See the pictures below.  A pair of sunglasses I recently purchased has the polarization axis in one lens offset about 20 degrees (by eyeball estimation) from the other.  I don't have much experience with other polarized sunglasses, but this seems very obviously wrong to me.  And it's very noticeable while wearing the glasses.  When facing into glare from certain directions, one eye filters our considerably more glare than the other, which is very distracting and uncomfortable.

I can see arguments for making the polarization axis angle just about anything (though horizontal or vertical seem most logical), but what I cannot see is any argument for having it different in each lens on the same pair of glasses.

Thoughts?




There are several reasons for using the Hamiltonian formalism:

Statistical physics.  The standard thermal states weight of pure states is given according to
$$\text{Prob}(\text{state}) \propto e^{-H(\text{state})/k_BT}$$
So you need to understand Hamiltonians to do stat mech in real generality.

Geometrical prettiness.  Hamilton's equations say that flowing in time is equivalent to flowing along a vector field on phase space.  This gives a nice geometrical picture of how time evolution works in such systems.  People use this framework a lot in dynamical systems, where they study questions like 'is the time evolution chaotic?'.

The generalization to quantum physics.  The basic formalism of quantum mechanics (states and observables) is an obvious generalization of the Hamiltonian formalism.  It's less obvious how it's connected to the Lagrangian formalism, and way less obvious how it's connected to the Newtonian formalism.



[Edit in response to a comment:]
This might be too brief, but the basic story goes as follows:
In Hamiltonian mechanics, observables are elements of a commutative algebra which carries a Poisson bracket $\{\cdot,\cdot\}$.   The algebra of observables has a distinguished element, the Hamiltonian, which defines the time evolution via $d\mathcal{O}/dt = \{\mathcal{O},H\}$.  Thermal states are simply linear functions on this algebra.  (The observables are realized as functions on the phase space, and the bracket comes from the symplectic structure there.  But the algebra of observables is what matters:  You can recover the phase space from the algebra of functions.)
On the other hand, in quantum physics, we have an algebra of observables which is not commutative.  But it still has a bracket $\{\cdot,\cdot\} = -\frac{i}{\hbar}[\cdot,\cdot]$ (the commutator), and it still gets its time evolution from a distinguished element $H$, via $d\mathcal{O}/dt = \{\mathcal{O},H\}$.   Likewise, thermal states are still linear functionals on the algebra.
