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So, this a question from a Phd qualifying examination. Given the following Lagrangan $$L=frac12dotqtextsin^2q,$$ what is the Hamiltonian for this system? So, finding the canonical momentum $p$ and then using $H=pdotq -L$, I find that it is identically zero. Now, the answer 'Zero' was a choice in the choices of answers given but the correct answer is that the Hamiltonian is undefined. I'm not sure why this should be the answer, though this is a pretty weird dynamical system. The Euler Lagrange equations just give an identity and the Hamiltonian equations also seem rather off. The Hamiltonian equations would are $$dotp=dotq=0. $$
I'm hoping someone shed some light on this, it seems that any random Lagrangian doesn't constitute a dynamical system, so what I should I be looking for?

The reason the Hamiltonian is undefined is that converting a Lagrangian to a Hamiltonian requires:

• Finding $p$
  


• Writing $H=pdot q-L$
  


• Expressing $H$ in terms of $p$ and $q$, eliminating all dependence on $dot q$.

For the third step to be possible, you need to be able to define a coordinate transformation between the $(q,dot q)$ coordinate system and the $(q,p)$ coordinate system. This requires that $(q,p)$ is a good coordinate system, in that a state of the system can be uniquely represented by a pair of $(q,p)$ coordinates. In turn, this means that the transformation $(q,dot q)rightarrow (q,p)$ must have nonzero Jacobian.

However, it's easy to see this is not the case. We know that $(q,p)=(q,frac12sin^2 q)$. Thus, the Jacobian is

$$
J=left|beginsmallmatrix1 & 0\sin(q)cos(q) & 0endsmallmatrixright| = 0
$$

That means we CAN'T write $p$ as a function of $q,dot q$, and there can't be a Hamiltonian $H(p,q)$ that is a function of only the coordinates $(p,q)$ because the coordinates $(p,q)$ don't specify the state of the system.

1. OP's Lagrangian $$L=fracddtleft(fracq4-frac18sin(2q)right)tag1$$ is a total time derivative, so the Euler-Lagrange (EL) equation is a triviality $0=0$, , cf. e.g. this Phys.SE post, and hence not suitable to determine physical trajectories. In other words, the Lagrangian (and Hamiltonian) formulations are ill-defined.




2. The Hamiltonian is defined as the Legendre transformation of the Lagrangian, and hence formally defined as
   $$ H(q,p)~:=~sup_vinmathbbR(vp-L(q,v))~=~sup_vinmathbbRvleft(p-frac12sin^2(q)right)~=~left{beginarraylinftytext if p~neq~frac12sin^2(q),\ 0text otherwise. endarrayright. tag2$$