From 9e0685c7f9d5fa8338af65d91ffcfd6f1182f3a0 Mon Sep 17 00:00:00 2001 From: Billy Quarles <4674360+saturnaxis@users.noreply.github.com> Date: Sun, 27 Aug 2023 13:42:08 -0400 Subject: [PATCH] added link to Lagrangian method --- Tutorials/forced-eccentricity.ipynb | 2 +- docs/Tutorials/forced-eccentricity.html | 46 +++++++++---------- .../Tutorials/forced-eccentricity.ipynb | 2 +- docs/searchindex.js | 2 +- 4 files changed, 26 insertions(+), 26 deletions(-) diff --git a/Tutorials/forced-eccentricity.ipynb b/Tutorials/forced-eccentricity.ipynb index 902848c..f8fe92a 100644 --- a/Tutorials/forced-eccentricity.ipynb +++ b/Tutorials/forced-eccentricity.ipynb @@ -125,7 +125,7 @@ "x_2(t) &= -B_1^+ e^{+i\\omega_1 t} - B_1^- e^{-i\\omega_1 t} + B_2^+ e^{+i\\omega_2 t} + B_2^- e^{-i\\omega_2 t}.\n", "\\end{align}\n", "\n", - "Depending on initial and boundary conditions, Euler's equation could be used to transform the above into a linear combination of $\\sin$ and $\\cos$ functions. For more details, see the [lecture](https://scholar.harvard.edu/files/schwartz/files/lecture3-coupled-oscillators.pdf) by Matthew Schwartz at Harvard or [libretexts](https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HA__Classical_Mechanics/8%3A_Small_Oscillations/8.4%3A_Coupled_Oscillators_and_Normal_Modes) by Tom Weideman at UC Davis, or lectures on YouTube from Jeffery Chasnov (see below).\n", + "Depending on initial and boundary conditions, Euler's equation could be used to transform the above into a linear combination of $\\sin$ and $\\cos$ functions. For more details, see the [lecture](https://scholar.harvard.edu/files/schwartz/files/lecture3-coupled-oscillators.pdf) by Matthew Schwartz at Harvard or [libretexts](https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HA__Classical_Mechanics/8%3A_Small_Oscillations/8.4%3A_Coupled_Oscillators_and_Normal_Modes) by Tom Weideman at UC Davis, or lectures on YouTube from Jeffery Chasnov (see below). There is also a guide for the generalized method from [Lagrangian dynamics](http://teacher.pas.rochester.edu/PHY235/LectureNotes/Chapter12/Chapter12.pdf) provided by the University of Rochester that may be helpful.\n", "\n", "\n", "
\n", diff --git a/docs/Tutorials/forced-eccentricity.html b/docs/Tutorials/forced-eccentricity.html index 2c7eaeb..63ad560 100644 --- a/docs/Tutorials/forced-eccentricity.html +++ b/docs/Tutorials/forced-eccentricity.html @@ -439,20 +439,20 @@

5.1. A review of coupled oscillations

The mass on the left is displaced by a distance \(x_1\) to the right. It’s left spring will apply a force to bring it back to the equilibrium, but the middle spring will compress along with the rightmost spring. The compression of the middle and right spring will apply a force on the other mass on the right. Because the masses are connected, there will be a similar set of forces acting on the mass on the right. Altogether the forces on the left and right mass are:

-
-(5.1)#\[\begin{align} +
+(5.1)#\[\begin{align} \mathbf{F}_{\rm left} &= -kx_1 + \kappa(x_2-x_1) = m\ddot{x}_1, \\ \mathbf{F}_{\rm right} &= -kx_2 + \kappa(x_1-x_2) = m\ddot{x}_2. \end{align}\]

Notice that each of these equations have a part that we expect from an isolated spring and a part that describes the interaction via the connecting spring with spring constant \(\kappa\). Furthermore, the interaction on the left mass is equal and opposite to the interaction force on the right mass. The above equations can be rearranged to get:

-
-(5.2)#\[\begin{align} +
+(5.2)#\[\begin{align} m \ddot{x}_1 &= -(\kappa + k)x_1 + \kappa x_2, \\ m \ddot{x}_2 &= \kappa x_1 - (\kappa + k)x_2, \end{align}\]

or in matrix form as:

-
-(5.3)#\[\begin{align} +
+(5.3)#\[\begin{align} \begin{pmatrix} m\ddot{x}_1 \\ m\ddot{x}_2 \end{pmatrix} &= @@ -464,14 +464,14 @@

