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Given $r>0$ and $f:\mathbb{R}^n\to \mathbb{R}$, $d_rf$ is the function defined by \begin{equation}d_rf(x_1,x_2,\dots,x_n)=f(rx_1,rx_2,\dots,rx_n)\end{equation} and is called the $r$-dilation of $f$, and $f$ is said to be homogeneous of degree $D$ if $d_rf=r^Df$. Similar notion is defined for distributions by duality.

I do not know much about differential equations, and do not actually see why this notion is so important. Wikipedia says homogeneous functions are good because equations involving them can be solved by separation of variables. But I am not sure how that works and how that is related to the homogeneous degree of a function.

Can someone explain this? Thanks!

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One important theorem about homogeneous functions is Euler's homogeneous function theorem which essentially states that we have $$ \sum_{i=1}^n x_i \frac{ \partial f}{\partial x_i} = D f$$ where $D$ is the degree. To prove this, let $X_i = r x_i$, differentiate the homogeneity condition with respect to $r$, apply the chain rule and let $r=1.$

Expressions of the form of the LHS of the sum are encountered in Lagrangian Dynamics. A quantity called the Langrangian is defined by $L= T-V$ where $T$ is the total kinetic energy of the system and $V$ is the total potential energy (all are variables of time and position/velocity of each object). The energy is known to be given by $$E = \sum_{k=1}^n \dot{q_k} \frac{ \partial L}{\partial \dot{q_k} } -L$$ and the Lagrangian consists of up to 3 parts, of degree 2,1, and 0. Applying Euler's result gives a result which can be summarised by this quick rule: Take the Lagrangian, keep the terms quadratic in velocities the same, make the terms linear in velocities disappear, and change the sign of the terms constant in velocity, and you will have the energy.

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