Leonhard Euler
by Jong Rak Koh
Leonhard Euler was a Swiss Mathematician and physicist who made important contributions to math and physics. Euler is often considered one of the greatest mathematicians to have ever lived.
Early Life
Euler was born in Basel, Switzerland on April 15th, 1707. His father was a minister and the family naturally expected Euler to also go in to ministry. However, his father sparked a curiosity in math for Euler and he entered University of Basel at the age of 14, with Johann Bernoulli as his mentor.
Contributions to Physics
Arguably the greatest mathematician in history, Euler made lots of contribution to math such as the number “[math]\displaystyle{ e }[/math]”, the concept of a function, summation notation “Σ”, imaginary unit notation “[math]\displaystyle{ i }[/math]”, and popularizing [math]\displaystyle{ \pi }[/math]; however, Euler also made lots of important contributions to physics.
Euler-Bernoulli beam equation
The theory validates the beam deflection calculation for laterally loaded beams. The equation provides a relationship between the deflection of the beam and the applied load intensity.
- [math]\displaystyle{ \frac{\mathrm{d}^2}{\mathrm{d} x^2}\left(EI \frac{\mathrm{d}^2 w}{\mathrm{d} x^2}\right) = q\, }[/math]
where
- [math]\displaystyle{ w }[/math] = Out-of-plane displacement of the beam
- [math]\displaystyle{ E }[/math] = modulus of elasticity (Young's Modulus),
- [math]\displaystyle{ I }[/math] = area moment of inertia, and
- [math]\displaystyle{ q }[/math] = distributed load (force per unit length)
Work in Astronomy
Understanding the nature of comets
In addition to his work with classical mechanics, Euler was recognized by Paris Academy Prizes over the course of his career for calculating, with great accuracy, the orbits of comets and other celestial bodies.
Calculating the parallax of the sun
Euler calculated the parallax of the sun, calculating the difference in the apparent position of the object and the actual position of the object. Euler’s calculation of the parallax later led to the development of more accurate longitude tables.
Work in optics
While Newton argued that light was made of particles, Euler argued that light behaved more like waves. In Nova theoria lucis et colorum (1746), Euler argued that diffractions can be more easily argued with the wave theory rather than the previous “pulse theory”. Euler’s wave theory remained the dominant theory about light until the quantum theory of light.
Structural Engineering
Euler also published a formula for calculating the force where the strut would fail that is often used in structural engineering.
- [math]\displaystyle{ F=\frac{\pi^2 EI}{(KL)^2} }[/math]
where
- [math]\displaystyle{ F }[/math] = maximum or critical force (vertical load on column),
- [math]\displaystyle{ E }[/math] = modulus of elasticity (Young's Modulus),
- [math]\displaystyle{ I }[/math] = area moment of inertia,
- [math]\displaystyle{ L }[/math] = unsupported length of column,
- [math]\displaystyle{ K }[/math] = column effective length factor, whose value depends on the conditions of end support of the column, as follows.
- For both ends pinned (hinged, free to rotate), [math]\displaystyle{ K = 1.0 }[/math].
- For both ends fixed, [math]\displaystyle{ K = 0.50 }[/math].
- For one end fixed and the other end pinned, [math]\displaystyle{ K = 0.699\ldots }[/math].
- For one end fixed and the other end free to move laterally, [math]\displaystyle{ K = 2.0 }[/math].
- [math]\displaystyle{ K L }[/math] is the effective length of the column.
Fluid Dynamics
Euler in 1757 published a set of equations for flow of an ideal fluid with no viscosity (Inviscid flow) that are now known as the Euler Equations.
- [math]\displaystyle{ \rho\left( \frac{\partial}{\partial t}+{ \mathbf {u}}\cdot\nabla \right){ \mathbf {u}}+\nabla p=0 }[/math]
where
- [math]\displaystyle{ u }[/math] = fluid velocity
- [math]\displaystyle{ p }[/math] = pressure, and
- [math]\displaystyle{ \rho\ }[/math] = fluid density
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Further reading
http://eulerarchive.maa.org/ http://micro.magnet.fsu.edu/optics/timeline/people/euler.html https://muse.jhu.edu/journals/perspectives_on_science/v016/16.4.pedersen.html http://arxiv.org/ftp/arxiv/papers/1406/1406.7397.pdf http://blog.mechguru.com/machine-design/how-to-apply-the-euler-bernoulli-beam-theory-for-beam-deflection-calculation/ http://www.colorado.edu/engineering/CAS/courses.d/AVMM.d/AVMM.Ch08.d/AVMM.Ch08.pdf
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