# Example 7-10 Lagrangian Dynamics Marion and Thornton

## 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

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

BvU
Homework Helper
What would the Euler-Lagrange equations give you for ##\phi## ?

(sin(2φ)/2)*(dθ/dt)-(d^2/dt^2)(φ)=0
I though of integrating it but θ is also dependent on t

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even if I replace dθ/dt from what I get from euler-L of θ I would still get cosθ in the above equation!

BvU
Homework Helper
(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.

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+++++++

BvU