What is Variational principle: Definition and 47 Discussions

In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.

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  1. Matthew_

    I Issues in understanding screening effects in the Jellium model

    In the Jellium model, it is customary to evaluate the exchange term of the Hartree-Fock equation for plane waves ##\varphi_{\mathbf{k}_i}## as a correction to the energy of the non-interacting electron gas obtaining $$\hat{U}^{ex} \varphi_{\mathbf{k}_i}=-e^2 \left( \int \dfrac{\mathrm{d}^3k}{2...
  2. Baela

    A If the solution of a field vanishes on-shell does it mean anything?

    Let us consider an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. The solution of the field ##c## is given by the expression ##f(a,b)##. On taking into account the relations obtained from the solutions for ##a## and ##b##, we find that ##f(a,b)=0##. If the...
  3. Baela

    A Are equations of motion invariant under gauge transformations?

    We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations? If yes, can you show a mathematical proof (instead of just saying in words)?
  4. lua

    Upper bound for first excited state - variational principle

    I'm solving problem number 5 from https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/mit8_05f13_ps2/. (a) Here I got: $$ \beta = \frac{\hbar^{\frac{1}{3}}}{(\alpha m)^\frac{1}{6}} $$ and: $$ E = \left ( \frac{\alpha \hbar^4}{m^2} \right )^\frac{1}{3}e $$ (b) Using Scilab I...
  5. LCSphysicist

    Getting geodesic from variational principle

    The metric is $$ds^2 = \frac{dr^2 + r^2 d\theta ^2}{r^2-a^2} - \frac{r^2 dr^2}{(r^2-a^2)^2}$$ I need to prove the geodesic is: $$a^2 (\frac{dr}{d \theta})^2 + a^2 r^2 = K r^4$$ My method was to variate the action ##\int\frac{(\frac{dr}{d\theta})^2 + r^2 }{r^2-a^2} - \frac{r^2...
  6. Samama Fahim

    I Schrodinger Equation from Ritz Variational Method

    (This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method) 1) Are ##\frac{\delta}{\delta \psi^{*}}## derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get ##\int d^3x (\hat{H}\psi)## in equation 11.35a...
  7. jayzhao

    What is the function to extremise for finding geodesics on a Helicoid?

    I've got that length of a curve on the surface is: $$L=\int_{-\infty}^{\infty}\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}dx$$ So the function to extremise is: $$f(\rho,\rho')=\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}$$ Where...
  8. 1

    Using Variational Principle to solve ground state energy

    First I picked an arbitrary state ##|ϕ⟩=C_1|φ_1⟩+C_2|φ_2⟩+C_3|φ_3⟩## and went to use equation 1. Realizing my answer was a mess of constants and not getting me closer to a ground state energy, I abandoned that approach and went with equation two. I proceeded to calculate the following matrix...
  9. K

    I How it's known that the variational principle works for relativity

    This question is actually about relativity and quantum field theories. I have the impression that we just use the variational principle, and given the right lagrangian, they lead to equations that we know, are correct. That seems to me a good reason for "believing" that the variationa principle...
  10. M

    I Integral in a variational principle problem

    Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't...
  11. Gio83

    A Derive the Bianchi identities from a variational principle?

    Einstein's field equations (EFEs) describe the pointwise relation between the geometry of the spacetime and possible sources described by an energy-momentum tensor ##T^{ab}##. As well known, such equations can be derived from a variational principle applied to the following action: $$S=\int\...
  12. J

    Classical mechanics: Jacobi variational principle

    An isolated mechanical system can be represented by a point in a high-dimensional configuration space. This point evolves along a line. The variational principle of Jacobi says that, among many imagined trajectories between two points, only the SHORTEST is real and is associated with situations...
  13. S

    I Schroedinger Equation from Variational Principle

    Landau's nonrelativistic quantum mechanics has a "derivation" of Schroedinger's equation using what he calls "the variational principle". Apparently such a principle implies that: $$\delta \int \psi^{\ast} (\hat{H} - E) \psi dq = 0$$ From here I can see that varying ##\psi## and...
  14. rezkyputra

    Deriving Einstein Eq. from Variational Principle

    Homework Statement Okay, in Carrol's Intro to Spacetime and Geometry, Chapter 4, Eq. 4.63 to 4.65 require a derivation of a difference between Christoffel Symbol. I did the calculation and found my answer to be somewhat correct in form, but the indices doesn't match up Homework Equations So...
  15. D

    Deriving geodesic equation using variational principle

    I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got this. $$ \frac{d^2 x^u}{dt^2} + \Gamma^u_{mk} \frac{dx^m}{dt} \frac{dx^k}{dt} =...
  16. S

    Intuition behind Hamilton's Variational Principle

    Background: I am an upper level undergraduate physics student who just completed a course in classical mechanics, concluding with Lagrangian Mechanics and Hamilton's Variational Principle. My professor gave a lecture on the material, and his explanation struck me as a truism. Essentially, he...
  17. T

    Resources for variational principle to solve coulomb problem in D dimension

    Hi, i have been struggling to find some good resources on variational principle , I have got an instructor in advanced quantum course who just have one rule for teaching students- "dig the Internet and I don't teach you anything".. So I digged a lot and came up with a lot reading but I need...
  18. C

    Variational Principle for Spatially Homogeneous Cosmologies/KK-theory

    These questions applies to both spatially homogenous cosmological models, and multidimensional Kaluza-Klein theories: Suppose we have a manifold M, of dimension m, for which there is a transitive group of isometries acting on some n-dimensional homogeneous subspace N of M. Thus there exists a...
  19. B

    What is the Variational Principle for Estimating Energy of First Excited State?

