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In Goldstein (Third Edition, Page 339) the equation 8.20 is as follow:

$$ H = \dot q_ip_i - L = \dot q_ip_i-[L_0(q_i,t)+L_1(q_i,t)\dot q_k+L_2(q_i,t)\dot q_k\dot q_m].\tag{8.20}$$

Can someone explain why the second equality hold? Moreover, $H$ is in no way a tensor, how come it has the last term indexed by $km$?

Note: by equation 2.55, the Lagrange can be decomposed as such: $$L(q,\dot q,t)=L_0(q,t)+L_1(q,\dot q,t)+L_2(q,\dot q,t).\tag{2.55}$$ where $L_2$ is a homogeneous function of the second degree, in $\dot q$, while $L_1$ is homogeneous of the first degree in $\dot q$.

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The material surrounding eq. (8.20) in the 3rd edition is new (as compared to the 2nd edition), and filled with typos and breaks previously established notations from eqs. (1.73) & (2.55). The errors are not reported so far (February 2019) on the errata homepage. The errors can be fixed in various straightforward ways.

(This is another example where the 3rd edition of Goldstein is inferior to the 2nd edition, cf. this Phys.SE post.)

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