coordenadas espaciales

8/18/2019 coordenadas espaciales

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Generalized Coordinates

Rectangular coordinates are the most straightforward coordinates, but not necessarily the easiest."It would be very desirable and convenient to have a general method for setting up equations ofmotion directly in terms of any convenient set of generalized coordinates. Furthermore, it isdesirable to have uniform methods of writing down, and perhaps solving, the equations of motion

in terms of any coordinate system."

1. he number of generalized coordinates must equal the number of rectangular coordinates.!his is not always three. If we have two obects we are following, each has three coordinates sothe total number of coordinates is si#$%

&. Constraints are conditions which restrict the possible set of values of the coordinates.!'#ample( in &)* circular motion, the radius is constrained +by something to be constant, withonly the angle, -, being allowed to vary.%

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Examples:olar coordinates are an e#ample of a &)* system of coordinates that are not rectangular(

r / r+#,y / √+#0y0 → q1 / q1+#,y and # / r cos+- → # / #+q1,q&

- / -+#,y / arctan+y2#→ q& / q&+#,y and y / r sin+- → y / y+q1,q&.

3nother e#ample involving time( rotating polar coordinates

r / r+#,y,t / √+#0y0 → q1 / q1+#,y,t and # / #+r,-,t / r cos+-ωt

- / -+#,y,t / arctan+y2# ) ωt → q& / q&+#,y,t and y / y+r,-,t / r sin+-ωt

4note( sign difference due to fact that if we let 5 / -ωt, then - / 5)ωt6

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7. 8e also need generalized velocities for energy considerations +for use in conservation ofenergy and for momentum considerations +for use in 9ewtons :econd ;aw(


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Example: Rotating olar Aoordinates

# / r cos+ωt- → #? / +∂#2∂rr? +∂#2∂--? +∂#2∂t

/ r?cos+ωt- -?+)r sin+ωt- ω+)r sin+ωt-

# component of r? and of r-? and of r ω .

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B. he =' involves not ust #?, but #?0. 8hat does this looC liCeD

#?0 / !


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EXAMPLE :

For the case of one particle, use the above to show what the Cinetic energy looCs liCe in the

rotating polar coordinates. In particular,

a find 311, 31&, 3&1 and 3&&K show that this system is orthogonal +31& / 3&1 / JK

b find G1 and G&K

c find oK

d write down the final =inetic 'nergyK identify each term in the e#pression.

For rotating polar coordinates +&)*(

q1 / r / r+#,y,t / √+#0y0 and # / #+r,-,t / r cos+-ωt

q& / - / -+#,y,t / arctan+y2# ) ωt and y / y+r,-,t / r sin+-ωt

=inetic 'nergy( / mi+#?0 y?0 z?06 /


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b G1 / m!+∂#2∂q1+∂#2∂t +∂y2∂q1+∂y2∂t%

/ m!4cos+-ωt6@4 )rP sin+-ωt6 4 sin+-ωt6@4 rP cos+-ωt 6 % / J.

G& / m!+∂#2∂q&+∂#2∂t +∂y2∂q&+∂y2∂t%

/ m! 4)r sin+-ωt6@4 )rP sin+-ωt6 4 r cos+-ωt6@4 rP cos+-ωt6%

/ mr &P .

c o / mi!+∂#i2∂t0 +∂yi2∂t0 +∂zi2∂t0%6

/ >m! 4)rP sin+-ωt60 4 rP cos+-ωt60 % / >mr &P& .

d / mi+#?0 y?0 z?06 / mr?&

J J > mr &

-?&

J mr &

P-? > mr &

P&

/ > mr?& > mr &+-?P&

where r? is the radial speed, and r+-?P is the tangential speed.

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WRITTEN HOMEWORK PROBLEMS #5a and #6a:

+Qou should be able to do part a of problem and part a of problem S now.

Problem #5: (Problem 9-1 in Symon)

Aonsider the coordinates u and w that are defined in terms of the polar coordinates r and -(u = ln(r/a) – θ cot(ζ) and w = ln(r/a) θ tan(ζ)

where a and ζ are constants.

:Cetch the curves of constant u and w( this is already done for you ) see notes on ne#tsection and2or the e#cel spreadsheet E:piral Aoordinates on the course web page..

a Find the Cinetic energy for a particle of mass m in terms of u, w, du2dt, and dw2dt . b Find e#pressions for Tu and Tw in terms of the polar force components Fr and F- .c Find pu and pw .

@@ d e#tra credit( find the forces Tu and Tw required to maCe the particle move with constantspeed, ds2dt, along a spiral of constant u/uo .

!"$: %ou can try to &ind t'e inere tran&ormation to *et+(u,w) and y(u,w), and t'en roceed a indicated aboe. r it

may be eaier to *et t'e 0inetic ener*y in olar &orm (r,θ) &ind

t'e inere tran&ormation: r(u,w) and θ(u,w) t'en &ind

r2(u,w,u2,w2) and θ2(u,w,u2,w2) t'en write t'e 0inetic ener*y in

term o& u,w,u2,w2.


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Problem #3: (Problem 9-4 in Symon)

For plane +&)* parabolic coordinates f and h defined such that( # / f U h and y / &+fh12&

+see the e#cel spreadsheet Earabolic Aoordinates accessed from the course web page to see an

imageK

a find the e#pression for the Cinetic energy in terms of f, h, f?, and h?K

b find the e#pression for the momenta( pf and ph K

c write out the ;agrange equations in these coordinates if we assume the particle is not acted on

by any force.

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