• Support PF! Buy your school textbooks, materials and every day products Here!

Finding the Lagrangian Matrix for Two-Spring Systems

  • #1

Homework Statement


The problem is attached. I'm working on the second system with the masses on a linear spring (not the first system).
I think I solved part (a), but I'm not sure if I did what it was asking for. I'm not sure exactly what the question means by the... L=.5Tnn-.5Vnn. Namely, I'm not sure what the n's are and what they represent, so I'm worried that I'm missing something. I also am not sure about the equillibrium position...
Then I'm pretty sure I can do part (b) and (c), but I'm wondering if I just find the eigenvector and values like a normal linear algebra problem. Do I simply just take the matrix I have and operate on it???
Screen Shot 2018-10-27 at 10.41.39 PM.png

Homework Equations


L=T-V
e^-iwt
d/dt(dL/dqdot)-(dL/dq)

The Attempt at a Solution


Attached (for the second system). I'm pretty sure I can figure out (1) if I figure out (2).
Pt1.png
pt2.png
 

Attachments

Answers and Replies

  • #2
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,580
1,187
You didn't do what was asked for in part (a). It's asking you to find two matrices, ##\mathbf T## and ##\mathbf V##, for one thing.

In the Lagrangian, you have for the kinetic energy ##T=\frac 12 m(\dot x_1^2 + \dot x_2^2)##. You can write this as
$$T = \frac 12 \begin{pmatrix}\dot x_1 & \dot x_2 \end{pmatrix} \begin{pmatrix} m & 0 \\ 0 & m \end{pmatrix}\begin{pmatrix}\dot x_1 \\ \dot x_2 \end{pmatrix}.$$ The matrix of masses is the matrix ##\mathbf T## you're being asked to find. You can express the potential energy similarly in terms of a matrix.
 

Related Threads for: Finding the Lagrangian Matrix for Two-Spring Systems

  • Last Post
Replies
1
Views
8K
  • Last Post
Replies
5
Views
7K
Replies
6
Views
3K
Replies
2
Views
3K
Replies
10
Views
7K
Replies
9
Views
619
Replies
2
Views
1K
Replies
2
Views
2K
Top