1. Why Generalized Coordinates?

A single particle in 3-D space is usually described by Cartesian variables $(x, y, z)$. Many real systems, however, do not move freely in all three directions. They may be constrained to:

• curved surfaces (e.g., a bead on a wire, motion on a sphere),
• restricted paths or linkages (mechanisms, robot arms),
• rotating frames (aircraft, spacecraft, planetary motion),
• conditions such as rolling without slipping or fixed distances.

In these situations Cartesian coordinates:

• introduce extra variables that are later removed by constraints,
• make the algebra messy and obscure the geometry,
• force constraint forces to appear explicitly in Newton’s equations.

Generalized coordinates avoid these issues by allowing any convenient, independent set of parameters to describe the configuration.

Definition. A generalized coordinate $q_i$ is an independent variable that, together with the other $q_j$, uniquely determines the configuration of the system.

Examples of generalized coordinates:

• Polar coordinates $(r, \theta)$ in a plane,
• Spherical coordinates $(r, \theta, \phi)$ in 3-D,
• Cylindrical coordinates $(d, \phi, z)$,
• Joint angles, link lengths, or slider positions in mechanisms,
• Parameters on a constraint surface (e.g., latitude and longitude).

If a system has $n$ degrees of freedom, we can always describe its configuration using $n$ generalized coordinates.

2. Position, Velocity, and Acceleration in Common Coordinate Systems

Consider a particle $P$ with inertial position vector $\mathbf{r}$ measured in the frame $\{\mathbf{i}_1, \mathbf{i}_2, \mathbf{i}_3\}$.

2.1 Cartesian Coordinates

\[ \mathbf{r} = x\,\mathbf{i}_1 + y\,\mathbf{i}_2 + z\,\mathbf{i}_3 \] \[ \dot{\mathbf{r}} = \dot{x}\,\mathbf{i}_1 + \dot{y}\,\mathbf{i}_2 + \dot{z}\,\mathbf{i}_3 \]

2.2 Spherical Coordinates $(r,\theta,\phi)$

With radial unit vector $\mathbf{e}_r$, \[ \mathbf{r} = r\,\mathbf{e}_r \] Velocity: \[ \dot{\mathbf{r}} = \dot r\,\mathbf{e}_r + r\dot\theta\,\mathbf{e}_\theta + r\dot\phi\cos\theta\,\mathbf{e}_\phi \] Acceleration: \[ \ddot{\mathbf{r}} = (\ddot r - r(\dot\theta^2 + \dot\phi^2\cos^2\theta))\,\mathbf{e}_r + (r\ddot\theta + 2\dot r\dot\theta - r\dot\phi^2\sin\theta\cos\theta)\,\mathbf{e}_\theta \] \[ \quad + (r\ddot\phi\cos\theta + 2\dot r\dot\phi\cos\theta - 2r\dot\theta\dot\phi\sin\theta)\,\mathbf{e}_\phi \]

2.3 Cylindrical Coordinates $(d,\phi,z)$

With radial unit vector $\mathbf{e}_d$ in the plane: \[ \mathbf{r} = d\,\mathbf{e}_d + z\,\mathbf{i}_3 \] Velocity: \[ \dot{\mathbf{r}} = \dot d\,\mathbf{e}_d + d\dot\phi\,\mathbf{e}_\phi + \dot z\,\mathbf{i}_3 \] Acceleration: \[ \ddot{\mathbf{r}} = (\ddot d - d\dot\phi^2)\,\mathbf{e}_d + (d\ddot\phi + 2\dot d\dot\phi)\,\mathbf{e}_\phi + \ddot z\,\mathbf{i}_3 \]

3. Transformations Between Coordinate Systems

Generalized coordinates are linked to Cartesian coordinates by transformation functions. For example:

Spherical $\rightarrow$ Cartesian

\[ x = r\cos\theta\cos\phi,\quad y = r\cos\theta\sin\phi,\quad z = r\sin\theta \]

Cylindrical $\rightarrow$ Cartesian

\[ x = d\cos\phi,\quad y = d\sin\phi,\quad z = z \]

These mappings show that different coordinate sets are simply different “parameterizations” of the same geometry.

