# What is meant by homogeneous in $x$ in $n$'th degree?

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

• – rob Oct 1 at 13:26
• Would Mathematics be a better home for this question? We can migrate it if you like, but there may be a duplicate there already. – rob Oct 1 at 13:27
• Thank you @rob no matter for me if you migrate it. I thought about the wiki-site, but I wasn't sure about it. Seems like the term "homogeneous" is used in many contexts and meanings. – jjepsuomi Oct 1 at 13:28

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