Centrifugal Force - Use of The Term in Lagrangian Mechanics

Use of The Term in Lagrangian Mechanics

Lagrangian mechanics formulates mechanics in terms of generalized coordinates {q_k}, which can be as simple as the usual polar coordinates or a much more extensive list of variables. Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the Euler–Lagrange equations. Among the generalized forces, those involving the square of the time derivatives {(dq_k   ⁄ dt )2} are sometimes called centrifugal forces.

The Lagrangian approach to polar coordinates that treats as generalized coordinates, as generalized velocities and as generalized accelerations, is outlined in another article, and found in many sources. For the particular case of single-body motion found using the generalized coordinates in a central force, the Euler–Lagrange equations are the same equations found using Newton's second law in a co-rotating frame. For example, the radial equation is:

where is the central force potential and μ is the mass of the object. The left side is a "generalized force" and the first term on the right is the "generalized centrifugal force". However, the left side is not comparable to a Newtonian force, as it does not contain the complete acceleration, and likewise, therefore, the terms on the right-hand side are "generalized forces" and cannot be interpreted as Newtonian forces.

The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference, but in the case of motion in a central potential the result is the same as the fictitious centrifugal force derived in a co-rotating frame. The Lagrangian use of "centrifugal force" in other, more general cases, however, has only a limited connection to the Newtonian definition.