What is the physical understanding of degeneracy, is it only a quantum concept? In many books, they refer summery to a conserved physical quantity, this is know as Noether's theorem in classical Mechanics. In quantum mechanics, the degeneracy of energy level is related the hamiltonian symmetry. The degeneracy is the fact that a unique eigenvalue has two or more eigenstates which are linearly independent. Physically, I can't grasp the fact that different states of the a system has the same energy. How could this happen. Is there a concrete example to accept this evidence?    
 A: Here's an example that works for both classical and quantum mechanics. Imagine you have a particle of mass $m$ in free space, with momentum $\vec p =p\hat{x}$. Then the energy is $\frac{p^2}{2m}$. Say the particle instead has $\vec p = p\hat y$. Then the state of the particle is different, but the energy is the same. The energy operator is highly degenerate, since any momentum of the form $\vec{p}=p\cos(\theta)\hat x+p\sin(\theta)\hat y$ has the same energy $\frac{p^2}{2m}$. This, of course, comes from the rotation symmetry of the problem.
A: Some energy eigenstates of the hydrogen atom are degenerate.  The energy $E_n$ depends only on the quantum number $n$ but many states can share this $n$.  For instance, with $n=2$ there are $3$ states with $\ell=1$ plus another state with $\ell=0$ all having the same energy $-13.6/4$ eV.
Of course in classical mechanics the evolution of a system will take place on a surface of constant $H$ (when $L$ does not depend on time) so all points of that surface are examples of degenerate states.
