lagrange analysis for non-relativistic particle in the rotating reference frame

8/3/2019 Lagrange Analysis for Non-Relativistic Particle in the Rotating Reference Frame

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Lagrange Analysis for Non-Relativistic Particle in the

Rotating Reference Frame

Wen-Jie Tian

College of Physics and Information Technology,

Shaanxi Normal University, P.O.Box 339, Xian, 710062, P.R.China

Abstract: To analyze the dynamics for non-relativistic particles in rotating refer-

ence frames, centrifugal inertial force and Coriolis forces should be introduced to ensurethat Newtons second law holds mathematically in formalism. Under the same set of

generalized coordinates, we study the dynamics for non-relativistic particles in conser-

vative field via Lagrange mechanics, in both rest inertial reference (case I) and rotating

noninertial reference (case II). Its concluded that, an extra generalized potential en-

ergy (GPE) term, which is nonconservative, is required to keep the Lagrange function of

case II identical to that of case I. The generalized force corresponding to the very GPE

term agrees with the resultant of the centrifugal and Coriolis forces. Inversely, the GPE

terms generated by the two inertial forces are numerous, only one of which is effective.

Calculations go on in rectangular and spherical coordinates system respectively.

Key Words:Lagrange Function; Reference Frames; Generalized Potential Energy; In-

ertial Force.

1 Introduction

As to the dynamics for non-relativistic particles at general motion in rotating reference

frames, the original method is the introduction of centrifugal and Coriolis force and

the employment of Newtons second law. On the other hand, Lagrange mechanics,

as a generalized formulation of Newtons work, has been a powerful tool in analyzing

dynamics of all complicated kinds of processes[1][2][3][4]5. Hence this paper aims at a

Lagrange analysis of the above thesis, i.e. dynamics in rotating references. Concrete

computing goes in both rectangular and spherical coordinate systems, and for non-

relativistic particles only.

[email protected]; [email protected]

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2 Scenario in Rectangular Coordinates

Given the inertial coordinates system R[O; i, j, k] at rest, here is another coordinates

system R[O;i,j, k], whose origin and z-axis are coincident with those of R, yet ro-

tates around z-axis relative to R at a constant angular speed of . Theres a particle

with mass m located in the conservative field in R, with potential energy V(x,y,z) (rel-

ative to R). m is at general motion in R, as well as an associated rotation speed with R.

Firstly, lets employ the inertial reference R to study the dynamics of m, but take

coordinates {x,y,z} in R as generalized ones for Lagrange analysis. Thus, the absolute

velocity of m is:

v = xi + yj + zk + k

xi + yj + zk

= (x y)i + (y + x)j + zk(1)

The particle is non-relativistic, so the kinetic energy is:

T =1

2mv v

=1

2m

(x y)2 + (y + x)2 + z2

=1

2m

x2 + y2 + z2

+1

2m2

x2 + y2

+ m (xy xy)

(2)

and the Lagrange function is:

L = T V

=1

2m

x2 + y2 + z2

+1

2m2

x2 + y2

+ m (xy xy) V(x ,y ,z)

(3)

ThusL

x= m2x + my

V

x(4a)

L

x= mx my (4b)

L

y= m2y mx

V

y(4c)

Ly

= my + mx (4d)

L

z=

V

z(4e)

L

z= mz (4f)

According to Lagranges dynamical equation for conservative fields

d

dt

L

qi

L

qi= 0 (5)

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The concrete components will be

mx = V

x+ m2x + 2my (6a)

my = Vy

+ m2y 2mx (6b)

mz = V

z(6c)

Hence Eqs6 are the absolute dynamical equations for m relative to R.

On the other hand, if the observer locates in the rotating reference R, where the

angular velocity vanishes due to co-moving, one will find the kinetic and potential

energy to be

T =

1

2 m

x2

+ y2

+ z2

, V = V(x,y,z) (7)

respectively. Its apparent that T V from Eq7 isnt identical to Eq3. As is known,

for the same set of generalized coordinates, the Lagrange function is determined and

unique, independent from the motional states of the coordinates systems. Thus, via

comparison between Eq3 and Eq7, one could conclude the Lagrangian function for R

reference to be:

L =1

2m

x2 + y2 + z2

V(x,y,z)

1

2m2

x2 + y2

m (xy xy)

