Problem about Lagrangian mechanics

  • Thread starter Another
  • Start date
  • #1
101
5
Homework Statement:
Find the Lagrangian of system.
In the question say the wedge can move without friction on a smooth surface.
Why does the potential energy of the wedge appear in Lagrangian?
Relevant Equations:
##\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q}##
CM 2. 20.png

In Solution https://www.slader.com/textbook/978...-3rd-edition/67/derivations-and-exercises/20/

In the question say the wedge can move without friction on a smooth surface.

Why does the potential energy of the wedge appear in Lagrangian?

(You can see the Larangian of this system at below.)



CM 2. 20 .1.png

CM 2. 20 .2.png
 

Answers and Replies

  • #2
etotheipi
The potential energy of the wedge is constant anyway, so isn't going to affect the equation of motion.

But in any case the expression ##U = Mgy_M## isn't correct, because the centre of mass of the wedge is not the coordinate ##y_M## (that is the coordinate of the left corner). In any case they will still stumble upon the correct answer, because the mistake happens to be constant and drops out.
 
  • Like
Likes Steve4Physics
  • #3
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
20,137
11,475
It looks like strange thing to do. I wouldn't have the terms involving ##y_M## and ##\dot y_M## as the wedge is constrained to move in the x-direction only.
 
  • #4
Steve4Physics
Homework Helper
Gold Member
940
736
They are just following a systematic approach and plugging-in all values of potential energy, even the constant one which is going to disappear. If you do this for all problems, it helps you to establish a standard approach and makes it less likely you'll forget some term(s) in more complex problems. Not essential but probably a useful practice for some students.

As etotheipi points out, they've done it incorrectly, which is a good example of irony.
 
  • #5
wrobel
Science Advisor
Insights Author
919
673
Moreover, the Lagrangian is defined up to an additive function
$$\dot f(t,q)=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial q^i}\dot q^i$$
that is the Lagrangians ##L## and ##L'=L+\dot f(t,q)## generate the same equations
 
  • Like
Likes vanhees71 and etotheipi
  • #6
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,299
1,887
They are just following a systematic approach and plugging-in all values of potential energy, even the constant one which is going to disappear. If you do this for all problems, it helps you to establish a standard approach and makes it less likely you'll forget some term(s) in more complex problems. Not essential but probably a useful practice for some students.
I don't think this was the motivation here. The problem asks for the forces of constraint as well, so you don't want to impose the constraint right from the start. It's not entirely clear to me from the problem statement, however, if it was asking for just the forces of constraint on just the particle or for all of the forces of constraint within the system.
 

Related Threads on Problem about Lagrangian mechanics

  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
2K
Replies
1
Views
1K
Replies
1
Views
2K
  • Last Post
Replies
3
Views
1K
Replies
0
Views
4K
  • Last Post
Replies
3
Views
1K
Top