motion in central-force field

motion in central-force field

Let us consider a body with

m

in a gravitational force field (

http://planetmath.org/VectorField

) exerted by the origin and directed always from the body towards the origin.  Set the plane through the origin and the

velocity vector

v

→

of the body.  Apparently, the body is forced to move constantly in this plane, i.e. there is a question of a planar motion.  We want to derive the

trajectory

of the body.

Equip the plane of the motion with a

polar coordinate system

r

,

φ

and denote the

position vector

of the body by

r

→

.  Then the velocity vector is

v

→

=

d

⁢

r

→

d

⁢

t

=

d

d

⁢

t

⁢

(

r

⁢

r

→

0

)

=

d

⁢

r

d

⁢

t

⁢

r

→

0

+

r

⁢

d

⁢

φ

d

⁢

t

⁢

s

→

0

,

(1)

where

r

→

0

and

s

→

0

are the

unit vectors

in the

direction

of

r

→

and of

r

→

rotated 90 degrees anticlockwise (

r

→

0

=

i

→

⁢

cos

⁡

φ

+

j

→

⁢

sin

⁡

φ

,  whence

r

→

0

d

⁢

t

=

(

-

i

→

⁢

sin

⁡

φ

+

j

→

⁢

cos

⁡

φ

)

⁢

d

⁢

φ

d

⁢

t

=

d

⁢

φ

d

⁢

t

⁢

s

→

0

).  Thus the kinetic energy of the body is

E

k

=

1

2

⁢

m

⁢

|

d

⁢

r

→

d

⁢

t

|

2

=

1

2

⁢

m

⁢

(

(

d

⁢

r

d

⁢

t

)

2

+

(

r

⁢

d

⁢

φ

d

⁢

t

)

2

)

.

Because the gravitational force on the body is exerted along the position vector, its moment is 0 and therefore the angular momentum

L

→

=

r

→

×

m

⁢

d

⁢

r

→

d

⁢

t

=

m

⁢

r

2

⁢

d

⁢

φ

d

⁢

t

⁢

r

→

0

×

s

→

0

of the body is

constant

; thus its magnitude is a constant,

m

⁢

r

2

⁢

d

⁢

φ

d

⁢

t

=

G

,

whence

d

⁢

φ

d

⁢

t

=

G

m

⁢

r

2

.

(2)

The

central

force

F

→

:=

-

k

r

2

⁢

r

→

0

(where

k

is a constant) has the

scalar potential

U

⁢

(

r

)

=

-

k

r

.  Thus the total energy

E

=

E

k

+

U

⁢

(

r

)

of the body, which is constant, may be written

E

=

1

2

⁢

m

⁢

(

d

⁢

r

d

⁢

t

)

2

+

1

2

⁢

m

⁢

r

2

⁢

(

G

m

⁢

r

2

)

2

-

k

r

=

m

2

⁢

(

d

⁢

r

d

⁢

t

)

2

+

G

2

2

⁢

m

⁢

r

2

-

k

r

.

This equation may be revised to

(

d

⁢

r

d

⁢

t

)

2

+

G

2

m

2

⁢

r

2

-

2

⁢

k

m

⁢

r

+

k

2

G

2

=

2

⁢

E

m

+

k

2

G

2

,

i.e.

(

d

⁢

r

d

⁢

t

)

2

+

(

k

G

-

G

m

⁢

r

)

2

=

q

2

where

q

:=

2

m

⁢

(

E

+

m

⁢

k

2

2

⁢

G

2

)

is a constant.  We introduce still an auxiliary angle

ψ

such that

k

G

-

G

m

⁢

r

=

q

⁢

cos

⁡

ψ

,

d

⁢

r

d

⁢

t

=

q

⁢

sin

⁡

ψ

.

(3)

Differentiation

of the first of these equations implies

G

m

⁢

r

2

⋅

d

⁢

r

d

⁢

t

=

-

q

⁢

sin

⁡

ψ

⁢

d

⁢

ψ

d

⁢

t

=

-

d

⁢

r

d

⁢

t

⋅

d

⁢

ψ

d

⁢

t

,

whence, by (2),

d

⁢

ψ

d

⁢

t

=

-

G

m

⁢

r

2

=

-

d

⁢

φ

d

⁢

t

.

This means that

ψ

=

C

-

φ

, where the constant

C

is determined by the initial conditions.  We can then solve

r

from the first of the equations (3), obtaining

r

=

G

2

k

⁢

m

⁢

(

1

-

G

⁢

q

k

⁢

cos

⁡

(

C

-

φ

)

)

=

p

1

-

ε

⁢

cos

⁡

(

φ

-

C

)

,

(4)

where

p

:=

G

2

k

⁢

m

,

ε

:=

G

⁢

q

k

.

By the

http://planetmath.org/node/11724

parent entry, the result (4) shows that the trajectory of the body in the gravitational field (

http://planetmath.org/VectorField

) of one point-like sink is always a

conic section

whose focus the sink causing the field.

As for the of the conic, the most interesting one is an

ellipse

.  It occurs, by the

http://planetmath.org/node/11724

parent entry, when

ε

<

1

.  This condition is easily seen to be

equivalent

with a negative total energy

E

of the body.

One can say that any planet revolves around the Sun along an ellipse having the Sun in one of its foci — this is

Kepler’s first law

.

References

1

Я. Б. Зельдович &  А. Д. Мышкис:

Элементы  прикладной  математики

.  Издательство
‘‘Наука’’.  Москва (1976).

Title

motion in central-force field

Canonical name

MotionInCentralforceField

Date of creation

2013-03-22 18:52:41

Last modified on

2013-03-22 18:52:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Derivation

Classification

msc 15A72

Classification

msc 51N20

Synonym

Kepler’s first law

Related topic

CommonEquationOfConics

Related topic

PropertiesOfEllipse

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