Mechanics of Planar Particle Motion - Fictitious Forces in Polar Coordinates - Two Terminologies - Lagrangian Approach

Lagrangian Approach

See also: Lagrangian, Lagrangian mechanics, Generalized coordinates, and Euler-Lagrange equations

To motivate the introduction of "coordinate" inertial forces by more than a reference to "mathematical convenience", what follows is a digression to show these forces correspond to what are called by some authors "generalized" fictitious forces or "generalized inertia forces". These forces are introduced via the Lagrangian mechanics approach to mechanics based upon describing a system by generalized coordinates usually denoted as {q_k}. The only requirement on these coordinates is that they are necessary and sufficient to uniquely characterize the state of the system: they need not be (although they could be) the coordinates of the particles in the system. Instead, they could be the angles and extensions of links in a robot arm, for instance. If a mechanical system consists of N particles and there are m independent kinematical conditions imposed, it is possible to characterize the system uniquely by n = 3N - m independent generalized coordinates {q_k}.

In classical mechanics, the Lagrangian is defined as the kinetic energy, of the system minus its potential energy, . In symbols,

Under conditions that are given in Lagrangian mechanics, if the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation, a particular family of partial differential equations.

Here are some definitions:

Definition:

is the Lagrange function or Lagrangian, q_i are the generalized coordinates, are generalized velocities,

  are generalized momenta,

  are generalized forces,

  are Lagrange's equations.

It is not the purpose here to outline how Lagrangian mechanics works. The interested reader can look at other articles explaining this approach. For the moment, the goal is simply to show that the Lagrangian approach can lead to "generalized fictitious forces" that do not vanish in inertial frames. What is pertinent here is that in the case of a single particle, the Lagrangian approach can be arranged to capture exactly the "coordinate" fictitious forces just introduced.

To proceed, consider a single particle, and introduce the generalized coordinates as {q_k} = (r, θ). Then Hildebrand shows in polar coordinates with the q_k = (r, θ) the "generalized momenta" are:

leading, for example, to the generalized force:

with Q_r the impressed radial force. The connection between "generalized forces" and Newtonian forces varies with the choice of coordinates. This Lagrangian formulation introduces exactly the "coordinate" form of fictitious forces mentioned above that allows "fictitious" (generalized) forces in inertial frames, for example, the term Careful reading of Hildebrand shows he doesn't discuss the role of "inertial frames of reference", and in fact, says " presence or absence depends, not upon the particular problem at hand but upon the coordinate system chosen." By coordinate system presumably is meant the choice of {q_k}. Later he says "If accelerations associated with generalized coordinates are to be of prime interest (as is usually the case), the terms may be conveniently transferred to the right … and considered as additional (generalized) inertia forces. Such inertia forces are often said to be of the Coriolis type."

In short, the emphasis of some authors upon coordinates and their derivatives and their introduction of (generalized) fictitious forces that do not vanish in inertial frames of reference is an outgrowth of the use of generalized coordinates in Lagrangian mechanics. For example, see McQuarrie Hildebrand, and von Schwerin. Below is an example of this usage as employed in the design of robotic manipulators:

In the above equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in where the coefficients may depend on . These are further classified into two types. Terms involving a product of the type are called centrifugal forces while those involving a product of the type for i ≠ j are called Coriolis forces. The third type is functions of only and are called gravitational forces.

— Shuzhi S. Ge, Tong Heng Lee & Christopher John Harris: Adaptive Neural Network Control of Robotic Manipulators, pp. 47-48

For a robot manipulator, the equations may be written in a form using Christoffel symbols Γ_ijk (discussed further below) as:

where M is the "manipulator inertia matrix" and V is the potential energy due to gravity (for example), and are the generalized forces on joint i. The terms involving Christoffel symbols therefore determine the "generalized centrifugal" and "generalized Coriolis" terms.

The introduction of generalized fictitious forces often is done without notification and without specifying the word "generalized". This sloppy use of terminology leads to endless confusion because these generalized fictitious forces, unlike the standard "state-of-motion" fictitious forces, do not vanish in inertial frames of reference.