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We have Rayleigh's dissipation function, defined as

$$\mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right)$$

Also we have transformation equations to generalized coordinates as

$$\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}, \ldots, q_{3 N-k}, t\right) \\ & \vdots \\ \mathbf{r}_{N} &=\mathbf{r}_{N}\left(q_{1}, q_{2}, \ldots, q_{3 N-k}, t\right) \end{aligned}$$

How can I prove that the dissipation function is homogeneous of degree 2 in $\dot{q}$?

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The function $~f(x,y)=x^2+y^2~$ is homogenous of degree 2 if $f(p\,x,p\,y)=~(p\,x)^2+(p\,y)^2=p^2\,f(x,y)~$

assume

$$F=\frac{k_x}{2}(\,v_{x1}^2+v_{x2}^2)$$

with $~x_i=x_i(q_k)~$ $$v_{xi}=\sum_{k=1}^{n_k} \frac{\partial x_i}{\partial q_k}\,\dot{q}_k$$

$\Rightarrow$

$$\bar F=\frac{2\,F}{k_x}=f(x,y)=\bigg(\underbrace{\sum_{k=1}^{n_k} \frac{\partial x_1}{\partial q_k}\,\dot{q}_k}_{x}\bigg)^2+\bigg(\underbrace{\sum_{k=1}^{n_k} \frac{\partial x_2}{\partial q_k}\,\dot{q}_k}_y\bigg)^2=x^2+y^2$$

$$f(p\,x,p\,y)=\bigg({p\sum_{k=1}^{n_k} \frac{\partial x_1}{\partial q_k}\,\dot{q}_k}_{x}\bigg)^2+\bigg({p\sum_{k=1}^{n_k} \frac{\partial x_2}{\partial q_k}\,\dot{q}_k}_y\bigg)^2=p^2\,f(x,y)$$

thus, the function $~f(x,y)~$ is homogenous of degree 2

edit

\begin{align*} F&=\sum_i\bigg(\sqrt{\frac{k_x}{2}}\,v_{xi}\bigg)^2+\sum_i\bigg(\sqrt{\frac{k_y}{2}}\,v_{yi}\bigg)^2+\sum_i\bigg(\sqrt{\frac{k_z}{2}}\,v_{zi}\bigg)^2\\ F(X,Y,Z)&=\bigg(\underbrace{\sqrt{\frac{k_x}{2}}\sum_i\sum_k\frac{\partial x_i}{\partial q_k}\dot{q}_k}_{X}\bigg)^2+ \bigg(\underbrace{\sqrt{\frac{k_y}{2}}\sum_i\sum_k\frac{\partial y_i}{\partial q_k}\dot{q}_k}_{Y}\bigg)^2+ \bigg(\underbrace{\sqrt{\frac{k_z}{2}}\sum_i\sum_k\frac{\partial z_i}{\partial q_k}\dot{q}_k}_{Z}\bigg)^2\\&=X^2+Y^2+Z^2\\ F(p\,X,p\,Y,p\,Z)&=p^2\,\bigg(X^2+Y^2+Z^2\bigg) \end{align*}

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  • $\begingroup$ Great. Thank you. Got it. If you don't mind you can use kx ky and the result will be even more general. Thank you once again brother. $\endgroup$
    – Kashmiri
    Sep 27 at 4:34

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