Generalized coordinates

Key takeaways

Generalized coordinates belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Generalized coordinates to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Generalized coordinates from memory before moving on to harder problems.

Reference excerpt

In analytical mechanics, generalized coordinates are a set of parameters used to represent the configuration of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference configuration. The generalized velocities are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes these parameters from the traditional use of the term "coordinate" to refer to Cartesian coordinates. An example of a generalized coordinate would be to describe the position of a pendulum using the angle of the pendulum relative to vertical, rather than by the x and y position of the pendulum. Although there may be many possible choices for generalized coordinates for a physical system, they are generally selected to simplify calculations, such as the solution of the equations of motion for the system. If the coordinates are independent of one another, the number of independent generalized coordinates is defined by the number of degrees of freedom of the system. Generalized coordinates are paired with generalized momenta to provide canonical coordinates on phase space.

Constraints and degrees of freedom

Generalized coordinates are usually selected to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of Lagrange's equations of motion. However, it can also occur that a useful set of generalized coordinates may be dependent, which means that they are related by one or more constraint equations.

Holonomic constraints

For a system of N particles in 3D real coordinate space, the position vector of each particle can be written as a 3-tuple in Cartesian coordinates:

r 1 = ( x 1 , y 1 , z 1 ) , r 2 = ( x 2 , y 2 , z 2 ) , ⋮ r N = ( x N , y N , z N ) {\displaystyle {\begin{aligned}&\mathbf {r} _{1}=(x_{1},y_{1},z_{1}),\\&\mathbf {r} _{2}=(x_{2},y_{2},z_{2}),\\&\qquad \qquad \vdots \\&\mathbf {r} _{N}=(x_{N},y_{N},z_{N})\end{aligned}}}

Any of the position vectors can be denoted rk where k = 1, 2, …, N labels the particles. A holonomic constraint is a constraint equation of the form for particle k

f ( r k , t ) = 0 {\displaystyle f(\mathbf {r} _{k},t)=0}

which connects all the 3 spatial coordinates of that particle together, so they are not independent. The constraint may change with time, so time t will appear explicitly in the constraint equations. At any instant of time, any one coordinate will be determined from the other coordinates, e.g. if xk and zk are given, then so is yk. One constraint equation counts as one constraint. If there are C constraints, each has an equation, so there will be C constraint equations. There is not necessarily one constraint equation for each particle, and if there are no constraints on the system then there are no constraint equations. So far, the configuration of the system is defined by 3N quantities, but C coordinates can be eliminated, one coordinate from each constraint equation. The number of independent coordinates is n = 3N − C. (In D dimensions, the original configuration would need ND coordinates, and the reduction by constraints means n = ND − C). It is ideal to use the minimum number of coordinates needed to define the configuration of the entire system, while taking advantage of the constraints on the system. These quantities are known as generalized coordinates in this context, denoted qj(t). It is convenient to collect them into an n-tuple

q ( t ) = ( q 1 ( t ) , q 2 ( t ) , … , q n ( t ) ) {\displaystyle \mathbf {q} (t)=(q_{1}(t),\ q_{2}(t),\ \ldots ,\ q_{n}(t))}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Generalized coordinates

Start with the simplest possible case. Write down what Generalized coordinates claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Generalized coordinates before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Generalized coordinates ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Generalized coordinates

In research: Generalized coordinates appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Generalized coordinates in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Generalized coordinates is common in secondary-school and first-year university syllabi. It links to neighbouring topics Dynamical systems, Lagrangian mechanics, Mechanical quantities, so understanding it makes those chapters shorter.
In everyday life: Look for Generalized coordinates outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Generalized coordinates in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Generalized coordinates means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Generalized coordinates out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Generalized coordinates in simple terms?

In analytical mechanics, generalized coordinates are a set of parameters used to represent the configuration of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference configuration.

Why does Generalized coordinates matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Generalized coordinates?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Generalized coordinates.