Euler-lagrange Definition and 18 Discussions

In the calculus of variations and classical mechanics, the Euler-Lagrange equations is a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero.
In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations. In classical mechanics, it is equivalent to Newton's laws of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.

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  1. P

    Hamiltonian of this system?

    Let a mass m charged with q, attached to a spring with constant factor k = mω ^2 in an electric field E(t) = E0(t/τ) x since t=0. (Equilibrium position is x0 and the deformation obeys ξ = x - x0) What would the hamiltonian and motion equations be in t ≥ 0, in terms of m and ω?? Despise magnetic...
  2. polytheneman

    What is the derivative of velocity with respect to position?

    I'm reading a book on Classical Mechanics (No Nonsense Classical Mechanics) and one particular section has me a bit puzzled. The author is using the Euler-Lagrange equation to calculate the equation of motion for a system which has the Lagrangian shown in figure 1. The process can be seen in...
  3. JuanC97

    I Do 4-divergences affect the eqs of motion for nth order perturbed fields?

    Intuitively, I'd say that adding a 4-divergence to the Lagrangian should not affect the eqs of motion since the integral of that 4-divergence (of a vector that vanishes at ∞) can be rewritten as a surface term equal to zero, but... In some theories, the addition of a term that is equal to zero...
  4. T

    A Lagrangian to the Euler-Lagrange equation

    Hello all, I understand the formation of the Lagrangian is: Kinetic Energy minus the potential energy. (I realize one cannot prove this: it is a "principle" and it provides a verifiable equation of motion. Moving on... One inserts the Lagrangian into the form of the "Action" and minimizes it...
  5. Another

    Euler-Lagrange equation

    My question : I am wondering about definition of a function. when ##y_x = (\frac{b+y}{a-y})^2## Why in this book is defined solution ##y = y(x)## in from ## y = y(θ(x))## . And have a relationship in the form ## y = \frac{1}{2} (a-b) - \frac{1}{2} (a+b) cosθ ## ...
  6. MattIverson

    Finding the Lagrangian Matrix for Two-Spring Systems

    Homework Statement The problem is attached. I'm working on the second system with the masses on a linear spring (not the first system). I think I solved part (a), but I'm not sure if I did what it was asking for. I'm not sure exactly what the question means by the... L=.5Tnn-.5Vnn. Namely, I'm...
  7. redtree

    I Derivation of the Euler-Lagrangian

    I have a question about a very specific step in the derivation of Euler-Lagrangian. Sorry if it seems simple and trivial. I present the question in the course of the derivation. Given: \begin{equation} \begin{split} F &=\int_{x_a}^{x_b} g(f,f_x,x) dx \end{split} \end{equation}...
  8. C

    Null geodesics given line element

    Hi, I'm the given the following line element: ds^2=\Big(1-\frac{2m}{r}\large)d\tau ^2+\Big(1-\frac{2m}{r}\large)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2) And I'm asked to calculate the null geodesics. I know that in order to do that I have to solve the Euler-Lagrange equations. For...
  9. P

    Euler-Lagrange Equations for geodesics

    Homework Statement The Lagrange Function corresponding to a geodesic is $$\mathcal{L}(x^\mu,\dot{x}^\nu)=\frac{1}{2}g_{\alpha \beta}(x^\mu)\dot{x}^\alpha \dot{x}^\beta$$ Calculate the Euler-Lagrange equations Homework Equations The Euler Lagrange equations are...
  10. V

    I Euler-Lagrange query

    On the following post, where it says q=θ for the Euler-Lagrange equation where does the 2mr˙r˙θ come from?
  11. C

    Lagrangian mechanics: Bar connected to a spring

    Homework Statement Mass 1 can slide on a vertical rod under the influence of a constant gravitational force and and is connected to the rod via a spring with the spring konstant k and rest length 0. A mass 2 is connected to mass 1 via a rod of length L (forms a 90 degree angel with the first...
  12. P

    Minimisation Problem (Euler-Lagrange equation)

    Homework Statement Homework Equations d/dx ∂F/∂y'=∂F/∂y The Attempt at a Solution I have no problem with the first bit, but the second bit is where I get stuck. Since the question says the speed is proportional to distance, I have taken v(x)=cx where c is some...
  13. V

    Initial Conditions Applied to a Lagrangian

    Homework Statement The scenario is a pendulum of length l and mass m2 attached to a mass of m1 which is allowed to slide along the horizontal with no friction. The support mass moves along in the X direction and the pendulum swings through the x-y plane with an angle θ with the vertical. After...
  14. samgrace

    Calculus of variations

    Hello, here is my problem.'][/PLAIN] [Broken] My confusion lies in, why those particular partial derivatives are chosen to be acted upon the auxiliary function and then how they are put together to get the Euler-Lagrange equation? My guess is...
  15. bananabandana

    Euler Lagrange Derivation (Taylor Series)

    Mod note: Moved from Homework section 1. Homework Statement Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way: $$ L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] +...
  16. E

    Finding the Lagrangian for an elastic collision

    Homework Statement a. Suppose two particles with mass $m$ and coordinates $x_1$, $x_2$ collides elastically, find the lagrangian and prove that the linear momentum is preserved. b. Find new coordiantes (and lagrangian) s.t. the linear momentum is conjugate to the cyclical coordinate. Homework...
  17. VintageGuy

    Tensor indices (proving Lorentz covariance)

    Homework Statement [/B] So, I need to show Lorentz covariance of a Proca field E-L equation, conceptually I have no problems with this, I just have to make one final step that I cannot really justify. Homework Equations "Proca" (quotation marks because of the minus next to the mass part, I...
  18. J

    Euler Lagrange exercise

    Homework Statement On very hot days there sometimes can be a mirage seen hovering as you drive. Very close to the ground there is a temperature gradient which makes the refraction index rises with the height. Can we explain the mirage with it? Which unit do you need to extremalise? Writer the...