= T V U

(8)

where

U = 1

2m2

x2 + y2

m (xy xy) (9)

is the generalized potential energy(GPE). This term is said to be generalized because U

depends on not only the generalized coordinates, but also their one-order derivatives,

i.e. the generalized velocities, while Eq5 dedicates only to systems that are workless,

holonomic and conservative. An energy term in Lagrange function such as U conjugates

with certain generalized force (GF) Q. However, in R reference it comes up with

centrifugal and Coriolis forces due to Newton mechanism at the same time. As is tobe verified below, the GF Q conjugating with U agrees with the resultant of these two

inertial forces, or in other words, the two inertial forces leads to the GPE U. The

components of Q conjugating with U above read:

Qx =d

dt

U

x

U

x= m2x + 2my (10a)

Qy =d

dt

U

y

U

y= m2y + 2mx (10b)

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Qz =d

dt

U

z

U

z= 0 (10c)

Or

Q = m2x + 2myi + m

2y 2mxj (11)The resultant of the centrifugal and Coriolis forces are:

m ( r) 2m v

=

m2xi + m2yj

2myi + 2mxj

=

m2x + 2my

i +

m2y 2mx

j

(12)

which is identical to Q perfectly. Hence, as for the non-relativistic particle m at motion

relative to the rotating reference R, one runs into two inertial forces by Newton mech-

anism, or into GPE U. The GF Q associated with U agrees with the resultant of the two

inertial forces, and inversely, due to non-conservation, [m ( r) 2m v] =0, the two inertial forces result in the non-conservative GPE U.

Now lets conduct the same process in spherical coordinates system.

3 Scenario in Spherical Coordinates

Relative to the reference frame R at rest, we establish a spherical coordinates system

RS[O; er, e, e] based on the Cartesian coordinates RC[O;i,j, k] in the rotating refer-

ence R. RS shares the origin ofRC, and rotates around R

s z-axis at a constant angularspeed . Here the particle m, who still shares the rotation of RS and locates in the

conservative field in RS, has potential energy of V(r). To perform Lagrangian analysis

proceedingly, we still take R for reference, but utilize {r,,} in RS as the generalized

coordinates. Then the absolute linear velocity and the absolute angular velocity for m

will be

v = r er + r e + r sin e + r er (13)

= (cos er sin e) (14)

respectively. Insert Eq14 into Eq13 to get:

v = r er + r e + (r sin + r sin ) e (15)

Hence the kinetic energy and Lagrangian function for m read:

T =1

2mv v

=1

2m

r2 + r22 + (r sin + r sin )2

=1

2m

r2 + r22 + r22 sin2 + 2r2 sin2 + 2r2 sin2

(16)

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L = T V

=1

2m

r2 + r22 + r22 sin2

+1

2m

2r2 sin2 + 2r2 sin2

V(r)(17)

Hence

Lr

= mr2 + mr2 sin2 + m2r sin2 + 2mr sin2 Vr

(18a)

L

r= mr (18b)

L

=

1

2mr22 sin2 +

1

2m2r2 sin2 + mr2 sin2 (18c)

L

= mr2 (18d)

L

= 0 (18e)

L

= mr2 sin2 + mr2 sin2 (18f)

Then the components of Lagrange equation Eq5 are

mr = V

r+ mr2 + mr2 sin2 + m2r sin2 + 2mr sin2 (19a)

mr2 = 2mrr +1

2mr22 sin2 +

1

2m2r2 sin2 + mr2 sin2 (19b)

mr2 sin2 = 2mrr sin2 mr2 sin2 2mrr sin2 mr2 sin2 (19c)

Hence Eqs19 are the absolute dynamical equations for m relative to RS.

On the other hand, if the observer locates in the rotating coordinates RS, where

vanishes too, the kinetic energy and potential energy are to be

T =1

2m

r2 + r22 + r22 sin2

, V = V(r) (20)

For the generalized coordinates {r,,}, the Lagrange function is determined and

unique. To make Eq20 in agreement with Eq17, we put forward the GPE term U

U = 1

2 m

2

r2

sin2

+ 2r2

sin2

(21)

to attain

L = T V U

=1

2m

r2 + r22 + r22 sin2

V(r)

1

2m

2r2 sin2 + 2r2 sin2 (22)

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According to Eq21, its clear that U, being non-conservative, depends not only on r, but

also on the generalized coordinate and the generalized velocity . The components of

the conjugating GF Q are

Qr = ddt

Ur

U

r= m2r sin2 + 2mr sin2 (23a)

Q =d

dt

U

U

=

1

2m2r2 sin2 + mr2 sin2 (23b)

Q =d

dt

U

U

= 2mrr sin2 mr2 sin2 (23c)

while the centrifugal force is

m ( r) = m2r sin2 er +1

2m2r sin2 e (24)

