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I'm reading about classical mechanics by Goldstein, and in the section about Hamiltonian mechanics it is stated that in the expression:

$$H(q,p,t)=\dot{q}_ip_i-L(q,\dot{q}, t)$$

the Lagrangian function $L$ can be braken down into three components:

$$L=L_0(q_i,t) + L_1(q_i, t)\dot{q}_k + L_2(q_i,t)\dot{q}_k\dot{q}_m$$

(no sum on $i$ in the square brackets) where $L_0$ is the part of the Lagrangian that is independent of the generalized velocities, $L_1$ represents the coefficients of the part of the Lagrangian that is homogeneous in $\dot{q}_i$ in the first degree, and $L_2$ is the part that is homogeneous in $\dot{q}_i$ in the second degree.

It is not fully clear to me what is meant by:

$L_1$ represents the coefficients of the part of the Lagrangian that is homogeneous in $\dot{q}_i$ in the first degree, and $L_2$ is the part that is homogeneous in $\dot{q}_i$ in the second degree.

What does these mean ? Mathematical definition? The book is not so clear on these definitions

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In case someone else bumps into the same question, I think I found the answer:

The Euler's theorem states that if $f$ is a homogeneous function of degree $n$ in the variables $x_i$, then

$$\sum_i x_i\frac{\partial f}{\partial x_i}=nf.$$

So for example, if $f$ is a function of two variables $x_1, x_2$ and it is homogeneous, say to 3rd degree, in these variables, then:

$$x_1\frac{\partial f(x_1, x_2)}{\partial x_1}+x_2\frac{\partial f(x_1, x_2)}{\partial x_2}=3f(x_1, x_2).$$

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