QuantumPhysics.dvi

The rhs is a total time derivative only if∂L/∂t= 0, namely the Lagrangian hasno explicit

time dependence. Assuming this to be the case, we see thatX=L, the associated conserved

charge is readily computed, and found to be the Hamiltonian,

H=

∑

i

piq ̇i−L (9.10)

which is what we expect: energy conservation results from time translation invariance.

9.3 Group and Lie algebra structure of classical symmetries

Transformations that permute the elements of a set form a group, which is generally a

subgroup of the full permutation group of the set in question. Generally, symmetry transfor-

mations act as permutations on the space of all solutions of an equation, and therefore also

form a group. It is actually in this context that groups were first discovered by Galois in

his study of the behavior of solutions to polynomial equations underpermutations of these

solutions.

Continuous symmetries depend differentiably on parameters and generally form Lie groups,

while their infinitesimal versions form Lie algebras. This may be seen asfollows. Suppose a

given Lagrangian system has two infinitesimal transformationsδ 1 qk(t) andδ 2 qk(t), both of

which generate infinitesimal symmetries with associated conservedchargesC 1 andC 2 , both

of which are conserved. The sum of the two infinitesimal transformationsδ 1 qk(t) +δ 2 qk(t)

is then also a symmetry with associated conserved chargeC 1 +C 2. But furthermore, as is

characteristic of a Lie algebra, there is an antisymmetriccommutatorof transformations

δ 1

(

δ 2 qk(t)

)

−δ 2

(

δ 1 qk(t)

)

(9.11)

which also produces a symmetry, with associated conserved charge{C 1 ,C 2 }. To see this, we

work out the composition rule in terms of Poisson brackets, using the fact thatδaqk(t) =

{qk(t),Ca}, and we find,

δ 1

(

δ 2 qk(t)

)

−δ 2

(

δ 1 qk(t)

)

=

{

C 1 ,{C 2 ,qk(t)}

}

−

{

C 2 ,{C 1 ,qk(t)}

}

=

{

{C 1 ,C 2 },qk(t)

}

(9.12)

The last line was obtained from the first by the use of the Jacobi identity which always holds

for the Poisson bracket.

As a final remark, the chargesC, discussed above always generate transformations of

the system, even if these transformations are not necessarily symmetries. For example, in a

mechanical system with degrees of freedomqkandpℓfork,ℓ= 1, 2 ,3 and canonical Poisson

brackets{qk,pℓ}=δkℓ, the (orbital) angular momentum generator, defined by

La=

∑

k,ℓ

εakℓqkpℓ (9.13)