Example 7-10 Lagrangian Dynamics Marion and Thornton

In summary, the homework statement says that a particle of mass is on a frictionless hemisphere with a radius of a. The Lagrange equations are used to determine the constraint force and the point at which the particle detaches from the hemisphere.
  • #1
MARX
49
1

Homework Statement


A particle of mass m is on top of a frictionless hemisphere centered at the origin with radius a"
Set up the lagrange equatinos determine the constraint force and the point at which the particle detaches from the hemisphere

Homework Equations


L=T-U

The Attempt at a Solution


this is NOT HW
solution is there in full (example from book) I am just trying to understand
why is φ not one of the proper generalized coordinates? can't the particle move sideways as well when released? shouldn't the GC be (r,θ,φ) ie all spherical and the constraint equation r-a=0.
THanks
 
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  • #2
What would the Euler-Lagrange equations give you for ##\phi## ?
 
  • #3
(sin(2φ)/2)*(dθ/dt)-(d^2/dt^2)(φ)=0
I though of integrating it but θ is also dependent on t
 
Last edited:
  • #4
even if I replace dθ/dt from what I get from euler-L of θ I would still get cosθ in the above equation!
 
  • #5
(Sorry for the late reaction) ##\quad##I find, with $$ \ T = {1\over 2} m\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\,\dot\phi^2 \ $$ (here, top of p7) and $$V = -mr\cos\theta\ $$ that $$ {\partial {\mathcal L}\over \partial \phi} = 0 $$ so that $${ d\over dt } 2r^2 \sin^2\theta\,\dot\phi = 0 \ ,$$in other words: ##L_z=## constant (##L## being the angular momentum). The particle is let go with ##L_z=0## so it stays at that value.
 
  • #6
yes absolutely simple one line answer
forgive my stupidity I copied the T term from Wolfram Alpha and there they use θ for φ and vice vera- (I was too lazy to do this trivial math considering I'm a fanatic I solve every single example and problem in every chapter)
damn mathematicians-lol(kidding love them)
that explains why my differential equation above gets complicated and doesn't reveal conservation
I completely get it the angular momentum does not change in that direction
Lz is explicitly determined and happens to be a constant multiple of the euler-Lagrangian variation and that's why it never moves sideways I get it supports intuition in this case
Thank you so much for your help and clarification appreciated A+++++++
 
  • #7
OK, on to the next ...
 
  • #8
Sure
I am on problem 7-20 so far able to solve them all
actually I do have a question (clarification) on problem 7-10 should I post here or different thread
THanks
 

Related to Example 7-10 Lagrangian Dynamics Marion and Thornton

1. What is Lagrangian dynamics?

Lagrangian dynamics is a mathematical method used to describe the motion of a system of particles in classical mechanics. It was developed by Joseph-Louis Lagrange in the 18th century and is based on the principle of least action, which states that the path followed by a particle between two points is the one that minimizes the action functional.

2. What is the significance of Example 7-10 in Marion and Thornton's book?

Example 7-10 in Marion and Thornton's book is a classic example used to illustrate the application of Lagrangian dynamics in solving problems related to the motion of particles. It involves a system of two masses connected by a string, and shows how Lagrangian dynamics can be used to determine the equations of motion and solve for the system's behavior.

3. What is the difference between Lagrangian mechanics and Newtonian mechanics?

Lagrangian mechanics and Newtonian mechanics are two different approaches to describing the motion of particles in classical mechanics. While Newtonian mechanics is based on the concept of forces and uses Newton's laws of motion, Lagrangian mechanics is based on the principle of least action and uses the Lagrangian function to determine the equations of motion.

4. What are the advantages of using Lagrangian dynamics in problem-solving?

One of the main advantages of using Lagrangian dynamics is its ability to simplify complex systems and reduce the number of equations needed to describe their motion. It also provides a more elegant and general approach to solving problems in classical mechanics, as it is not limited to specific coordinate systems or types of forces.

5. What are some real-world applications of Lagrangian dynamics?

Lagrangian dynamics has many applications in physics and engineering, including celestial mechanics, fluid dynamics, and robotics. It is also used in fields such as economics, where it is used to study the behavior of complex systems. Additionally, it has been applied in the development of mathematical models for predicting the behavior of financial markets.

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