What is Variation method: Definition and 14 Discussions

In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations.
Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa.

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  1. Pouramat

    Variation principle -- looking for resources to read and understand

    Summary:: Can anyone introduce an informative resource with solved examples for learning variation principle? For example I cannot do the variation for the electromagnetic lagrangian when ##A_\mu J^\mu## added to the free lagrangian and also some other terms which are possible: $$ L =...
  2. egozenovius

    I Recovering some math notions: Variations

    I have a paper and on that paper I only can read: Let $$f:\mathbb{S^{1}} \to \mathbb{R^2}$$ be a function and $$f_{\epsilon}=f+\epsilon hn$$ and $$\mathbb{S^1}$$ is the unit circle. $$\dot{f_\epsilon}=\dot{f}+\epsilon\dot{h}n+\epsilon h\dot{n}$$ $$\delta\dot{f}=\dot{h}n+h\dot{n}$$ can you...
  3. E

    Did I use the correct identities in my variation method calculations?

    <Moderator's note: Moved from a technical forum and thus no template.> Problem: One dimensional quartic oscillator, V(x) = cx^4 (c is a constant) Use the trial function e^(-aplha(x^2)/2) to determine the value of the appropriate variational integral W. I've attached a picture of my work. I...
  4. A

    A Helium atom, variation method and virial theorem

    I need to calculate the energy of the ground state of a helium athom with the variational method using the wave function: $$\psi_{Z_e}(r_1,r_2)=u_{1s,Z_e}(r1)u_{1s, Z_e}(r2)=\frac{1}{\pi}\biggr(\frac{Z_e}{a_0}\biggr)^3e^{-\frac{Z_e(r_1+r_2)}{a_0}}$$ with ##Z_e## the effective charge considered...
  5. J

    I Determining the accuracy of the Variational Method

    The Variational Method allows us to obtain an upper bound on energy of the ground state (and sometimes excited states). Is there any way of determining an upper bound on the error of the energy obtained by the variational method without an analytic or numerical solution to the problem? i.e. Is...
  6. dreens

    I Variational Equations, Chaos Indicators

    I work with an electromagnetic molecule trap, and I'd like to determine which orbits are chaotic. To this end, I intend to study the evolution of a perturbation on a trajectory with time. I'd like to compute something called the fast lyapunov indicator for various trajectories y(t), where I...
  7. M

    What is the Linear Variation Method in Molecular Quantum Mechanics?

    In the chapter 9-5 "The Linear Variation Method" p. 363 from the book: Basic Principles and Techniques of Molecular Quantum Mechanics by Ralph Christoffersen, the first thing he does is to minimize the energy, E = c†Hc/c†Sc, by requiring its derivative with respect to the...
  8. F

    Variation Method: Finding Ground State Energy of 1D Harmonic Oscillator

    Homework Statement Use the variation method to find a approximately value on the ground state energy at the one dimensional harmonic oscillator, H = -ħ^2/(2m) * d^2/dx^2 + 1/2mω^2*x^2 Homework Equations H = -ħ^2/(2m) * d^2/dx^2 + 1/2mω^2*x^2 u(x) = Nexp(-ax^2) <H> = <u|Hu> The Attempt at a...
  9. Rorshach

    Particle in a potential- variation method

    Homework Statement Okay, I have no idea about the method they want me to solve it with. What in this case is the indicator that a function is appropriate? A particle mass m affects a potential of the form ##V(x)=V_0 \frac{|x|}{a}## where ##V_0## and ##a## are positive constants. a) Draw a...
  10. A

    Variation Method for Higher Energy States

    The variation method for approximating the the ground state eigenvalue, when applied to higher energy states requires that the trial function be orthogonal to the lower energy eigenfunctions.In that respect this book I am referring(by Leonard Schiff) mentions the following function as the...
  11. G

    Use the variation method with trial Wavefunction (Szabo and Oslund ex 1.18)

    Homework Statement The Schrodinger equation (in atomic units) of an electron moving in one dimension under the influence of the potential -delta(x) [dirac delta function] is: (-1/2.d2/dx2-delta(x)).psi=E.psi use the variation method with the trial function psi'=Ne-a.x2 to show that...
  12. O

    QM Variation Method: Show Equations from ci Parameters

    Homework Statement Show that variation principle (parameters ci) leads to equations \sum\limits_{i = 1}^n {\left\langle i \right|H\left| j \right\rangle c_j = Ec_i {\rm{ where }}} \left\langle j \right|H\left| i \right\rangle = \int {d\textbf{r}^3 \chi _j^* \left( \textbf{r} \right)\left(...
  13. P

    Variation Method: Proving \int \phi^{*} \hat{H} \phi d\tau>E_1

    Homework Statement This is the problem 8.10 from Levine's Quantum Chemistry 5th edition: Prove that, for a system with nondegenerate ground state, \int \phi^{*} \hat{H} \phi d\tau>E_{1}, if \phi is any normalized, well-behaved function that is not equal to the true ground-state wave function...
  14. T

    Variation method (Quantum Mech.)

    Hi, Here's the problem: Homework Statement Quantum particle moving in 1D. Potential energy function is V(x) = C|x|^{3}. Using the variational method, find an approx. ground-state wave function for the particle. The Attempt at a Solution Using \psi = Ae^{-ax^{2}}, I find that A =...