noethers theorem

1. we proved the noether's theorem( for every continuous symmetry with end points fixed characterized by m parameters, there are exactly m conserved quantities characterised by Q_k, where Q_k=(delta L /delta q_dot)*(delta q / delta alpha_k), alpha ks are parameters.) this we did by writing the transformed q_i's as functions of q_j and alpha_k, effecting the variation of the action,setting it to zero for a symmetry and using the lagrange equations.
2.next, we also considered the cases where the end points vary and noted that such a variation of end points causes the lagrangian to remain otherwise unchanged.this variation causes a variation of the coordiantes at the boundary, but along the trajectory.the limits of integration change.for the action to remain unchanged, the change of lagrangian due to this must equal the change of lagrangian due to the coordinate variation.a short calcualtion then yielded the conservation of energy.
3.we also saw that the alpha_k 's form a lie algebra
4.we derived the energy momentum tensor of the field.we noted that the tensor was not unique and so certain tensors(anti-symmetric in the last two indices) can be added to it to symmetrize it.
5.we derived its form for the classical electromagnetic field, checked that the tensor was symmetric and traceless.
6.it was also noted that this tensor cannot always be diagonalised; since under such a transformation the metric (which is not an identity metric, it is a minkowski metric) must also be preserved.