Generalized Coordinates and Generalized Forces

The Idea in Plain Language

Imagine a pendulum. In Cartesian coordinates, you must track the horizontal position \(x\) and vertical position \(y\), and you must explicitly include the tension in the string as a force to keep the bob on its circular path. However, the tension does no work because it is always perpendicular to the motion. If instead you describe the system using the angle \(\theta\), you have reduced the problem from two coordinates to one. The tension disappears from your equations of motion because it is a constraint force, and the only force you need to consider is gravity. This angle \(\theta\) is a generalized coordinate. It is any parameter that uniquely defines the configuration of the system without violating its constraints. The corresponding "force" associated with changing this angle is the generalized force. It is not a physical force in the Newtonian sense, but a measure of how much work the applied forces would do if the coordinate were to change by a small amount. This approach allows us to focus solely on the forces that actually drive the motion.

Defining Generalized Coordinates

Consider a system of \(N\) particles. The position of the \(i\)-th particle is given by the vector \(\mathbf{r}_i\). If the system is subject to \(k\) independent holonomic constraints, the number of degrees of freedom is \(f = 3N - k\). We can describe the entire configuration of the system using \(f\) independent parameters \(q_1, q_2, \dots, q_f\), known as generalized coordinates. These coordinates can be anything: lengths, angles, or even combinations thereof, as long as they are independent and sufficient to specify the position.

The relationship between the Cartesian coordinates and the generalized coordinates is given by:

Here, the time dependence \(t\) is included to account for time-dependent constraints, such as a pendulum pivot that moves. The velocity of the \(i\)-th particle is obtained by differentiating with respect to time:

This expression is crucial because it links the generalized velocities \(\dot{q}_j\) to the physical velocities \(\mathbf{v}_i\). It allows us to express the kinetic energy \(T\), which is a function of \(\mathbf{v}_i\), as a function of \(q_j\) and \(\dot{q}_j\).

Deriving the Generalized Force

The generalized force \(Q_j\) associated with the coordinate \(q_j\) is defined through the principle of virtual work. For a virtual displacement \(\delta \mathbf{r}_i\) consistent with the constraints, the total virtual work done by the applied forces \(\mathbf{F}_i^{(a)}\) is:

Since the virtual displacement is a change in the generalized coordinates at a fixed time, we have:

Substituting this into the work equation and rearranging terms, we get:

By definition, the term in the parentheses is the generalized force \(Q_j\):

Thus, the virtual work can be written compactly as \(\delta W = \sum_{j=1}^{f} Q_j \delta q_j\). This shows that \(Q_j\) is the coefficient of the virtual displacement \(\delta q_j\) in the work expression. It is a scalar quantity that may have units of force, torque, or other physical quantities depending on the nature of \(q_j\).

Connection to D'Alembert's Principle

D'Alembert's principle states that for a system in dynamic equilibrium, the sum of the applied forces and the inertial forces (reversed effective forces) does no virtual work. Mathematically, this is expressed as:

where \(\dot{\mathbf{p}}_i = m_i \mathbf{a}_i\) is the rate of change of momentum. Using the expression for \(\delta \mathbf{r}_i\) in terms of generalized coordinates, we can project this principle onto each coordinate \(q_j\). The term involving the applied forces becomes \(Q_j \delta q_j\). The term involving the inertial forces requires more careful manipulation of the kinetic energy \(T\).

It can be shown that the inertial contribution to the virtual work for the \(j\)-th coordinate is:

Therefore, D'Alembert's principle leads directly to the equation of motion for the \(j\)-th generalized coordinate:

This is a powerful result because it provides the equations of motion without explicitly solving for constraint forces. The left-hand side depends only on the kinetic energy, while the right-hand side is the generalized force. This form is the precursor to the Euler-Lagrange equations, where the potential energy \(V\) is introduced to define the Lagrangian \(L = T - V\).