# Notation on tensors and Hamiltonians

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.

• 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. Jun 23 at 15:03
• In the last sentence, do you mean $T(t,q): T_p^*M \rightarrow \mathbb{R}$? Jun 23 at 15:07
• No I don't think so, I want $T(t,q)$ to be a function of the fiber $T_q^*M$.@Deane 3 hours ago