Where do the conservation laws come from?

Question

I know the conservation of energy comes from Noether's theorem via the time-translational symmetry, and if I remember correctly, the conservation of momentum comes from space-translational symmetry.

My question is: for a given system, what is the starting point of identifying conservation laws, and how do you know that there's not more that you haven't identified? Do you just start by identifying symmetries?

As an example, consider a system of hard, spherical, elastically colliding billiard balls. I suppose the conserved quantities are energy, momentum, and angular momentum; for each particle these correspond to the terms $\frac{1}{2}mv^2$, $mv$, and $m\omega$. How do we know there isn't something like $kmv^3$? (I know this isn't one, but it's just to demonstrate my point).

And why is the conserved quantity always a scalar? Are there cases where the conserved quantity is not calculated by summing other scalars?

Answer 1

Noether's theorem is one way to relate conservation laws to symmetries, but it is not the only or necessarily the most common method of doing so. For example, in quantum mechanics a continuous symmetry is associated with a family of unitary operators $U(\lambda)$ that shift the eigenvalues associated with some operator (e.g. the position operator, in which case $U(\lambda) \equiv T(\lambda)$ translates states by $\lambda$).

One can show that such an operator is expressible in terms of a self-adjoint (Hermitian) generator $G$ via the relation $U(\lambda) = e^{iG\lambda}$. When the Hamiltonian is invariant invariant with respect to this transformation (i.e. $U^\dagger H U = H$ or equivalently the commutator $\left[U, H\right]$ vanishes) time evolution does not change the eigenvalues of $U$. The same is true of $G$ and, as such, the eigenvalues of $G$ (and $U$) are conserved in this circumstance. For spacial translations, $G$ is proportional to the momentum operator $p$ and momentum is the conserved quantum number.

Conserved quantities are not always scalar. Momentum is, for example, a vector.

In response to your final question, finding all the symmetries of a given system can be an exceptionally difficult problem. Finding such symmetries is a common method of trying to render many-body problems in quantum mechanics "integrable" (exactly solvable) or tractable via numerical methods. There is no systematic method that works generally. Occasionally, hidden symmetries are discovered in the context of well-known problems, and this can be a big deal when it happens.

Answer 2

Two comments:

A conservation law, being a function of the degrees of freedom of a system, relates variables and thereby restricts them to some exten. More independend conservation laws means more restrictions and depending on the system, you can't have infinite constraints while still maintaining propagation of the variables.

Secondly, given such a constant of motion (Energy, momentum), earch function of it is also a constant. E.g. the quantity $(\sum_im_iv_i^2)^7$ ist constant if $\sum_i\tfrac{1}{2}m_iv_i^2$ is.

Answer 3

Conservation laws come exclusively from equations of motion. If you found your solutions $y_i(t)$ as functions of the initial data $y_i(t|y_k(t_0))$, then you can resolve them to express the initial data via the "current data" $y_k(t_0)=f_k(y_i)$ and each such expression is a "conservation law" - a combination of dynamical variables that is constant in time. So the true origin of conservation laws is existence of solutions of equations. The number of independently conserving quantities is the number of independently given initial data, although the form of conserving quantities is not fixed - any combination of conserved quantities is a conserved quantity. We are mostly familiar with "additive" conservation laws where a sum of some variable combinations of different entities is written.

Symmetries do not lead to conservation laws. They help construct some conservation laws from the Lagrangian, and nothing else. In case of a symmetry a given conserved quantity can take a simpler form, nothing more.