The Lagrangian in generalized coordinates 1 2 L(0, 4, 8, 4) = gm R cos(0) + − m R² (¿² sin² (0) + Ö²) - 2 (b) The generalized momenta are Pe=0m R² P₁ = m R² 4 sin²(0) (c) The Hamiltonian is CSC csc² (0) p² + P² H(0, 4, Po, P4)= - gm R cos(0) 2m R²
The Lagrangian in generalized coordinates 1 2 L(0, 4, 8, 4) = gm R cos(0) + − m R² (¿² sin² (0) + Ö²) - 2 (b) The generalized momenta are Pe=0m R² P₁ = m R² 4 sin²(0) (c) The Hamiltonian is CSC csc² (0) p² + P² H(0, 4, Po, P4)= - gm R cos(0) 2m R²
Related questions
Question
Show how to get Hamiltonian (answer highlighted in pictures
Transcribed Image Text:Consider a small, point-like particle of mass m that
slides under the influence of gravity, without friction
inside of a hemispherical bowl, of radius R as pic-
tured. Use the polar and the azimuthal angles 0 and
as generalised coordinates to describe the location of
the particle.
(a) Using the two spherical angles 0 and 4 construct the
Lagrangian.
(b) Determine the generalized momenta po
and
·P4-
Side view
Ө
R
(c) Construct the Hamiltonian.
Top view
Transcribed Image Text:The Lagrangian in generalized coordinates
1
4, ·
L(0, +, i, )=gm R cos(0) + − m R (sin(0) + ở
-
2
(b) The generalized momenta are
Pe=0m R²
P₁ = m R² sin² (0)
(c) The Hamiltonian is
csc² (0) p² + p
H(0, 4, Pe, P)
- gm R cos(0)
2m R²
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
