A particle of mass m moves on a plane. Locate its position using polar coordinates (r,0). It is attached to a spring with one end fixed at the origin. Its kinetic energy is 1 m ( ;? + r? 0²) and its potential T = energy is U = -k (r - r) where k > 0 is constant. %3D (a) Obtain its Lagrangian. (b) Use two Euler-Lagrange equations to show that (1) r0 is constant , (2) m(* -rè?) = -k(r-r,).
A particle of mass m moves on a plane. Locate its position using polar coordinates (r,0). It is attached to a spring with one end fixed at the origin. Its kinetic energy is 1 m ( ;? + r? 0²) and its potential T = energy is U = -k (r - r) where k > 0 is constant. %3D (a) Obtain its Lagrangian. (b) Use two Euler-Lagrange equations to show that (1) r0 is constant , (2) m(* -rè?) = -k(r-r,).
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:A particle of mass m moves on a plane. Locate its
position using polar coordinates (r,0). It is attached
to a spring with one end fixed at the origin.
1
Its kinetic energy is
1
m (;?
2
+ r? 6?)
T
and its potential
|
is U = k(r- r)? where k > 0 is constant.
energy
2
(a) Obtain its Lagrangian.
(b) Use two Euler-Lagrange equations to show that
(1) r0 is constant ,
(2) m(*-ro?) = -k(r–r,).
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