# 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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12. ### 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. ### 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. ### 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...

21. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...