Generalised Coordinates and Constraints

A system with $n$ degrees of freedom can be described by $n$ generalised coordinates $q_1, q_2, \ldots, q_n$Which may be angles, arc lengths, or any other set of parameters that Uniquely determines the configuration.

The Cartesian coordinates are functions of the generalised coordinates (and possibly time):

$\mathbf{r}_i = \mathbf{r}_i(q_1, q_2, \ldots, q_n, t), \quad i = 1, \ldots, N$

The velocities are:

$\dot{\mathbf{r}}_i = \sum_{j=1}^n \frac{\partial \mathbf{r}_i}{\partial q_j}\dot{q}_j + \frac{\partial \mathbf{r}_i}{\partial t}$

Example. A simple pendulum has one degree of freedom. We can use the angle $\theta$ from the Vertical as the generalised coordinate, rather than the Cartesian coordinates $(x, y)$ of the bob.

Holonomic constraints relate the coordinates by equations:

$f(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N, t) = 0$

A holonomic constraint reduces the number of degrees of freedom.

Non-holonomic constraints involve inequalities or non-integrable differential relations:

$\sum_j a_j(q, t)\, dq_j + a_t(q, t)\, dt = 0$

Which cannot be integrated to yield a relation among the $q_j$ alone.

Scleronomic constraints do not depend explicitly on time. Rheonomic constraints do.

Example. A bead sliding on a fixed wire: the constraint $f(x,y) = 0$ is holonomic and scleronomic. A bead on a wire that moves with time: holonomic and rheonomic.

Problem. Classify the following constraints: (a) a particle on the surface of a sphere of radius $R$(b) a rolling disk (vertical), (c) a particle constrained to $z \geq 0$(d) a pendulum whose pivot oscillates as $x_0(t) = A\cos(\omega t)$.

Problem. A rod of length $l$ and negligible mass has masses $m_1$ and $m_2$ at its ends. The rod slides on a frictionless horizontal table. Find suitable generalised coordinates.

A virtual displacement $\delta \mathbf{r}_i$ is an infinitesimal change in position consistent with The constraints at a fixed instant in time ($\delta t = 0$).

Definition (Virtual Work). The virtual work of the forces is:

$\delta W = \sum_{i=1}^N \mathbf{F}_i \cdot \delta \mathbf{r}_i$

Definition (Ideal Constraints). Constraints are ideal if the virtual work of the constraint forces is zero:

$\sum_{i=1}^N \mathbf{C}_i \cdot \delta \mathbf{r}_i = 0$

Where $\mathbf{C}_i$ is the constraint force on particle $i$.

Theorem 2.1 (Principle of Virtual Work). A system is in static equilibrium if and only if the virtual work of the applied forces vanishes for all virtual displacements consistent with the constraints.

Proof. In static equilibrium, $\mathbf{F}_i + \mathbf{C}_i = \mathbf{0}$ for each particle. Therefore:

$\sum_i (\mathbf{F}_i + \mathbf{C}_i) \cdot \delta\mathbf{r}_i = 0$

For ideal constraints, $\sum_i \mathbf{C}_i \cdot \delta\mathbf{r}_i = 0$So $\sum_i \mathbf{F}_i \cdot \delta\mathbf{r}_i = 0$. Conversely, if the virtual work of applied forces vanishes for all admissible virtual displacements, the system must be in equilibrium (otherwise one could choose a virtual displacement in the direction of net force to get non-zero work). $\blacksquare$

Theorem 2.2 (D’Alembert’s Principle). For a system of $N$ particles:

$\sum_{i=1}^N (\mathbf{F}_i - m_i \ddot{\mathbf{r}}_i) \cdot \delta \mathbf{r}_i = 0$

Where $\mathbf{F}_i$ includes both applied and constraint forces. For ideal constraints, the Constraint forces do no virtual work, so only the applied forces contribute.

Proof. D’Alembert’s principle extends the principle of virtual work to dynamics by treating $-m_i \ddot{\mathbf{r}}_i$ as a “fictitious force” (the inertia force). Starting from Newton’s second law $\mathbf{F}_i + \mathbf{C}_i = m_i \ddot{\mathbf{r}}_i$:

$\sum_i (\mathbf{F}_i + \mathbf{C}_i - m_i\ddot{\mathbf{r}}_i) \cdot \delta\mathbf{r}_i = 0$

This is true. For ideal constraints $\sum_i \mathbf{C}_i \cdot \delta\mathbf{r}_i = 0$Giving:

$\sum_i (\mathbf{F}_i - m_i\ddot{\mathbf{r}}_i) \cdot \delta\mathbf{r}_i = 0$

$\blacksquare$

flowchart TD
    A[2_Generalised Coordinates And Constraints] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Generalised coordinates are the minimum number of independent parameters needed to describe a system’s configuration. A pendulum needs only one angle, not two Cartesian coordinates. Constraints are the rules that reduce the degrees of freedom, like a bead confined to a wire. Holonomic constraints can be written as equations, making them mathematically tractable. D’Alembert’s principle extends virtual work to dynamics by treating inertia as a fictitious force. This insight leads directly to Lagrangian mechanics, where constraint forces disappear entirely because they do no virtual work.

Mistake 1: Confusing virtual displacements with actual displacements A virtual displacement $\delta\mathbf{r}_i$ is an infinitesimal change consistent with the constraints at a fixed instant in time ($\delta t = 0$). An actual displacement occurs over a finite time interval and may violate constraints that change with time. For rheonomic constraints, virtual and actual displacements differ because the constraint surface itself moves.

Mistake 2: Assuming all constraints reduce the degrees of freedom by one Non-holonomic constraints (inequalities or non-integrable differential relations) do not reduce the number of degrees of freedom in the same way holonomic constraints do. A holonomic constraint $f(q_1, \ldots, q_n) = 0$ reduces the degrees of freedom by one, but a non-holonomic constraint like $z \geq 0$ restricts the accessible region without reducing the dimensionality of the configuration space.

Mistake 3: Applying D’Alembert’s principle without checking that constraints are ideal D’Alembert’s principle requires that constraint forces do no virtual work. This holds for frictionless surfaces, rigid rods, and inextensible strings, but not for sliding friction or deformable boundaries. If the constraints are not ideal, the constraint forces contribute to the virtual work and must be included explicitly.

• Lagrangian Mechanics: The Lagrangian formulation uses generalised coordinates to eliminate constraint forces.

• Hamiltonian Mechanics: The Hamiltonian formalism extends Lagrangian mechanics to phase space representations.

• Calculus

• Linear Algebra

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.