Generalized coordinates basic question

[nomadreid]

From "A Student's Guide to Langrangins and Hamiltonians", Patrick Hamill, Cambridge, 2017 edition.
Apologies: since I do not know how to put dots above a variable in this box, I will put the dots as superscripts. Similarly for the limits in a sum.
On page 6,
"we denote the coordinates by q_i and the corresponding velocities by q^⋅_i."
Further he terms p_i the generalized momenta.
Then, on page 97, using L as the Lagrangian,
" H(q_i, p_i, t) = ∑_i=1ⁿp_iq^⋅_i - L(q_i, q^⋅_i,t)" [equation (4.8)]
The function H is called the Hamiltonian...Keep in mind that the Hamiltonian must be expressed in terms of the generalized momentum. An expression for H involving velocities is wrong." (Italics in the original.)

But it is not clear to me why equation 4.8 is not also in terms of velocities, i.e., q^⋅_i

(Perhaps this should be a mathematics threads rather than a QM thread; I place it here because of the context.)

 

[haushofer]

I don't get that sentence either, since there is a 1 to 1 correspondence between momenta and velocities.

 

[haushofer]

By the way, a dot is obtained by the code $\dot{q}_i$, "\dot{q}_i".

 

[vanhees71]

But the book is in fact right. For the Hamiltonian formulation of Hamilton's principle the Hamiltonian must be expressed in terms of generalized coordinates and their canonical momenta, not generalized velocities. The Lagrangian formulation deals with the generalized coordinates and their assiciated generalized velocities. Of course you can use any form of "mixture", which is due to Routh. It's rarely found in modern textbooks, and I've never found any application which is easier solved using this most general description than with Lagrange or Hamilton. If you are interested in it, you find it in Sommerfeld's Lectures on Theoretical Physics, vol. 1 (which book series I cannot recommend enough; for me it's the master piece in theoretical-physics textbook writing which sets the very high standard of any text-book writing in this field).

 

[stevendaryl]

Obviously, you can convert back and forth between momenta and velocities. But in Hamiltonian dynamics, the Hamiltonian serves a role beyond its numerical value (which is of course, not changed by substituting velocities for momenta or vice-versa). The Hamiltonian equations of motion are (I'm just going to use a single variable, $x$ and the corresponding momentum, $p$. The generalization to multiple generalized coordinates and momenta is straight-forward.)

1. $\frac{dx}{dt} = \frac{\partial H}{\partial p}$
2. $\frac{dp}{dt} = - \frac{\partial H}{\partial x}$

In order to compute the partial derivatives on the right side of the equalities, you have to write $H$ in terms of $x$ and $p$

So there's really four steps in obtaining the hamiltonian:

1. First, compute the momentum as a function of position and velocity, using the Lagrangian: $p = \frac{\partial L}{\partial \dot{x}}$
2. Use that to compute the velocity as a function of position and momentum: $\dot{x} = F(x,p)$
3. Then, compute the "mixed" expression: $H(x,p,\dot{x}) = p \dot{x} - L$
4. Finally, use the result of 2 to rewrite all occurrences of $\dot{x}$ in terms of $p$ and $x$, to get an expression: $H(p,x)$ that only involves $p$ and $x$

 

[nomadreid]

Thanks, stevendaryl, for the very helpful and complete answer.
Thanks also vanhees71; I will keep the recommended book in mind if I find a way to freely access it.