which is independent of time along the trajectory.
A basic example occurs when the Lagrangian is independent of the position
variablesqj, depending only on the velocities ̇qj, for example in the case of a free
particle, whenV(qj) = 0. In such a case one has invariance of the Lagrangian
under the Lie groupRdof space-translations. TakingXto be an infinitesimal
translation in thej-direction, one has as conserved quantity
∂L
∂q ̇j=pjFor the case of the free particle, this will be
∂L
∂q ̇j=mq ̇jand the conservation law is conservation of thejth component of momentum.
Another example is given (ind= 3) by rotational invariance of the Lagrangian
under the groupSO(3) acting by rotations of theqj. One can show that, for
Xan infinitesimal rotation about thekaxis, thekth component of the angular
momentum vector
q×p
will be a conserved quantity.
The Lagrangian formalism has the advantage that the dynamics depends
only on the choice of action functional on the space of possible trajectories, and
it can be straightforwardly generalized to theories where the configuration space
is an infinite dimensional space of classical fields. Unlike the usual Hamiltonian
formalism for such theories, the Lagrangian formalism allows one to treat space
and time symmetrically. For relativistic field theories, this allows one to exploit
the full set of space-time symmetries, which can mix space and time directions.
In such theories, Noether’s theorem provides a powerful tool for finding the
conserved quantities corresponding to symmetries of the system that are due to
invariance of the action under some group of transformations.
On the other hand, in the Lagrangian formalism, since Noether’s theorem
only considers group actions on configuration space, it does not cover the case
of Hamiltonian group actions that mix position and momentum coordinates.
Recall that in the Hamiltonian formalism the moment map provides functions
corresponding to group actions preserving the Poisson bracket. These functions
will give the same conserved quantities as the ones one gets from Noether’s
theorem for the case of symmetries (i.e., functions that Poisson-commute with
the Hamiltonian function), when the group action is given by an action on
configuration space.
As an important example not covered by Noether’s theorem, our study of
the harmonic oscillator exploited several techniques (use of a complex structure
on phase space, and of theU(1) symmetry of rotations in theqpplane) that are
unavailable in the Lagrangian formalism, which just uses configuration space,
not phase space.