4. Kinetic Energy in Different Coordinates

For a particle of mass $m$, the kinetic energy is always \[ T = \frac{1}{2}m \lVert \dot{\mathbf{r}} \rVert^2. \] Using the velocities above, we obtain:

• Cartesian: \[ T = \frac{m}{2}\left( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right) \]
• Spherical: \[ T = \frac{m}{2}\left( \dot r^2 + r^2\dot\theta^2 + r^2\dot\phi^2\cos^2\theta \right) \]
• Cylindrical: \[ T = \frac{m}{2}\left( \dot d^2 + d^2\dot\phi^2 + \dot z^2 \right) \]

The formulas look different because each coordinate system encodes the geometry in a different way, but they describe the same physical motion.

5. Holonomic Constraints & Virtual Displacements

Many systems do not move freely in all directions; instead their motion is restricted to a subset of configuration space defined by constraints.

A holonomic constraint can be written as \[ \psi(q_1, q_2, \ldots, q_n, t) = 0, \] which defines a smooth constraint surface on which the system must remain.

Examples:

• a bead sliding on a circular wire,
• a pendulum with fixed length,
• motion confined to a smooth surface,
• a robot link constrained by a joint limit.

If the constraint does not depend explicitly on time it is scleronomic; if time appears explicitly it is rheonomic.

Virtual Displacements

A virtual displacement $\delta\mathbf{R}_i$ is an infinitesimal change in configuration at a fixed time that respects all constraints. The corresponding virtual work of the applied forces is \[ \delta W = \sum_i \mathbf{F}_i \cdot \delta\mathbf{R}_i. \]

For holonomic constraints, the associated constraint forces are normal to the constraint surface, so they do no virtual work. This fact will allow us to eliminate constraint forces entirely from the Lagrange equations.

6. Newton’s Equations with Constraint Surfaces

For a particle constrained to a holonomic surface $\psi(x,y,z,t)=0$, Newton’s equation can be written as \[ \mathbf{f} + \mathbf{f}_c = m\ddot{\mathbf{R}}, \] where $\mathbf{f}$ is the resultant of all “working” forces and $\mathbf{f}_c$ is the constraint force.

Because the constraint force is normal to the surface, it can be expressed as \[ \mathbf{f}_c = \lambda \nabla \psi, \] where $\lambda(t)$ is a scalar Lagrange multiplier.

Writing components in Cartesian coordinates gives

\[ m\ddot x = f_x + \lambda \frac{\partial \psi}{\partial x},\quad m\ddot y = f_y + \lambda \frac{\partial \psi}{\partial y},\quad m\ddot z = f_z + \lambda \frac{\partial \psi}{\partial z}, \] together with the constraint equation \[ \psi(x,y,z,t) = 0. \]

These four equations determine the four unknown functions $x(t), y(t), z(t)$, and $\lambda(t)$.

7. Multiple Constraints: Sum of Constraint Normals

If a particle is simultaneously constrained by two independent holonomic equations $\psi_1 = 0$ and $\psi_2 = 0$, the total constraint force is the sum of the contributions from each surface: \[ \mathbf{f}_c = \lambda_1 \nabla \psi_1 + \lambda_2 \nabla \psi_2. \]

Newton’s equations become \[ m\ddot x = f_x + \lambda_1 \frac{\partial \psi_1}{\partial x} + \lambda_2 \frac{\partial \psi_2}{\partial x}, \] with similar expressions for $y$ and $z$. The constraints themselves provide two additional algebraic equations. Altogether, the system determines the motion along the curve formed by the intersection of the two constraint surfaces.

