Why Reformulate Mechanics? Constraints and Coordinates
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You should be comfortable with Newton’s laws of motion, basic vector calculus, and the concept of work and energy. Familiarity with Cartesian coordinates and simple trigonometry is assumed.
Why this matters
Newtonian mechanics is powerful but often cumbersome for complex systems.
When dealing with systems subject to constraints, such as a pendulum or a bead on a wire, Newton’s second law requires solving for unknown constraint forces that do no work. This lecture introduces the motivation for analytical mechanics: a framework that naturally handles constraints by choosing generalized coordinates, eliminating the need to calculate constraint forces explicitly, and paving the way for the Lagrangian and Hamiltonian formulations.
The Problem with Forces
Consider a simple pendulum. In Cartesian coordinates, the bob is constrained to move on a circle of radius \( L \). To apply Newton’s second law, we must write equations for both the \( x \) and \( y \) components of acceleration. However, the string exerts a tension force \( T \) that is always perpendicular to the motion. We do not know \( T \) a priori; we must solve for it simultaneously with the position. This introduces an extra unknown variable that is physically irrelevant to the trajectory of the bob.
Furthermore, if we have a system of many particles connected by rigid rods or sliding on surfaces, the number of constraint forces can become overwhelming. The equations of motion become a tangled web of coupled differential equations involving forces we cannot easily measure or predict. This suggests that the choice of coordinates is not merely a convenience but a fundamental aspect of the formulation. We need a method that respects the geometry of the allowed motion from the start.
Generalized Coordinates
Instead of using Cartesian coordinates \( (x, y, z) \) for each particle, we can describe the system using a set of independent variables \( q_1, q_2, \dots, q_N \) that fully specify the configuration. These are called generalized coordinates. For the pendulum, a single angle \( \theta \) is sufficient. The constraint \( x^2 + y^2 = L^2 \) is automatically satisfied by the definition \( x = L \sin \theta \) and \( y = -L \cos \theta \) (assuming the pivot is at the origin and the bob hangs down).
The key advantage is that the number of generalized coordinates \( N \) equals the number of degrees of freedom of the system. For a system of \( n \) particles in 3D space with \( k \) independent holonomic constraints, the number of degrees of freedom is \( 3n - k \). By choosing coordinates that respect the constraints, we reduce the dimensionality of the problem. The kinetic energy \( T \) and potential energy \( V \) can be expressed as functions of these generalized coordinates and their time derivatives \( \dot{q}_i \).
This scalar representation of energy is the foundation of the Lagrangian formulation. We no longer need to resolve forces into components; we only need to know how the energy changes as the system moves through its configuration space.
Virtual Displacements and Constraints
To understand why this works, we must distinguish between real and virtual displacements. A real displacement is the actual change in position over a finite time interval \( dt \). A virtual displacement, denoted \( \delta \mathbf{r} \), is an infinitesimal change in position consistent with the constraints at a fixed instant of time. It is a mathematical tool, not a physical motion.
For holonomic constraints, which can be expressed as equations \( f(\mathbf{r}_1, \dots, \mathbf{r}_n, t) = 0 \), the virtual displacements are tangent to the constraint surface. The work done by constraint forces during a virtual displacement is zero for ideal constraints. This is the principle of virtual work: \( \sum \mathbf{F}_{\text{constraint}} \cdot \delta \mathbf{r} = 0 \). This property allows us to ignore constraint forces when deriving the equations of motion, as they do no virtual work.
Non-holonomic constraints, such as a disc rolling without slipping on a plane, cannot be expressed as simple equations of position. They involve velocities and are more complex. Analytical mechanics primarily focuses on holonomic systems, where the generalized coordinate approach is most effective.
From Newton to Lagrange
Newton’s laws are vector equations: \( \mathbf{F} = m \mathbf{a} \). They are based on forces. The Lagrangian approach is scalar: it is based on energy. The Lagrangian is defined as \( L = T - V \). The equations of motion are derived from the principle of stationary action, which states that the actual path taken by the system between two points in time is the one for which the action \( S = \int L \, dt \) is stationary.
This shift from forces to energy is profound. It means that the dynamics of a system are determined by the geometry of its configuration space and the energy landscape, rather than by the specific forces acting on it. The equations keep the same form in any choice of generalized coordinates, a property that is crucial for understanding symmetries and conservation laws. It also provides a natural bridge to quantum mechanics, where the action plays a central role in the path integral formulation.
Generalized coordinates are not unique. You can choose any set of independent variables that describe the configuration. However, the choice affects the form of the kinetic energy and the complexity of the equations of motion. A poor choice can lead to singularities or unnecessarily complicated expressions. The goal is to choose coordinates that exploit the symmetries of the system, simplifying the analysis. C
Worked example
Problem. A bead of mass \( m \) slides without friction on a vertical circular hoop of radius \( R \). The hoop is fixed in space. Determine the generalized coordinate and express the kinetic and potential energies in terms of this coordinate.
- Choose the generalized coordinate. Since the bead is constrained to the hoop, one angle \( \theta \) is sufficient. Let \( \theta \) be the angle from the vertical downward direction.
- Express the position in Cartesian coordinates: \( x = R \sin \theta \), \( y = -R \cos \theta \) (taking the center of the hoop as the origin and \( y \) upward).
- Calculate the velocity components: \( \dot{x} = R \cos \theta \, \dot{\theta} \), \( \dot{y} = R \sin \theta \, \dot{\theta} \).
- Compute the kinetic energy: \( T = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) = \frac{1}{2} m R^2 \dot{\theta}^2 (\cos^2 \theta + \sin^2 \theta) = \frac{1}{2} m R^2 \dot{\theta}^2 \).
- Compute the potential energy: \( V = m g y = -m g R \cos \theta \). (Note: \( V=0 \) at the center, so at the bottom \( y=-R \), \( V=-mgR \).)
Units check. \( T \): \( \mathrm{kg} \cdot (\mathrm{m}^2) \cdot (\mathrm{rad/s})^2 = \mathrm{kg} \cdot \mathrm{m}^2/\mathrm{s}^2 = \mathrm{J} \). \( V \): \( \mathrm{kg} \cdot (\mathrm{m/s^2}) \cdot \mathrm{m} = \mathrm{J} \). Both are in Joules, as required.
Three quick tests
- Why do we prefer generalized coordinates over Cartesian coordinates for constrained systems?
Answer
Generalized coordinates automatically satisfy the constraints, reducing the number of variables and eliminating the need to solve for unknown constraint forces. - What is the difference between a real displacement and a virtual displacement?
Answer
A real displacement occurs over a finite time interval and follows the actual trajectory. A virtual displacement is an infinitesimal change at a fixed instant, consistent with the constraints, used for mathematical analysis. - For a double pendulum, how many degrees of freedom does the system have?
Answer
Two. Each pendulum bob can be described by one angle, so two generalized coordinates are needed.
Problem set for PU-201 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.