    Homework Statement A particle of mass m is in a potential of V(x) = Kx4 and the wave function is given as ψ(x)= e^-(ax2) use the variational principle to estimate the ground state energy. Part B: The true ground state energy wave function for this potential is a symmetric function of x...
  20. M

    Variational Principle and Vectorial Identities

    Hello there, I am struggling in proving the following. The principle of Minimum energy for an elastic body (no body forces, no applied tractions) says that the equilibrium state minimizes $$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}u)$$ among all vectorial functions u satisfying the...
  21. S

    Maxwells equations from variational principle

    1. Hey, I have to find Maxwells equations using the variational principle and the electromagnetic action: S=-\intop d^{4}x\frac{1}{4}F_{\mu\nu}F^{\mu\nu} by using \frac{\delta s}{\delta A_{\mu(x)}}=0 therefore \partial_{\mu}F^{\mu\nu}=0 3. I have had a go at the...
  22. J

    Solving differential equation from variational principle

    I have the following differential equation which I obtained from Euler-Lagrange variational principle \frac{\partial}{\partial x}\left(\frac{1}{\sqrt{y}}\frac{dy}{dx}\right)=0 I also have two boundary conditions: y\left(0\right)=y_{1} and y\left(D\right)=y_{2} where D, y_{1} and y_{2} are...
  23. J

    Variational Principle of 3D symmetric harmonic oscillator

    Homework Statement Use the following trial function: \Psi=e^{-(\alpha)r} to estimate the ground state energy of the central potential: V(r)=(\frac{1}{2})m(\omega^{2})r^{2} The Attempt at a Solution Normalizing the trial wave function (separating the radial and spherical part)...
  24. N

    Need guidance on where to start on the study of the variational principle

    I was experimenting with some physics and the mathematics started to get a bit tougher than what I'm used to. I had a professor who looked at what I'm doing, offered to guide me, and told me to do some research on the variational principle. At the moment, I am in Calculus II. I did a couple...
  25. P

    Christoffel symbol from Variational Principle

    Homework Statement It's not exactly a homework question. I can find Christoffel Symbols using general definition of Christoffel symbol. But, when I try to find Christoffel Symbols using variational principle, I end up getting zero. I have started with the space-time metric in a weak...
  26. X

    Tough exponential integral (QM, Variational Principle)

    Homework Statement http://img4.imageshack.us/img4/224/32665300.png The Attempt at a Solution http://img684.imageshack.us/img684/2920/scan0003xo.jpg I've uploaded my work so far since its much faster than typing and I'm stuck on the last line trying to solve the integral. The first...
  27. A

    Variational Principle Integration

    Homework Statement The problem statement is a bit length so I have attached a picture of the problem instead. The issue I am having pertains to part (b). Homework Equations The Attempt at a Solution The main issue I am having is with what my Hamiltonian should look like when I do...
  28. S

    What is varing in the variational principle of GR

    Consider the variational principle used to obtain that in the vacuum the Einstein tensor vanish. So we set the lagrangian density as L(g,\partial g)=R and asks for the condition 0 = \delta S =\delta\int{d^4 x \sqrt{-g}L} proceeding with the calculus I finally have to vary R such that...
  29. tom.stoer

    Variational principle in quantum mechanics

    I have a question regarding the variational principle in quantum mechanics. Usually we have a Hamiltonian H and we construct a state |ψ> using some trial states. Then we minimize E = <ψ|H|ψ> and get an upper bound for the ground state energy. In many cases the state |ψ> is then used to...
  30. H

    The connection between variational principle and differential equations

    It is very well known that the result of varying some functional gives a differential equation which solutions minimizes the given functional. What about the other way around? Can one find a functional that is minimized given a differential equation? Is there a procedure for this? The reason...
  31. B

    Variational Principle on First Excited State

    I am trying to prove the variational principle on 1st excited state, but have some questions here. The theory states like this: If <\psi|\psi_{gs}>=0, then <H>\geq E_{fe}, where 'gs' stands for 'grand state' and 'fe' for 'first excited state'. Proof: Let ground state denoted by 1, and...
  32. B

