Math 2.0

Mechanics of particles and systems Fact

Euler's laws of motion

Builds on Continuum mechanicsStressNewton's laws of motionAngular momentumCross product

Statement

Let a continuum body move in ℝ3\R^3 with mass density ρ\rho, velocity field vv and body force per unit mass bb. Let t(x,n)t(x, n) be its traction field of stress. For every part of the body, let PtP_t be the region it occupies at time tt, with outward unit normal nn on ∂Pt\partial P_t. Euler's laws of motion postulate that, for every part and every time tt:

  1. (balance of linear momentum) ddt∫Ptρv dV=∫∂Ptt(x,n) dA+∫Ptρb dV;\frac{d}{dt} \int_{P_t} \rho v\, dV = \int_{\partial P_t} t(x, n)\, dA + \int_{P_t} \rho b\, dV;
  2. (balance of angular momentum about the origin) ddt∫Ptx×ρv dV=∫∂Ptx×t(x,n) dA+∫Ptx×ρb dV,\frac{d}{dt} \int_{P_t} x \times \rho v\, dV = \int_{\partial P_t} x \times t(x, n)\, dA + \int_{P_t} x \times \rho b\, dV,

where ×\times is the cross product.

Remarks

These are axioms of continuum mechanics, not consequences of other results; they extend Newton's laws of motion from point masses to bodies. The first law says that the rate of change of the total linear momentum of a part equals the total external force on it. The second says the same for angular momentum and torque. For a rigid body, the first law reduces to m x¨c=Fm\, \ddot{x}_c = F, where mm is the mass, xcx_c the centre of mass and FF the resultant external force. In a point-mass theory with central forces the second law follows from the first. For continua it is an independent postulate, and it is what makes the Cauchy stress tensor symmetric.

Going deeper

For a student

Applied to small pieces, the first law yields Cauchy's stress theorem and then the local Cauchy momentum equation. The second law then reduces to the symmetry of the stress tensor. Together with conservation of mass and a constitutive equation, these laws give a closed system of field equations for the motion.

Reading path

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Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler%27s_laws_of_motion
About this page
TagSBCV6VZG6D
URIm2://SBCV6VZG6D
Kindstatement, fact
WritingW0 stub
MSC70A05, 74A99
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AboutContinuum mechanics
UsesContinuum mechanics, Stress, Newton's laws of motion, Angular momentum, Cross product
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Changed3 November 2026

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