# Goldstein expression for the Lagrangian

I was looking for help in order to proove 2 relations that Goldstein has put in his book.

$$L(q, \dot{q}, t)=L_{0}(q, t)+\tilde{\mathbf{\dot{q}}} \mathbf{a}+\frac{1}{2} \tilde{\boldsymbol{\dot{q}}} \mathbf{T} \dot{\mathbf{q}}\tag{8.23}$$ $$H(q, p, t)=\frac{1}{2}(\tilde{\mathbf{p}}-\tilde{\mathbf{a}}) \mathbf{T}^{-1}(\mathbf{p}-\mathbf{a})-L_{0}(q, t)\tag{8.27}$$ I understand how they came mathematically, but I'm not sure about what do these relations imply physically, I don't get how I canninterpret the vector generalized velocities and also the momenta vectors. Also, I assume that my Lagrangian its in the form:

$$L\left(q_{i}, \dot{q}_{i}, t\right)=L_0(q, t)+\dot{q}_{i} a_{i}(q, t)+\dot{q}_{i}^2T_{i}(q, t)\tag{8.22}$$ I will very grateful for any advice in the prove, or interpretation of the vectors.

PS: I know how to prove the second relationship from the first one, so I'm more interested in the first one. Also, these eqs came from Classical Mechanics by H Goldstein 3rd Edition (8.22), (8.23), (8.27).

• It seems OP's questions mainly concern various typos in the 3rd edition of Goldstein. Commented Jan 13, 2021 at 18:37

Equation 8.23 is just 8.22 rewritten in vector/matrix notation. Perhaps the problem you are having is that there must be an implied assumption that the coefficients $$a_i$$ are not just arbitrary numbers, but to be physically meaningful must be related so they transform as a vector. Similiarly, the $$T_i$$ coefficients transform as a matrix.