5.1. A review of coupled oscillations

Let’s use the following trial functions:

-
-(5.4)#\[\begin{align} +
+(5.4)#\[\begin{align} x_1(t) &= B_1 e^{i\omega t}, \\ x_2(t) &= B_2 e^{i\omega t}, \end{align}\]

based on our prior knowledge concerning the isolated springs. Upon substitution, we get

-
-(5.5)#\[\begin{align} +
+(5.5)#\[\begin{align} -m\omega^2\begin{pmatrix} B_1 e^{i\omega t} \\ B_2 e^{i\omega t} \end{pmatrix} &= @@ -483,8 +483,8 @@

5.1. A review of coupled oscillations

or

-
-(5.6)#\[\begin{align} +
+(5.6)#\[\begin{align} \begin{bmatrix} (\kappa + k) - m\omega^2 & -\kappa \\ -\kappa & (\kappa + k) - m\omega^2 \end{bmatrix} @@ -496,38 +496,38 @@

5.1. A review of coupled oscillations

The trivial solution is \(B_1 = B_2 = 0\). But the non-trivial solution is solved via the characteristic equation (or determinant) via:

-
-(5.7)#\[\begin{align} +
+(5.7)#\[\begin{align} \left[(\kappa + k) - m\omega^2\right]^2 - \kappa^2 &= 0, \\ (m\omega^2)^2 - 2m(\kappa + k)\omega^2 + (\kappa + k)^2 - \kappa^2 &= 0, \\ x^2 -2(\kappa+k)x + 2\kappa k + k^2 & = 0, \end{align}\]

which has solutions via the quadratic equation:

-
-(5.8)#\[\begin{align} +
+(5.8)#\[\begin{align} x &= (\kappa + k) \pm \frac{1}{2}\sqrt{4(\kappa + k)^2 - 4(2\kappa k + k^2)}, \\ m\omega^2 &= (\kappa + k) \pm \kappa, \\ \omega &= \pm \sqrt{\frac{(\kappa + k) \pm \kappa}{m}}. \end{align}\]

Each root (from the quadratic equation) permits a separate eigenfrequency:

-
-(5.9)#\[\begin{align} +
+(5.9)#\[\begin{align} \omega_1 = \pm \sqrt{\frac{2\kappa + k}{m}}, \qquad& \text{and} \qquad& \omega_2 = \pm \sqrt{\frac{k}{m}}. \end{align}\]

To determine the coefficients \(B_1\) and \(B_2\), we substitute the eigenfrequencies (\(\omega_1\) and \(\omega_2\)) back into the characteristic equation to get:

-
-(5.10)#\[\begin{align} +
+(5.10)#\[\begin{align} \left[ k + \kappa - (k + 2\kappa) \right]B_1 - \kappa B_2 &= -\kappa (B_1 + B_2) = 0,&\ (\text{for } \omega_1;\ B_1 = -B_2) \\ \left( k+ \kappa - k\right)B_1 - \kappa B_2 &= \kappa (B_1 - B_2) = 0.& (\text{for } \omega_2;\ B_1 = B_2) \end{align}\]

The most general solution of the coupled harmonic oscillator problem is

-
-(5.11)#\[\begin{align} +
+(5.11)#\[\begin{align} x_1(t) &= B_1^+ e^{+i\omega_1 t} + B_1^- e^{-i\omega_1 t} + B_2^+ e^{+i\omega_2 t} + B_2^- e^{-i\omega_2 t}, \\ x_2(t) &= -B_1^+ e^{+i\omega_1 t} - B_1^- e^{-i\omega_1 t} + B_2^+ e^{+i\omega_2 t} + B_2^- e^{-i\omega_2 t}. \end{align}\]
-

Depending on initial and boundary conditions, Euler’s equation could be used to transform the above into a linear combination of \(\sin\) and \(\cos\) functions. For more details, see the lecture by Matthew Schwartz at Harvard or libretexts by Tom Weideman at UC Davis, or lectures on YouTube from Jeffery Chasnov (see below).

+

Depending on initial and boundary conditions, Euler’s equation could be used to transform the above into a linear combination of \(\sin\) and \(\cos\) functions. For more details, see the lecture by Matthew Schwartz at Harvard or libretexts by Tom Weideman at UC Davis, or lectures on YouTube from Jeffery Chasnov (see below). There is also a guide for the generalized method from Lagrangian dynamics provided by the University of Rochester that may be helpful.