# What is Euler equations: Definition and 34 Discussions

In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula.
Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler.

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1. ### Linearization Euler's equation

I'm trying to linearize (first order) the Euler's equation for a small perturbation ##\delta## Starting with ##mna (\frac{\partial}{\partial t} + \frac{\vec{v}}{a} \cdot \nabla ) \vec{u} = - \nabla P - mn \nabla \phi## (1) ##\vec{u} = aH\vec{x(t)} + \vec{v(x,t)}## Where a is the scale factor...
2. ### I Static sphere with gravitating fluid

Hi The rotating bucket problem with a fluid is well known as a homework. For the fun i wanted to adapt it to the case of a massive non-rotating sphere surrounded by a fluid. However i don't know if the calculations i made are correct or don't make sense at all (even if the result lead to an...
3. ### Loss of Hölder continuity by solutions of the Euler equations

Our thesis can be restated as follows: \exists C_\alpha\in\mathbb R_+ s.t. \forall w\in\mathbb R^2_+ \begin{align*} \lVert u(t,w)-u(t,w')\rVert^2 \leq C_\alpha^2\lVert w-w'\rVert^{2\alpha} \end{align*} where w=(y,z) and \alpha=\beta\gamma. We get an upper bound for each (squared) component of...
4. ### A Relativistic Hydrodynamics Eqn for Perfect Fluid: Insight Needed

Hello. Could anyone help me with some insight in an extra term appearing in the motion equations of a relativistic fluid? I say extra term, because it's not present on the motion for a test particle, as it follows: Let's propose Minkowski space-time, the motion equations for a fluid with zero...
5. ### When to consider body fixed reference frame

Hello I am studying mechanics and I have been reading about having the reference frame fixed at a certain point, body fixed and also the gyro equations. I an identify the gyro case easily as I am looking for an AAC body which rotates about an axis. I am confused about the other two cases in...
6. ### I Solving Euler's Dynamics Equations for a Gyro-Compass

Hello, I know it might sound silly but sometimes I get confused. Let's say I have a gyro-compass and I get 3 equations of torque for the 3 axes. I am expected to find the equation of motion and two of them are equated with 0. These are the Euler's dynamics equation with moving reference frame...
7. ### Reaction forces on a gyroscope

I know that ##\vec{v_c}=(\omega_1;-\omega_2;0)×(L;-r;0)=0## So ##\omega_2=\frac{r\omega_1}{L}## Then, using the system of coordinates shown in the picture and ##\Sigma M_z## I can find the reaction force in ##C##. But how can I find the reaction forces on ##A## and ##O##? I mean, what system...
8. ### Euler's equation of thermodynamics in free expansion (Joule expansion)

Hi everyone, I am confused when I apply Euler's equation on the free expansion of an ideal gas. Consider a free expansion (expansion of gas in vaccum) where the volume is doubled (V->2V) The classical free expansion of an ideal gas results in increase in entropy by an amount of nR ln(2), a...
9. ### Introductory application of the Newton Euler equations to a composite body

α is the second derivative of angle and w is the first derivative In the free body diagrams the only force on A is the normal force since it is only constrained not to move vertically. Have I drawn the free body diagram and kinetic diagram correctly? By relating the accelerations of the...
10. ### Euler Equations for Dynamics of rigid body

I have been studying the dynamics of free top from Morin's book. In his book when describing the dynamics,he writes down the equation of motion as shown in screenshot. However,I am not able to understand which term refers to which coordinate system. For eg: Here ##\omega## refers to angular...
11. ### MHB Euler equations having double roots as a solution

If the Euler equations have double roots as it's solution, second solution will be $y_2(x)=x^r\ln{x}$. what is its proof? or how it can be derived?
12. ### Sinusoids as Phasors, Complex Exp, I&Q and Polar form

Hi, I am going around in circles, excuse the pun, with phasors, complex exponentials, I&Q and polar form... 1. A cos (ωt+Φ) = Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt) Right hand side is polar form ... left hand side is in cartesian (rectangular) form via a trignometric identity? 2. But then...
13. ### I Deduce Geodesics equation from Euler equations

I am using from the following Euler equations : $$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$ with function ##f## is equal to : $$f=g_{ij}\dfrac{\text{d}u^{i}}{\text{d}s}\dfrac{\text{d}u^{j}}{\text{d}s}$$ and we have...
14. ### A Why does the Dzhanibekov effect occur in rotating tennis rackets?

Hello all, I can understand the mathematics of this phenomena First, one can solve the Euler equations of motion numerically, using Runge-Kutta and plot the motion. Also, the path of the angular velocity vector will like on the kinetic energy ellipsoid and the angular momentum vector...
15. ### How to convert Euler Equations to Lagrangian Form?

I am not entirely sure how to convert the conservation of mass and momentum equations into the Lagrangian form using the mass coordinate h. The one dimensional Euler equations given by, \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0...
16. ### Calculate the angular frequency w using the Euler equations

Homework Statement Consider the Earth as a rigid body with moment of inertia I1, I2 and I3. The Earth is symmetric around the z-axis (I1 = I2). Calculate the angular frequency w using the euler equations Homework EquationsThe Attempt at a Solution
17. ### Euler equations in rigid body: Taylor VS Kleppner - Kolenkow

Hello all. After reading both chapters on rigid body motion both in Kleppner - Kolenkow and Taylor books, I still do not undertand the physical meaning of Euler equations. Let me explain: In Kleppner - Kolenkow, they claim (page 321 - 322) that in Euler equations, Γ1, Γ2 and Γ3 are the...
18. ### Euler Equations & Rigid Lamina Moment of Inertia