And the Coriolis force is

2m v = 2m(cos er sin e) (r er + r e + r sin e)

= 2mr sin2 er + mr sin2 e + (2mr sin 2mr cos ) e(25)

Hence, the components of the resultant of the two inertial forces are:

Fr = m2r sin2 + 2mr sin2 (26a)

F =1

2m2r sin2 + mr sin2 (26b)

F = 2mr sin 2mr cos (26c)

In spherical coordinates, the transformational relations between the generalized force

and the ordinary force are:

Qr = Fr, Q = rF, Q = r sin F (27)

Comparing Eq23, Eq26 and Eq27, one immediately concludes that the GF Q is equal

to the resultant of the two inertial forces .

4 Discussions and Extensions

With rectangular and spherical coordinates respectively, we have performed Lagrange

analysis on the dynamical process of the non-relativistic particles at general motion

in uniformly rotating references. Based on energy terms, the complex analyzing of

ordinary forces is avoided. As is commented above, the problem of inertial forces

in Newtonian mechanism becomes the problem of non-conservative GPE in Lagrange

mechanism, and the conjugating GF identifies with the resultant of the inertial forces.

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However, further algebra as below shows that, non-conservative fields resulting from

centrifugal and Coriolis forces are countless, and U in Eq9 is simply a particular case,

yet the only effective one. Eq9 is made up of two different components, each of which

is self-symmetric:

U = U1 + U2, U1 = 1

2m2(x2 + y2), U2 = m(xy xy) (28)

U1 is the centrifugal potential energy resulting from the centrifugal inertial force, which

depends only on the generalized coordinates. Since the R reference rotates around the

fixed z-axis in R reference, the coordinate z vanishes in U1 and U1 is also z-axis-fugal

energy. U1 is the conservative component of U, U2 being the non-conservative com-

ponent. The conservative potential energy U1 results from centrifugal (or z-axis-fugal)

force, the GPE U2 resulting from Coriolis force.

As to Coriolis force, FC = 2mv , its similar to the Lorentz force for charged

particles at motion in static magnetic field[6], FL = qv B. Obviously, FL = 0,

FL is non-conservative, too. And B = A, where A is the magnetic vectorial

potential. Its proved that the generalized Lagrange potential energy corresponding to

FL = qv B = qv ( A) is [2]:

U = qv A (29)

Similarly, lets represent as = A. Considering = k, hence A1 = yi, A2 =

xj or A3 =1

2(yi + xj) could all lead to Ai = k = , (i = 1, 2, 3). Yet, A3

is actually half of the sum of A1 and A2, i.e. A3 =1

2

A1 + A2

, so with a resembling

method, such as the sum of 1n

A1 andn1n

A2, we could generate numerous candidates

of A, which satisfies = k = A, and FC = 2mv = 2mv ( A). Similar

to Eq29, we attain the generalized Lagrange potential energy corresponding to FC:

U2 = 2mv A (30)

For a unique FC, due to the diversity of A, U2 is not unique at all, so does U = U1 + U2.

However, only one of the numerous GPEs generated by FC is effective for the identity

of Lagrange functions in rest inertial reference R and rotating reference R, i.e. in Eq3

and Eq7. And the effective one, U in Eq9, is the particular case when A takes the value

of A3 above.

The formulation of this paper could act as a model and work in different problems.

As an example, stellar and galaxies in the distance could be selected to be inertial ref-

erence at rest, and the earth rotates around its polar axis uniformly. In some global

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motions such as ocean currents and trade winds, Coriolis force makes a considerable

difference. The formulation in this paper works well on such practical problems.

Reference[1]Marion J B, Classical Dynamics of Particles and Systems(Ed2). Academic Press.

1970. 189-227.

[2]JIN S N. Classical Mechanics. Shanghai: Fudan University Press, 1987. 198-220.

[3]Greenwood D T. Classical Dynamics. Prence-Hall Inc., 1977. 56-102.

[4]ZHOU Y B. Textbook for Theoretical Mechanics(Ed2). Beijing: High Education

Press, 1986. 241-298.

[5]Wu D Y. Classical Dynamics. Beijing: Science Press, 1982. 33-85.

[6]Guo S H. Electrodynamics(Ed2). Beijing: High Education Press, 1997. 98-107.

Appendix:

This paper is accepted by Science and Technology of Western China, by the Chinese

Academy of Science

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lagrange analysis for non-relativistic particle in the rotating reference frame

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