    Variational principle & lorentz force law

    Homework Statement Show that the Lorentz force law follows from the following variational principle: S=\frac{m}{2}\int\eta_{\mu\nu}u^\mu u^\nu ds-q\int A_\mu u^\mu ds Homework Equations Definition of Field Strength Tensor Integration by Parts Chain Rule & Product Rule for Derivatives The...
  33. M

    Confused by variational principle

    My notes give the variational principle for a geodesic in GR: c\tau_{AB} = c\int_A^B d\tau = c\int_A^B \frac{d\tau}{dp}dp = \int_A^B Ldp and then apply the Euler-Lagrange equations. By choosing p to be an "affine parameter" where \frac{d^2 p}{d\tau^2} the Euler-Lagrange equations are...
  34. J

    Quantum Field Theory - variational principle

    Quantum Field Theory -- variational principle In non-relativistic quantum mechanics, the ground state energy (and wavefunction) can be found via the variational principle, where you take a function of the n particle positions and try to minimize the expectation value of that function with the...
  35. L

    Variational Estimate of Hydrogen Atom Ground State Energy

    Obtain a variational estimate of the ground state energy of the hydrogen atom by taking as a trial function \psi_T(r) = \text{exp } \left( - \alpha r^2 \right) How does your result compare with teh exact result? You may assume that \int_0^\infty \text{exp } \left( - b r^2 \right) dr =...
  36. M

    Easy variational principle question that I can't integrate

    Homework Statement Use trial wavefunction exp(-bx^2) to get an upper limit for the groundstate energy of the 1-d harmonic oscillator The Attempt at a Solution This is always going to give an integral of x^2*exp(-x^2). How do you do it? :/
  37. D

    How Do Variational Principles Determine Critical Points in Fluid Dynamics?

    Homework Statement Consider the variation principle for the space-time functional of the variables \eta, \phi A( \eta, \phi) = \int \int \phi \partial _t \eta -\frac{1}{2}g \eta ^2 -\frac{1}{2} h ( \partial_x \phi ) ^2\ \mbox{d}x \mbox{d}t Derive the two coupled equations for the critical...
  38. Helios

    Variational principle & Emden's eqn

    I once tried to come up with a variational principle that would lead to Emden's equation. I think this is instructive. Start with the mass M = - 4 \pi a^{3} \rho_{c} \xi^{2} \Theta' rewrite this as M / 4 \pi a^{3} \rho_{c} + \xi^{2} \Theta' = 0 but just let X = M / 4 \pi a^{3}...
  39. M

    Variational principle convergence

    A text I am reading has used the variational principle not only to find the ground state of a system, but also to find some higher order states. (Specifically, it was used to derive the Roothaan equations, which are ultimately related to the LCAO method of orbital calculations.) I don't see how...
  40. cepheid

    Variational Principle: Find Best Bound State for 1D Harmonic Oscillator

    Homework Statement Find the best bound state on Egs for the one-dimensional harmonic oscillator using a trial wave function of the form \psi(x) = \frac{A}{x^2 + b^2} where A is determined by normalization and b is an adjustable parameter.Homework Equations The variational principle...
  41. N

    Variational Principle: Solving a Sawtooth Wave Potential

    Homework Statement If I'm given a potential say A(x/a-m) m an integer, (this is the sawtooth wave) What kind of trial function should I use to approximate this? Homework Equations The Attempt at a Solution I do recall this function arising in Fourier series. Should I actually...
  42. E

    QM variational principle

    [SOLVED] QM variational principle Homework Statement In order to use the variational principle to estimate the ground-state energy of the one-dimensional potential V(x) = Kx^4, where K is a constant, which of the following wave functions would be a better trial wave function: 1) \psi(x) =...
  43. L

    Some physics without variational principle ?

    I am in search for some part of physics that could not be derived from a variational principle. For the small part of physics I know (CM, Schrödinger), everything can be derived from a variational principle. I would like to know if this is a deep fingerprint of physics or a general...
  44. L

    Can all differential equations be derived from a variational principle?

    In classical mechanics, for conservative systems, it well knows that the differential laws of motion can be derived from a variational principle called "least action principle". I know also that some non-conservative systems can be derived from a variational principle: the damped harmonic...
  45. W

    What is the variational principle and how does it apply to physics?

    I need someone who can briefly (in easy way) explain me the variational principle or tell me where (in checked source) I can find this. I will be very greatful.
  46. E

    Schwinger variational principle

    What is this used for?..i don,t see any utility on using it..:frown: :frown: for Commuting and Anti-commuting operators we would have: \delta{<A|B>}=i<A|\delta{S_{AB}}|B> but i don,t see that it provides a way to obtain Schroedinguer equation or the propagator for the theorie...what is...
  47. E

    Quantum Mechanics - Ritz variational principle

    I was asked to do an assigment for a Chemical Physics class on the Ritz variational principle (used to calculate an approximation of an observable). We are working a simple potential, the one dimensional particle in the box (v=0 for 0<x<L, V= infinite elsewhere) and only considering the ground...
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