Quantum Mechanics for Mathematicians

but these are precisely Hamilton’s equations since the Euler-Lagrange equations
imply
∂L
∂q

=

d

dt

∂L

∂q ̇

= ̇p

While the Legendre transform method given above works in some situations,
more generally and more abstractly, one can pass from the Lagrangian to the
Hamiltonian formalism by taking as phase space the space of solutions of the
Euler-Lagrange equations. This is sometimes called the “covariant phase space”,
and it can often concretely be realized by fixing a timet= 0 and parametrizing
solutions by their initial conditions at such at= 0. One can also go directly
from the action to a sort of Poisson bracket on this covariant phase space (this
is called the “Peierls bracket”). For a general Lagrangian, one can pass to a
version of the Hamiltonian formalism either by this method or by the method of
Hamiltonian mechanics with constraints. Only for a special class of Lagrangians
though will one get a non-degenerate Poisson bracket on a linear phase space
and recover the usual properties of the standard Hamiltonian formalism.

35.2 Noether’s theorem and symmetries in the Lagrangian formalism

The derivation of the Euler-Lagrange equations given above can also be used to
study the implications of Lie group symmetries of a Lagrangian system. When
a Lie groupGacts on the space of paths, preserving the actionS, it will take
classical trajectories to classical trajectories, so we have a Lie group action on
the space of solutions to the equations of motion (the Euler-Lagrange equations).
On this space of solutions, we have, from equation 35.1 (generalized to multiple
coordinate variables),

δS[γ] =





∑d

j=1

∂L

∂q ̇j

δqj(X)



(t 1 )−





∑d

j=1

∂L

∂q ̇j

δqj(X)



(t 2 )

where nowδqj(X) is the infinitesimal change in a classical trajectory coming
from the infinitesimal group action by an elementXin the Lie algebra ofG.
From invariance of the actionSunderGwe must haveδS=0, so



∑d

j=1

∂L

∂q ̇j

δqj(X)



(t 2 ) =





∑d

j=1

∂L

∂q ̇j

δqj(X)



(t 1 )

This is an example of a more general result known as “Noether’s theorem”.
In this context it says that given a Lie group action on a Lagrangian system
that leaves the action invariant, for each elementXof the Lie algebra we will
have a conserved quantity
∑d

j=1

∂L

∂q ̇j

δqj(X)