35.2 Noether’s theorem and symmetries in the Lagrangian formalism

35.2 Noether’s theorem and symmetries in the Lagrangian formalism

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie group · group action

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 group $G$ acts on the space of paths, preserving the action $S$, 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),

$$\delta S[\gamma ]=\left(\sum _{j=1}^d\frac{\partial L}{\partial {\dot{q}}_j}\delta q_j(X)\right)(t_1)-\left(\sum _{j=1}^d\frac{\partial L}{\partial {\dot{q}}_j}\delta q_j(X)\right)(t_2)$$

where now $\delta q_j(X)$ is the infinitesimal change in a classical trajectory coming from the infinitesimal group action by an element $X$ in the Lie algebra of G. From invariance of the action $S$ under G we must have $\delta S=0$ , so

$$\left(\sum _{j=1}^d\frac{\partial L}{\partial {\dot{q}}_j}\delta q_j(X)\right)(t_2)=\left(\sum _{j=1}^d\frac{\partial L}{\partial {\dot{q}}_j}\delta q_j(X)\right)(t_1)$$

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 element $X$ of the Lie algebra we will have a conserved quantity

$$\sum _{j=1}^d\frac{\partial L}{\partial {\dot{q}}_j}\delta q_j(X)$$

which is independent of time along the trajectory.

A basic example occurs when the Lagrangian is independent of the position variables $q_j,$ depending only on the velocities ${\dot{q}}_j$ , for example in the case of a free particle, when $V(q_j)=0$ . In such a case one has invariance of the Lagrangian under the Lie group ${\mathbf{R}}^d$ of space-translations. Taking X to be an infinitesimal translation in the j-direction, one has as conserved quantity

$$\frac{\partial L}{\partial {\dot{q}}_j}=p_j$$

For the case of the free particle, this will be

$$\frac{\partial L}{\partial {\dot{q}}_j}=m{\dot{q}}_j$$

and the conservation law is conservation of the jth component of momentum. Another example is given (in $d=3)$ by rotational invariance of the Lagrangian under the group $SO(3)$ acting by rotations of the $q_j$ . One can show that, for X an infinitesimal rotation about the $k$ axis, the kth component of the angular momentum vector

$$\mathbf{q}\times \mathbf{p}$$

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 the $U(1)$ symmetry of rotations in the qp plane) that are unavailable in the Lagrangian formalism, which just uses configuration space, not phase space.

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