Example 7-10 Lagrangian Dynamics Marion and Thornton

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1. May 29, 2017

MARX

1. The problem statement, all variables and given/known data
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

2. Relevant equations
L=T-U

3. 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

2. May 30, 2017

BvU

What would the Euler-Lagrange equations give you for $\phi$ ?

3. May 30, 2017

MARX

(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: May 30, 2017
4. May 30, 2017

MARX

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. Jun 1, 2017

BvU

(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. Jun 3, 2017

MARX

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. Jun 3, 2017

BvU

OK, on to the next ...

8. Jun 3, 2017

MARX

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