"A rigid lamina (i.e. a two dimensional object) has principal moments of inertia about the centre of mass given by ##I_1=u^2-1##, ##I_2=u^2+1##, ##I_3=2u^2## Choose the initial angular velocity to be ##ω = µN \hat{e_1} + N \hat{e_2}##. Define tan α = ω2/ω1, which is the angle the component of ω...
19. ### Frobenius method and Euler equations

Hi, I'm having trouble with this one. Homework Statement Find a particular solution of the second-order homogeneous lineal differential equation x^2y'' + xy' - y = 0 taking in account that x = 0 is a regular singular point and performing a power series expansion. Homework...
20. ### Euler Equations, Sod shock tube & conservation

Is momentum conserved? I am considering the Euler equations in conservative form and solving the Sod shock tube problem I have written a Godunov finite volume type solver. It solves for density ρ, momentum ρu, and total energy E; therefore, I would expect all of these quantities to be...
21. ### Euler equations for ideal fluids, approximations

Homework Statement The Euler equations for ideal compressible flow are given by \partial_t v + (v\cdot \nabla)v = g-\frac{1}{\rho}\nabla p \\ \partial_t \rho + \nabla \cdot(\rho v) = 0 In my book these are written in terms of the small-value expansions \rho = \rho_0 + \delta \rho, p = p_0 +...
22. ### Analytical solution of the Euler equations in 1D?

Hi. Since these equations are approaching three hundred years old I'm pretty sure someone must have solved them somewhere before. However I have not been able to find any text-books or papers that actually show me how to solve these equations. So I'm wondering if anyone here know where I can...
23. ### How Do You Linearize the Euler Equations for Fluid Motion?

1. Homework Statement Linearize the euler equations of fluid motion, write as a single partial differential equation for example the pressure pertubation Homework Equations The euler equations of fluid dynamics The Attempt at a Solution Not sure how I would be able to do this.
24. ### Application of the euler equations using fermats principle

Homework Statement The speed of light in a medium with index of refraction n is c/n, where c is the speed of light in vacuum. Notice that n ≥1: Suppose a light ray travels in the xy-plane between (x1; y1) and (x2; y2) in a non-uniform material so that n(x) is the refractive index of the...
25. ### Deriving equation from 3D Euler Equations.

Homework Statement I've got the 3D Euler equations \frac{\delta u}{\delta t} + (u\cdot \nabla)u = -\nabla p \nabla \cdot u = 0 I've been given that the impulse is \gamma = u + \nabla\phi Homework Equations And I need to derive \frac{D\gamma}{Dt} = -(\nabla u)^T \gamma + \nabla...
26. ### Modified Euler equations doubt

Homework Statement Hi there. I'm not sure if this question corresponds to this subforum, but I think you must be more familiarized with it. The thing is I don't know how to get from: M_x=(I_0-I)\dot\Psi^2\sin\theta\cos\theta+I_0\dot\Phi\dot\Psi\sin\theta to...
27. ### What Are Embedded Axis Frames in Euler Equations?

my book says that it is actually difficult to get the true motion of a body by using these equations because it says that euler equations are written in embedded axis frame ... what is an embedded axis frame?where is it different from normal frames that i used in before?after solving euler...
28. ### Odd result from an eigenvalue problem in the Euler equations

Given the Euler equations in two dimensions in a moving reference frame: \frac{\partial U}{\partial t} + \frac{\partial F\left(U\right)}{\partial x} = 0 U = \left(\rho , \rho u , \rho v , \rho e \right) F\left(U\right) = \left(\left(1-h\right)\rho u , \left(1-h\right)\rho u^2...
29. ### Derivations of Euler equations

Every textbook i find breezes over the following point: \delta\partial (x) =\partial \delta (x) where delta is just the variation. Someone asked me why that's true and i guessed the only thing i could say was that delta is an operation not a variable so this is more like an algebraric...
30. ### Euler equations with Y(1) = 0, Y(2) = 0

Is it possible for an Euler equation to satisfy the boundary conditions Y(1)=0, Y(2)=0? I have considered the three possibilities, distinct real roots, repeated roots and conjugate complex roots and cannot find any solutions. Are there any other possibilities to consider? Thanks
31. ### Deriving Nonrelativistic Euler Equations from Stress Energy Tensor

Hi, I would like to start from the stress energy tensor for the perfect fluid: T^{\mu\nu}=\begin{pmatrix} \rho c^2 & 0 & 0 & 0\cr 0 & p & 0 & 0\cr 0 & 0 & p & 0\cr 0 & 0 & 0 & p\cr\end{pmatrix} where \rho is the mass density and p is the pressure, and I would like to derive the...
32. ### Forward euler equations of motion

Homework Statement Hi, I'm trying to compute the equations of motion for a car as shown in the attached image. α = steering angle θ = orientation of the car relative to the world coordinate system Say you're given the linear velocity v and the steering angle α. How do you compute...
33. ### Proving Euler Equations and Solving Complex Numbers in Signal and Systems Course

Hi i am reading about signal and systems course . What i want to prove is not a problem that i have to solve is something that the books take for granted and i want to prove it so i ll be able at exams to reprove so i won't have to remember it, (if u don't believe me i can give u the course's...
34. ### MATLAB Matlab and the Euler Equations

Hi, I was wondering if Matlab was the sort of program I'd want to solve the Euler Equations (fluid dynamics). And if it is, I am sure this must be a very standard problem.. does anybody know of any tutorials for this sort of problem as I have never used matlab? :-p