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In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation, is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved ...
If the eigenvalues (the case of Euler equations) are all real the system is defined hyperbolic, and physically eigenvalues represent the speeds of propagation of information. [12] If they are all distinguished, the system is defined strictly hyperbolic (it will be proved to be the case of one-dimensional Euler equations). Furthermore ...
Since the parameter is usually time, Cauchy conditions can also be called initial value conditions or initial value data or simply Cauchy data. An example of such a situation is Newton's laws of motion, where the acceleration ″ depends on position , velocity ′, and the time ; here, Cauchy data corresponds to knowing the initial position and ...
In mathematics a Cauchy–Euler operator is a ... It is named after Augustin-Louis Cauchy and Leonhard Euler. The simplest example is ... Cauchy–Euler equation ...
Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to ...
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. They are named in honour of Leonhard Euler. Their general vector form is
These flows correspond closely to real-life flows over the whole of fluid mechanics; in addition, many valuable insights arise when considering the deviation (often slight) between an observed flow and the corresponding potential flow. Potential flow finds many applications in fields such as aircraft design.
In the above equations (,) is the mass density (current), ˙ is the material time derivative of , (,) is the particle velocity, ˙ is the material time derivative of , (,) is the Cauchy stress tensor, (,) is the body force density, (,) is the internal energy per unit mass, ˙ is the material time derivative of , (,) is the heat flux vector, and ...