Talk:Physics/A/String vibration

$$\lambda\to 0$$

Partial derivatives
$$f=\frac{1+x}{\sqrt{(1+x)^2+y^2}}$$
 * Subscripts denote partial derivatives evaluated at $$x=y=0$$. Formulas obtained from https://www.symbolab.com/solver/
 * Special:Permalink/2270863
 * I think this is now right. But am troubled by terms with ?:

$$f_{x}=0$$

$$f_{y}=0$$

$$f_{xx}=0$$

$$f_{xy}=0$$

$$f_{yy}=-1$$

-

$$g=\frac{1}{\sqrt{(1+x)^2+y^2}}$$

$$g_{x}=-1$$

$$g_{y}=0$$

$$g_{xx}=-2$$?

$$g_{xy}=0$$

$$g_{yy}=-1$$

Substitute

 * $$f=\frac{1+x}{\sqrt{(1+x)^2+y^2}}=1-\tfrac 1 2 y^2 + \mathcal O z^3$$
 * $$f=\frac{1+\xi}{\sqrt{(1+\xi)^2+\eta^2}}=1-\tfrac 1 2 \eta^2 + \mathcal O z^3$$
 * $$g=\frac{1}{\sqrt{(1+x)^2+y^2}}=1-x-x^2?-\tfrac 1 2 y^2+\mathcal O z^3$$
 * $$g=\frac{1}{\sqrt{(1+\xi)^2+\eta^2}}=1-\xi -\xi^2?-\tfrac 1 2\eta^2+\mathcal O z^3$$

not TRASH
$$m\frac{d^2 \vec r_j}{dt^2}= -\kappa_s\left(\ell_{j,j-1} - a\right) \left[\frac{\vec r_j-\vec r_{j-1}}{\ell_{j,j-1}}\right] +\kappa_s\left(\ell_{j,j+1} - a\right) \left[\frac{\vec r_{j+1}-\vec r_j}{\ell_{j+1,j}}\right]$$

$$m\frac{d^2 \vec r_j}{dt^2}= -\kappa_s\left(\ell_{j,j-1} - a\right) \left[\frac{\vec r_j-\vec r_{j-1}}{\ell_{j,j-1}}\right] +\kappa_s\left(\ell_{j,j+1} - a\right) \left[\frac{\vec r_{j+1}-\vec r_j}{\ell_{j+1,j}}\right]$$

$$\ell_{j,j- 1}=\left|\vec r_j-\vec r_{j- 1}\right| =\sqrt{(x_j-\vec x_{j\pm 1})^2+(y_j-\vec y_{j-1})^2}$$

$$\vec\ell_{j,j-1}=\vec r_j-\vec r_{j-1}$$

FOOTNOTE$$\vec\gamma_j = \xi_j \hat x \, + \,\eta_j \hat y \, + \,\zeta_j\hat z$$

$$\vec\gamma_j = \xi_j \hat x \, + \,\eta_j \hat y $$

2
$$\vec F_{76}=\vec F_\text{on 7}^{\,\text{ by 6}}$$

$$\vec\ell_{76}=\ell\hat x + \vec\gamma_7 -\vec\gamma_6$$

$$\ell = 1$$

$$\vec\ell_{76}=\hat x + \vec\gamma_7 -\vec\gamma_6 = (1+\Delta\xi_{76})\hat x + (\Delta\eta_{76})\hat y$$

Recall that for small $$\epsilon$$,

$$\sqrt{1+\epsilon} = 1 + \tfrac{1}{2}\epsilon - \tfrac{1}{8}\epsilon^2 + \tfrac{1}{16}\epsilon^3 - \tfrac{5}{128}\epsilon^4 + \tfrac{7}{256}\epsilon^5 - \ldots,,$$

$$\ell_{76}=\sqrt{1+2\Delta\xi_{76} + (\Delta\xi_{76})^2 + (\Delta\eta_{76})^2} =1+\Delta\xi_{76}+ \tfrac 1 2 (\Delta\eta_{76})^2 + \mathcal O(\Delta\xi)^2 + \mathcal O(\Delta\eta)^4 + ...$$

Equation
We create a four-term expression using (4) and (5). Then we move the terms involving energy density and the Poynting vector to the LHS to obtain:

Key equation


References (Rowland)
First mention of Rowland

First non-Rowland reference

Second mention of roland

2nw non-Rowland reference

Second mention of roland