Rayleigh dissipation function

Key takeaways

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

Reference excerpt

In physics, the Rayleigh dissipation function, named after Lord Rayleigh, is a function used to handle the effects of velocity-proportional frictional forces in Lagrangian mechanics. It was first introduced by him in 1873. If the frictional force on a particle with velocity v → {\displaystyle {\vec {v}}} can be written as F → f = − k v → {\displaystyle {\vec {F}}_{f}=-k{\vec {v}}} , where k {\displaystyle k} is a diagonal matrix, then the Rayleigh dissipation function can be defined for a system of N {\displaystyle N} particles as

R ( v ) = 1 2 ∑ i = 1 N ( k x v i , x 2 + k y v i , y 2 + k z v i , z 2 ) . {\displaystyle R(v)={\frac {1}{2}}\sum _{i=1}^{N}(k_{x}v_{i,x}^{2}+k_{y}v_{i,y}^{2}+k_{z}v_{i,z}^{2}).}

This function represents half of the rate of energy dissipation of the system through friction. The force of friction is negative the velocity gradient of the dissipation function, F → f = − ∇ v R ( v ) {\displaystyle {\vec {F}}_{f}=-\nabla _{v}R(v)} , analogous to a force being equal to the negative position gradient of a potential. This relationship is represented in terms of the set of generalized coordinates q i = { q 1 , q 2 , … q n } {\displaystyle q_{i}=\left\{q_{1},q_{2},\ldots q_{n}\right\}} as

F f , i = − ∂ R ∂ q ˙ i {\displaystyle F_{f,i}=-{\frac {\partial R}{\partial {\dot {q}}_{i}}}} . As friction is not conservative, it is included in the Q i {\displaystyle Q_{i}} term of Lagrange's equations,

d d t ∂ L ∂ q i ˙ − ∂ L ∂ q i = Q i {\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q_{i}}}}}-{\frac {\partial L}{\partial q_{i}}}=Q_{i}} . Applying of the value of the frictional force described by generalized coordinates into the Euler-Lagrange equations gives

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Rayleigh dissipation function

Start with the simplest possible case. Write down what Rayleigh dissipation function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Rayleigh dissipation function 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 Rayleigh dissipation function 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 Rayleigh dissipation function

In research: Rayleigh dissipation function appears in mathematics 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 Rayleigh dissipation function 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: Rayleigh dissipation function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functions and mappings, Lagrangian mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Rayleigh dissipation function 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 Rayleigh dissipation function in 20 minutes

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

Frequently asked questions

What is Rayleigh dissipation function in simple terms?

In physics, the Rayleigh dissipation function, named after Lord Rayleigh, is a function used to handle the effects of velocity-proportional frictional forces in Lagrangian mechanics. It was first introduced by him in 1873.

Why does Rayleigh dissipation function matter?

Because it connects several mathematics 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 Rayleigh dissipation function?

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 Rayleigh dissipation function.