Conservation Laws and What Happens if they go Wrong? I read this excellent article on the Conservation Laws and also I was taught in Schools that Conservation Laws cannot be proven and only verified. I was wondering what would actually happen if a Conservation Law turned out to be false?
I know it would question our measurements as well as our calculations as we use them almost unknowingly everywhere like unless stated every mechanics problem has the mass taken to be conserved so let's say the laws hold true here but break in the boundary cases as most things in Physics do like break when we approach the speed of light or the edge of the universe or some other drastic condition.
Are there any good discussions on what consequences it may have?
 A: The question "what happens if a conservation law is observed to be broken?" is not a hypothetical one $-$ it is a historical one.
The clearest example of this is the conservation of parity, a quantum-mechanical quantity which is related to the invariance of physical laws under spatial inversion. (There is a deep relation, called Noether's theorem, between conservation laws and symmetries.)
Up until the early 1950s, parity was assumed to be conserved in all of the fundamental physical laws, right up until this was explicitly tested for the weak nuclear force, and the experiment showed that parity is not conserved in weak interactions.
The results of this experiment forced a complete re-think of the underpinnings of quantum field theory, and the symmetry properties of the fundamental forces. It was a bit of a violent process at times, but it was a key part of the process of formulating the Standard Model of particle physics.

So, what happens if somebody reports an experiment that exhibits a breakdown of a conservation law that is thought to hold universally? First of all, the results need to be checked extremely thoroughly, to make sure that they're not caused by, say, a loose cable or a badly-calibrated clock, and then they need to be replicated independently by a separate experiment, to make sure that the physics is real. And, if the experiment holds up, then it will force us to re-think the theory that includes that conservation law.
The details of how that re-think needs to go, and how much of current theory it needs to include, of course, depend on what the experiment actually shows.
A: Noether's theorem is the thing to pay attention to here. This theorem basically says that if there is a symmetry in the underlying physics, there is also a conserved quantity. Traditionally the best-known symmetries are:

*

*Time invariance, i.e. the laws of physics do not change with time. If this is the case, then energy is conserved.

*Translational invariance, i.e. the laws of physics do not vary with position. If this is the case, then linear momentum is conserved.

*Rotational invariance, i.e. the laws of physics do not vary depending on the direction at which you look at the experiment. If this is the case, then angular momentum is conserved.

In the same way, if any of the three quantities above are not conserved, then we know that physics varies with time/position/direction.
There are other conserved quantities, such as electric charge, but the relevant symmetries are more technical.
A: My volume of Symon's Mechanics address' this question:

The conservation laws are in a sense not laws at all, but postulates which we insist must hold in any physical theory.  If, for example, for moving charged particles, we find that the total energy, defined as (T + V) [kinetic plus potential], is not constant, we do not abandon the law, but change its meaning by redefining energy to include electromagnetic  energy in such a way as to preserve the law.  We prefer always to look for quantities which are conserved, and agree to apply the names 'total energy', 'total momentum', and 'total angular momentum' only to such quantities.  The conservation of these quantities  is then not a physical fact, but a consequence  of our determination to define them in this way.  It is, of course, a statement of physical fact, which may or may not be true, to assert that such definitions of energy, momentum and angular momentum can always be found.  The assertion, has so far been true...

A further example would be the combination of the conservation of mass and conservation of energy of classical mechanics into the conservation of total relativistic energy in Special Relativity.
A: Maybe it's useful to think this question from the Hamiltonian formalism. The Hamiltonian $\cal{H}$ of a system is, in general, the total energy of the system, expressed with a particular set of variables. For example, for a point particle moving in the presence of a constatn gravitational field, the energy is $$E = \frac{mv^2}{2} + mgx$$
but these variables are not suitable for expressing the Hamiltonian. Instead of the velocity, the apropriate variable is momentum $$\cal{H} = \frac{p^2}{2m} + mgx$$
This choice is not random, but it's based on certain relations between momentum and position. In general, two variables meeting these relations are called canonical coordinates, and can be thought as one representing a "position" $\mathbb{q}$ and the other a "momentum" $\mathbf{p}$ (eg: position $x$ and linear momentum $p$, angle $\theta$ and angular momentum $L$, etc.)
Based on this, the equations of movement of the particle can be found from the Hamilton equations $$\frac{d\mathbf{p}}{dt} = -\frac{d\cal{H}}{d\mathbf{q}},\;\;\frac{d\mathbf{q}}{dt} = +\frac{d\cal{H}}{d\mathbf{p}}$$
Let's use this in the simplest case possible: a point mass moving with velocity $v$ in free space, without the action of any force. The Hamiltonian in this case is simply $\cal{H}=p^2/2m$, and applying Hamilton equations $$\frac{dp}{dt} = -\frac{d\cal{H}}{dq} = 0,\;\;\frac{dq}{dt} = +\frac{d\cal{H}}{dp} = \frac{p}{m}$$
Translating this to "normal" variables $q\rightarrow$ position, $p\rightarrow$ momentum, the second equation reads "the variation of position is the velocity", which sounds pretty reasonable, and the first is "the velocity is constant", as we already knew. However, from the first equation you also get the conservation of momentum. This is caused by the Hamiltonian being independent of the position, which can be thought as "no matter where you are in the space, the Hamiltonian remains the same". This "no matter where you are" is called a symmetry in the Hamiltonian, and represent the validity of the equations representing the system, even if you move it anywhere in the universe (where the conditions are still met). All the conservation laws can be represented as symmetries in the Hamiltonian, always involving two related canonical coordinates (position and linear momentum, angle and angular momentum, time and energy, etc.). The formalism of this is the Noether theorem.
