3 Just as in 4] from last HW set, consider the polar coordinate chart for R²: y: (0, 0) × (0, 27) → R²: (r, 8) → (r cos 0, r sin 0). (a) Write {dr (a), do(a)} in terms of {dx(a), dy(a)}. (b) Suppose that a cotangent vector peT R² is written as p = Pxdx(a) + Pydy(a) = prdr(a)+ pęd0(a). Find (px. Py) in terms of (pr. pe) using the result from (a). (c ' Find an expression for the cotangent lift T *V-'(r, 0, pr. Pe). (d) Consider a Hamiltonian L: T*R² → R that can be written in the Cartesian coordinates as m k H(x, y. Px- Py):=(P+ P5) x² + y² with some constants m, k > 0. Find an expression for the Hamiltonian in terms of the polar coordinates (r, 0) and their associated momentum components (pr, pe) by computing H oT*y-(r, 0, pr. pe).
3 Just as in 4] from last HW set, consider the polar coordinate chart for R²: y: (0, 0) × (0, 27) → R²: (r, 8) → (r cos 0, r sin 0). (a) Write {dr (a), do(a)} in terms of {dx(a), dy(a)}. (b) Suppose that a cotangent vector peT R² is written as p = Pxdx(a) + Pydy(a) = prdr(a)+ pęd0(a). Find (px. Py) in terms of (pr. pe) using the result from (a). (c ' Find an expression for the cotangent lift T *V-'(r, 0, pr. Pe). (d) Consider a Hamiltonian L: T*R² → R that can be written in the Cartesian coordinates as m k H(x, y. Px- Py):=(P+ P5) x² + y² with some constants m, k > 0. Find an expression for the Hamiltonian in terms of the polar coordinates (r, 0) and their associated momentum components (pr, pe) by computing H oT*y-(r, 0, pr. pe).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3 Just as in 4| from last HW set, consider the polar coordinate chart for R2:
V: (0, 00) × (0, 27) → R²;
(r, 0) → (r cos 0, r sin 0).
Write {dr (a), do(a)} in terms of {dx(a), dy(a)}.
Suppose that a cotangent vector p e TÄR² is written as
(a)
(b)
p = Pxdx(a) + pydy(a) = prdr (a) + pod0(a).
Find (px, Py) in terms of (pr. pe) using the result from (a).
(с.
* Find an expression for the cotangent lift T *V-(r, 0, pr, Pe).
Consider a Hamiltonian L: T*R? → R that can be written in the Cartesian coordinates as
(d)
m
k
Н(х, у, рх. Ру) :%3D
(p + p3)+
2
x² + y²
with some constants m,k > 0. Find an expression for the Hamiltonian in terms of the polar coordinates (r, 0)
and their associated momentum components (pr, pe) by computing H oT*¥-(r, 0, pr. pe).
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