physics

Primary constraint

Primary constraint is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Primary constraint rather than just read about it. In short: In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations of motion are satisfied, but need not hold if they are not satisfied The secondary constraints arise from the condition that the primary constraints should be preserved in time.

Key takeaways

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

Reference excerpt

In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations of motion are satisfied, but need not hold if they are not satisfied The secondary constraints arise from the condition that the primary constraints should be preserved in time. A few authors use more refined terminology, where the non-primary constraints are divided into secondary, tertiary, quaternary, etc. constraints. The secondary constraints arise directly from the condition that the primary constraints are preserved by time, the tertiary constraints arise from the condition that the secondary ones are also preserved by time, and so on. Primary and secondary constraints were introduced by Anderson and Bergmann and developed by Dirac. The terminology of primary and secondary constraints is confusingly similar to that of first- and second-class constraints. These divisions are independent: both first- and second-class constraints can be either primary or secondary, so this gives altogether four different classes of constraints.

Footnotes

References Anderson, James L.; Bergmann, Peter G. (1951). "Constraints in covariant field theories". Physical Review. Series 2. 83 (5): 1018–1025. Bibcode:1951PhRv...83.1018A. doi:10.1103/PhysRev.83.1018. MR 0044382. Dirac, Paul A. M. (1964). Lectures on quantum mechanics. Belfer Graduate School of Science Monographs Series. Vol. 2. New York: Belfer Graduate School of Science. ISBN 9780486417134. MR 2220894. {{cite book}}: ISBN / Date incompatibility (help) 2001 reprint by Dover.

Further reading Salisbury, D. C. (2006). Peter Bergmann and the invention of constrained Hamiltonian dynamics. arXiv:physics/0608067. Bibcode:2006physics...8067S.

Worked examples

Example 1 — a first encounter with Primary constraint

Start with the simplest possible case. Write down what Primary constraint 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 Primary constraint 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 Primary constraint 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 Primary constraint

In research: Primary constraint 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 Primary constraint 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: Primary constraint is common in secondary-school and first-year university syllabi. It links to neighbouring topics Hamiltonian mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Primary constraint 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.

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How to study Primary constraint in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Primary constraint 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 Primary constraint out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Primary constraint in simple terms?

Why does Primary constraint 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 Primary constraint?

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 Primary constraint.