mathematics

Newton–Euler equations

Newton–Euler equations is a mathematics 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 Newton–Euler equations rather than just read about it. In short: In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body. Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices.

Key takeaways

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

Reference excerpt

In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.

Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments) acting on the rigid body.

Center of mass frame With respect to a coordinate frame whose origin coincides with the body's center of mass for τ(torque) and an inertial frame of reference for F(force), they can be expressed in matrix form as:

( F τ ) = ( m I 3 0 0 I c m ) ( a c m α ) + ( 0 ω × ( I c m ω ) ) , {\displaystyle \left({\begin{matrix}{\mathbf {F} }\\{\boldsymbol {\tau }}\end{matrix}}\right)=\left({\begin{matrix}m{\mathbf {I} _{3}}&0\\0&{\mathbf {I} }_{\rm {cm}}\end{matrix}}\right)\left({\begin{matrix}\mathbf {a} _{\rm {cm}}\\{\boldsymbol {\alpha }}\end{matrix}}\right)+\left({\begin{matrix}0\\{\boldsymbol {\omega }}\times \left({\mathbf {I} }_{\rm {cm}}\,{\boldsymbol {\omega }}\right)\end{matrix}}\right),}

where

F = total force acting on the center of mass m = mass of the body I3 = the 3×3 identity matrix acm = acceleration of the center of mass vcm = velocity of the center of mass τ = total torque acting about the center of mass Icm = moment of inertia about the center of mass ω = angular velocity of the body α = angular acceleration of the body

Arbitrary reference frame With respect to a coordinate frame located at point P that is fixed in the body and not coincident with the center of mass, the equations assume the more complex form:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Newton–Euler equations

Start with the simplest possible case. Write down what Newton–Euler equations 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 Newton–Euler equations 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 Newton–Euler equations 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 Newton–Euler equations

In research: Newton–Euler equations 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 Newton–Euler equations 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: Newton–Euler equations is common in secondary-school and first-year university syllabi. It links to neighbouring topics Equations, Rigid bodies, so understanding it makes those chapters shorter.
In everyday life: Look for Newton–Euler equations 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 Newton–Euler equations in 20 minutes

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

Frequently asked questions

What is Newton–Euler equations in simple terms?

In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body. Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vec…

Why does Newton–Euler equations 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 Newton–Euler equations?

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 Newton–Euler equations.