8. Nonholonomic (Pfaffian) Constraints

Not all constraints can be expressed purely in terms of the coordinates. Some depend on velocities in a way that cannot be integrated into a position-level relation. These are called nonholonomic constraints.

A common representation is the Pfaffian form: \[ \sum_{j=1}^{n} A_{kj}(q,t)\,\dot q_j + B_k(q,t) = 0, \qquad k = 1,2,\ldots,m, \] or, in differential form, \[ \sum_{j=1}^{n} A_{kj}(q,t)\,dq_j + B_k(q,t)\,dt = 0. \]

Typical examples:

• rolling without slipping (wheeled vehicles, rolling disks),
• mobile robots whose wheels impose direction-dependent motion limits,
• non-slipping aircraft or spacecraft steering rules.

In general, these constraints cannot be integrated to obtain a holonomic constraint of the form $\psi(q,t)=0$.

9. Constrained Newtonian Dynamics of $N$ Particles

For a system of $N$ particles, we often use $n = 3N$ generalized coordinates (for example, the Cartesian coordinates of each mass). Suppose there are $m$ Pfaffian nonholonomic constraints as above. Then the constrained equations of motion may be written in the form \[ M_j \ddot q_j = f_j + c_j + \sum_{k=1}^{m} A_{kj}(q,t)\,\lambda_k, \qquad j = 1,2,\ldots,n, \] with the constraint equations \[ \sum_{j=1}^{n} A_{kj}(q,t)\,\dot q_j + B_k(q,t) = 0, \qquad k = 1,2,\ldots,m. \]

Here $M_j$ denotes the effective mass term, $f_j$ the generalized working forces, $c_j$ the contributions from constraint reactions or Coriolis-like terms, and $\lambda_k$ are Lagrange multipliers associated with each constraint.

Together, these relations form a set of coupled differential–algebraic equations (DAEs) for the motion of the system.

10. Lagrange Multiplier Rule (Optimization Analogy)

The use of multipliers in dynamics is closely related to the classical Lagrange multiplier method in constrained optimization. Consider the problem of extremizing (minimizing or maximizing) a scalar function \[ \phi(x,y,z) \] subject to the equality constraint \[ \psi(x,y,z) = 0. \]

The necessary conditions for a stationary point are that there exists a scalar $\lambda$ such that \[ \nabla \phi + \lambda \nabla \psi = \mathbf{0}, \qquad \psi(x,y,z) = 0. \]

In terms of an augmented function \[ \Phi(x,y,z,\lambda) = \phi(x,y,z) + \lambda \psi(x,y,z), \] the stationary conditions can be written compactly as \[ \frac{\partial \Phi}{\partial x} = \frac{\partial \Phi}{\partial y} = \frac{\partial \Phi}{\partial z} = \frac{\partial \Phi}{\partial \lambda} = 0. \]

This mirrors the way constraint forces appear in mechanics: the constraint reaction is proportional to the gradient $\nabla\psi$, and $\lambda$ plays the role of a generalized “force level” associated with the constraint.

11. Towards d’Alembert’s Principle and Lagrange’s Equations

We now have all the building blocks required for the full analytical-dynamics framework:

• generalized coordinates $q_j$ describing the configuration,
• expressions for positions $\mathbf{R}_i(t,q)$ and velocities $\dot{\mathbf{R}}_i(t,q,\dot q)$,
• holonomic and nonholonomic constraints,
• virtual displacements consistent with those constraints,
• constraint forces that do no virtual work (for holonomic cases),
• Lagrange multipliers representing the intensity of constraint reactions.

These concepts feed directly into d’Alembert’s principle \[ \sum_i \left( \mathbf{F}_i - m_i \ddot{\mathbf{R}}_i \right) \cdot \delta \mathbf{R}_i = 0, \] and from there to the Lagrange equations of motion \[ \frac{d}{dt}\left( \frac{\partial L}{\partial \dot q_j} \right) - \frac{\partial L}{\partial q_j} = Q_j, \] which will be developed in the next sections of Domain B.