I have been reading some notes and at point the author uses a notation I am not aware of and I wanted to figure out what it was. Let $T^*M$ be the cotangent bundle of a compact manifold $M$, and consider $H:T^*M\times S^{1}\rightarrow \mathbb{R}$ to be a time-dependent Hamiltonian function. Now for the purpose of what I am reading we will wanna consider Hamiltonians with specific properties. As an example the author states that Physical Hamiltonians of the form

$$H(q,p,t)=\frac{1}{2}|T(t,q)p-A(t,q)|^{2}+V(t,q)$$ where the symmetric tensor $T^*T$ is everywhere positive, satisfy the conditions. Now what is it meant by the symmetric tensor $T^*T$ is everywhere positive, is it we consider for fixed $q$ the function $T(t,q):T_{q}^*M\rightarrow \mathbb{R}$ and then take it's differential and we want that $dT(t,q)[p,p]\geq 0$ for every $p\in T_{q}^*M$?

Any insight is appreciated, thanks in advance.

  • 1
    $\begingroup$ Quibble on notation: If you write the domain of $H$ as $T^*M\times S^1$, then in the displayed equation, you should write $H(p,q,t)$. The order of the inputs should match. $\endgroup$
    – Deane
    Jun 23 at 15:03
  • $\begingroup$ In the last sentence, do you mean $T(t,q): T_p^*M \rightarrow \mathbb{R}$? $\endgroup$
    – Deane
    Jun 23 at 15:07
  • $\begingroup$ No I don't think so, I want $T(t,q)$ to be a function of the fiber $T_q^*M$.@Deane $\endgroup$
    – whatever
    3 hours ago


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