Conservation Laws and Symmetry The toughest of topics in physics, like Quantum Mechanics, Relativity, String theory, can be explained in layman words and many have done so. Though there is no substitute to the understanding a theory in all it's mathematical detail, the idea can more or less be driven home. However, I seem not to be able to find such an explanation for Noether's theorem.
I've tried my hand at Group Theory and found quite perplexing. I've got stuck with generators, and this and that .... 
So here is my question. Is it possible to explain without using heavy mathematics, why a conservation law arises from a symmetry under some transformation?
I'm in school, so my "heavy" might not be the same as yours. 
Thanks.
 A: A helpful yet elementary answer may do the trick, If you are familiar with the Euler-Lagrange equation then it will be straight forward and you can skip ahead a little. If not then you have to accept that there is an equation in physics that generalises classical mechanics called the Euler-Lagrange equation. For a particle moving in one dimension under a conservative force it is written,
\begin{equation}
\frac{d}{dt}\bigg(\frac{\partial T}{\partial \dot x}\bigg)+\frac{\partial V}{\partial x}=0
\end{equation} 
Where $T$ is the kinetic energy of the system and $V$ is the potential energy, $x$ is the particles position and $\dot x=\frac {d}{dt}x$ is the velocity of the particle. We define the momentum of the particle to be,
\begin{equation}
p:=\frac{\partial T}{\partial \dot x}
\end{equation}
And you will note that we can now write the Euler-Lagrange equation as,
\begin{equation}
\frac{d}{dt}(p)+\frac{\partial V}{\partial x}=0
\end{equation}
This is Newton's second law of motion. The momentum is changed by the action of a force on the particle, if there are no forces then the time derivative of the momentum is zero. If the time derivative is zero then the momentum does NOT change as time evolves and will have the same value at the end of the experiment as it did at the beginning. 
In this way the Euler-Lagrange equation has given us a conservation law for $p$ only when $\partial V/\partial x=0$. The invariance of the potential with respect to $x$ leads to a conservation law. 
In general we do not write the Euler-Lagrange equation for a one dimensional particle. The general form is written, 
\begin{equation}
\frac{d}{dt}\bigg(\frac{\partial L}{\partial \dot x}\bigg)-\frac{\partial L}{\partial x}=0
\end{equation}
Where $L(x,\dot x)=T(\dot x)-V(x)$ is the Lagrangian of the system. Check that this will give the above stated equation. In general if the Lagrangian for a particular system is not a function of $x$ then you can clearly see that,
\begin{equation}
\frac{\partial L}{\partial \dot x}=constant
\end{equation}
Since the time derivative vanishes. When the Lagrangian is not a function of $x$ we say that the Lagrangian has a symmetry. When the Lagrangian has a symmetry, there is a conservation law. 
A: It is difficult to understand conservation from symmetry. But the opposite is much simpler. conservation means the invariance of equation of motion in its form under certain transformation. and the invariance of equation of motions arises as an implication of the underlying symmetry. for example i am taking one equation X2 =1  the solutions are +or-1
which implies under the operation of inversion X become -X but X square remains the same. which means if X=X the solution is 1 and if X=-X solution is -1.in words before transformation the solution was +1 and after transformation the solution is -1. I am trying to explain the equation of motion is form invariant doesn't mean the solutions of equation is the same. but the meaning is the equation of motion form invariant means a set of solution  are not changing but under transformation individual solutions changes from one to another within the set of solutions.these consideration leads to group theories. here you can see the invariance is related to certain operations and the invariance arises if there is symmetry.         
