Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

12.6: Lagrangian Mechanics in a Non-Inertial Frame

( \newcommand{\kernel}{\mathrm{null}\,}\)

The above derivation of the equations of motion in the rotating frame is based on Newtonian mechanics. Lagrangian mechanics provides another derivation of these equations of motion for a rotating frame of reference by exploiting the fact that the Lagrangian is a scalar which is frame independent, that is, it is invariant to rotation of the frame of reference.

The Lagrangian in any frame is given by

L=12mvvU(r)

The scalar product vv is the same in any rotated frame and can be evaluated in terms of the rotating frame variables using the same decomposition of the translational plus rotational motion as used previously and given in equation (12.4.4).

Equation (12.4.4) decomposes the velocity in the fixed inertial frame vfix into four vector terms, the translational velocity Vfix of the translating frame, the velocity in the rotating-translating frame vrot, and rotational velocity (ω×rmov). Using equations ??? and (12.4.4), plus appendix equation 19.2.21 for the triple products, gives that the Lagrangian evaluated using vfixvfix equals

L=12m[VfixVfix+vrotvrot+2Vfixvrot+2Vfix(ω×rmov)+2vrot(ω×rmov)+(ω×rmov)2]U(r)

This can be used to derive the canonical momentum in the rotating frame

prot=Lvrot=m[Vfix+vrot+ω×rmov]

The Lagrange equations can be used to derive the equations of motion in terms of the variables evaluated in the rotating reference frame. The required Lagrange derivatives are

ddrLvrot=m[Afix+arot+(ω×vrot)+(˙ω×rmov)]rot

and

Lr=m[(ω×Vfix)(ω×vrot)ω×(ω×rmov)]rotU

where the scalar triple product, equation 19.2.21, has been used. Thus the Lagrange equations give for the rotating frame basis that

marot=Um[Afix+(ω×Vfix)+2(ω×vrot)+ω×(ω×rmov)+(˙ω×rmov)]rot

The external force is identified as Ffixed=U. Equation (12.3.7) can be used to transform between the fixed and the rotating bases.

Afix=[Afix+(ω×V)fix]rot

This leads to an effective force in the non-inertial translating plus rotating frame that corresponds to an effective Newtonian force of

Feffrot=marot=Fm[Afix+2ω×vrot+ω×(ω×rmov)+(˙ω×rmov)]

where Afix is expressed in the fixed frame. The derivation of Equation ??? using Lagrangian mechanics, confirms the identical formula ??? derived using Newtonian mechanics.

The four correction terms for the non-inertial frame basis correspond to the following effective forces.

  • Translational acceleration: Feffmov=mAfix is the usual inertial force experienced in a linearly accelerating frame of reference, and where Afix is with respect to the fixed frame.
  • Coriolis force: Feffcor=2mω×vrot This is a new type of inertial force that is present only when a particle is moving in the rotating frame. This force is proportional to the velocity in the rotating frame and is independent of the position in the rotating frame
  • Centrifugal force: Feffef=mω×(ω×rmov) This is due to the centripetal acceleration of the particle owing to the rotation of the moving axis about the axis of rotation.
  • Transverse (azimuthal) force: Feffaz=m˙ω×rmov This is a straightforward term due to acceleration of the particle due to the angular acceleration of the rotating axes.

The above inertial forces are correction terms arising from trying to extend Newton’s laws of motion to a non-inertial frame involving both translation and rotation. These correction forces are often referred to as “fictitious” forces. However, these non-inertial forces are very real when located in the non-inertial frame. Since the centrifugal and Coriolis terms are unusual they are discussed below.


This page titled 12.6: Lagrangian Mechanics in a Non-Inertial Frame is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?