8 Introduction to Celebrities in Fluid Mechanics
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8.8 Euler (1707–1783)
Leonhard Euler (1707–1783) was a Swiss mathematician and mechanic who
wrote the most in the eighteenth century. His works involved all fields of
mathematics at that time and many mathematical terms were named after
Euler, such as Euler integral, Euler number, various Euler formulas, etc. He
and his successor Lagrange completed the transition from comprehensive
method to analytical method in mathematics, but they are quite different in
style. Euler is famous for his concreteness and delicacy, while Lagrange is good
at abstraction and generalization. Euler applied mathematical methods to
mechanics, and made outstanding contributions in all fields of mechanics. He
was the founder of rigid body dynamics and hydrodynamics, and the founder
of stability theory of elastic system. In the two volumes of Analysis and Explanation of Mechanics or Motion Science published in 1736, he considered the
differential equations of motion of free particles and constrained particles.
In terms of mechanical principles, in the study of rigid body kinematics and
rigid body dynamics, he obtained the most basic results, among which the
finite motion of rigid body fixed point is equivalent to the rotation around
a certain axis of fixed point; the movement of rigid body fixed point can be
described by the change of three angles (called Euler angle); the relationship
between the change of angular velocity and the external moment when the
rigid body fixed point rotates; the motion law of the fixed-point rigid body
without external force moment and the differential equation of free rigid
body. Euler believed that the differential equations of particle dynamics could
be applied to liquids (1750). He used two methods to describe the motion
of fluid, that is, to describe the velocity field of fluid according to the fixed
point of space (1755) and the particle of fluid (1759), respectively. These
two methods are usually called Eulerian representation and Lagrangian representation. Euler laid the theoretical foundation for the motion of ideal fluid
(assuming that the fluid is incompressible and its viscosity can be ignored).
The continuity equation reflecting the conservation of mass (1752) and the
hydrodynamic equation reflecting the law of momentum change (1755) were
given. Euler studied the vibration of elastic systems such as strings and bars.
Together with Daniel Bernoulli, he analyzed the vibrations of the heavy
chains suspended at the top and the corresponding discrete models (lines with
a string of masses). With the help of Daniel Bernoulli, he got the exact solution of the deflection curve of the thin rod under elastic compression. The
minimum pressure which can make the thin rod produce this kind of deflection is called the Euler critical load. Euler had also studied applied mechanics
589
8.8 Euler (1707–1783)
Leonhard Euler (1707–1783) was a Swiss mathematician and mechanic who
wrote the most in the eighteenth century. His works involved all fields of
mathematics at that time and many mathematical terms were named after
Euler, such as Euler integral, Euler number, various Euler formulas, etc. He
and his successor Lagrange completed the transition from comprehensive
method to analytical method in mathematics, but they are quite different in
style. Euler is famous for his concreteness and delicacy, while Lagrange is good
at abstraction and generalization. Euler applied mathematical methods to
mechanics, and made outstanding contributions in all fields of mechanics. He
was the founder of rigid body dynamics and hydrodynamics, and the founder
of stability theory of elastic system. In the two volumes of Analysis and Explanation of Mechanics or Motion Science published in 1736, he considered the
differential equations of motion of free particles and constrained particles.
In terms of mechanical principles, in the study of rigid body kinematics and
rigid body dynamics, he obtained the most basic results, among which the
finite motion of rigid body fixed point is equivalent to the rotation around
a certain axis of fixed point; the movement of rigid body fixed point can be
described by the change of three angles (called Euler angle); the relationship
between the change of angular velocity and the external moment when the
rigid body fixed point rotates; the motion law of the fixed-point rigid body
without external force moment and the differential equation of free rigid
body. Euler believed that the differential equations of particle dynamics could
be applied to liquids (1750). He used two methods to describe the motion
of fluid, that is, to describe the velocity field of fluid according to the fixed
point of space (1755) and the particle of fluid (1759), respectively. These
two methods are usually called Eulerian representation and Lagrangian representation. Euler laid the theoretical foundation for the motion of ideal fluid
(assuming that the fluid is incompressible and its viscosity can be ignored).
The continuity equation reflecting the conservation of mass (1752) and the
hydrodynamic equation reflecting the law of momentum change (1755) were
given. Euler studied the vibration of elastic systems such as strings and bars.
Together with Daniel Bernoulli, he analyzed the vibrations of the heavy
chains suspended at the top and the corresponding discrete models (lines with
a string of masses). With the help of Daniel Bernoulli, he got the exact solution of the deflection curve of the thin rod under elastic compression. The
minimum pressure which can make the thin rod produce this kind of deflection is called the Euler critical load. Euler had also studied applied mechanics