Consider the simple Atwood machine moving under the influence of gravity as depicted. Assume that the pulley is massive and its moment of inertia is I. The rope is massless, its length is constant and it rolls without slipping. (a) Construct the Lagrangian in terms of the single generalized coordi- nate x. Assume that x is measured from the suspension point down- ward, and set the reference point of the potential to the suspension point as well. (b) Derive the Lagrangian equation of motion. (c) Find the acceleration of the masses m₁ and m2. (d) Calculate the generalized momentum. (e) Construct the Hamiltonian. (f) Derive the Hamiltonian equations of motion.
Consider the simple Atwood machine moving under the influence of gravity as depicted. Assume that the pulley is massive and its moment of inertia is I. The rope is massless, its length is constant and it rolls without slipping. (a) Construct the Lagrangian in terms of the single generalized coordi- nate x. Assume that x is measured from the suspension point down- ward, and set the reference point of the potential to the suspension point as well. (b) Derive the Lagrangian equation of motion. (c) Find the acceleration of the masses m₁ and m2. (d) Calculate the generalized momentum. (e) Construct the Hamiltonian. (f) Derive the Hamiltonian equations of motion.
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Show in detail how the lagragian is constructed. Specifically, the rotational energy term
Transcribed Image Text:Consider the simple Atwood machine moving under the influence of
gravity as depicted. Assume that the pulley is massive and its moment
of inertia is I. The rope is massless, its length is constant and it rolls
without slipping.
(a) Construct the Lagrangian in terms of the single generalized coordi-
nate x. Assume that x is measured from the suspension point down-
ward, and set the reference point of the potential to the suspension
point as well.
(b) Derive the Lagrangian equation of motion.
(c) Find the acceleration of the masses m₁ and m2.
(d) Calculate the generalized momentum.
(e) Construct the Hamiltonian.
(f) Derive the Hamiltonian equations of motion.
m₁
R
1-x
m2
(g) Using the latter confirm your result for the acceleration of the masses m₁
and m2.
(h) Show that the coordinate transformation X = x + c, where c is a constant, is a symmetry of the
system.
(i) Extra credit: Derive the Noether invariant for the above transformation.
(j) Is energy conserved by this system? Why? Check your answer by directly calculating H(x, px).
(a) There's a single generalized coordinate x.
Lagrangian in the generalized coordinate
L(x, x)=gm2 (1-x) + gm₁ x +
1
-
2
1
+
2
m2 x² +
Ix
2 R2
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