval.
That is the
particle follows a path through 4-space that minimizes the space-time interval.
In curved space-time, this path is called a
We may take this back to an integral over proper time.
While the factor of
may seem useless, it becomes useful if an external potential (other than gravity) is applied.