# Questions tagged [variational-calculus]

Variational calculus is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find extrema of functionals: mappings from a set of functions to the real numbers. The archetype application in physics is Lagrangian mechanics, seeking extrema of action functionals.

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### Equation of motion in quadratic gravity

I am going through the paper https://arxiv.org/abs/1502.01028 which considers the quadratic gravity with the action \begin{align} S = \int d^4x \sqrt{-g} (R - \alpha C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\...
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### Variational derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} \dot{u}(x,t) - ca(u')^2$$ with respect to $u$, ($\delta L / \delta u$). Following my lecture ...
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### Magnitude of the variations $\delta q_i$ in the principle of stationary action

To determine the equation of motion using the principle of stationary action, one has to consider the variation of the action due to variations $\delta q_i$ in all the generalized coordinates $q_i$. ...
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### Variational operator confusion

Let $L=L(X, \dot X)$ such that the first variation of $L$ is given by $$\delta L=\frac{\partial L}{\partial X}\delta X+\frac{\partial L}{\partial \dot X}\delta \dot X.\tag{1}$$ This is pretty standard ...
1 vote
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### Confusion with the variational operator $\delta$ and finding variations

I have recently started studying String Theory and this notion of variations has come up. Suppose that we have a Lagrangian $L$ such that the action of this Lagrangian is just $$S=\int dt L.$$ The ...
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### Time dependent Schrodinger equation through variation principle - questions about derivation

I'm reading a text which discusses time dependent variation principle (Geometry of the Time-Dependent Variational Principle in Quantum Mechanics by Kramer and Saraceno), and there is some part of a ...
1 vote
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### Variation of the metric determinant

I know this question has been answered here for example but I want to make emphasis in a new aspect. Consider the variation of the metric determinant $\delta g$ with respects to variations of the ...
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### If $F^2 = g_{pq} \dot{x^p}\dot{x^q}$ , where $g_{pq}$ is a metric tensor, then find $\frac{\partial F}{\partial{\dot{x^k}}}$

I am trying to find the geodesics in a Riemannian space, using Tensor analysis. I am also using the Principle of Variation. I want to minimize the geodesics integral whose integrand is $F$. Then, ...
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### Bosonic closed string effective action

Neil Lambert in his lecture notes https://nms.kcl.ac.uk/neil.lambert/SBQG.pdf in section 3.9 states that imposing conformal invariance at one-loop imposes the following equations on the spacetime ...
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### Euler-Bernoulli equation for a periodically supported static beam

The Euler-Bernoulli equation for a homogeneous beam is $$EI w^{(4)}(x) = q(x),$$ where $w$ is beam height and $q$ is load density. Inspired by the deflection in a multi-support cantilever bridge ...
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### How to calculate the variation of the metric on a compact manifold?

For example, given a torus with a modular parameter $\tau$ and an action \begin{equation} I=\frac{g}{2}\int_\mathcal{M} d^2 z \sqrt{-g}\ g_{ij}(z) \partial^i\phi \partial^j\phi \end{equation} ...
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1 vote
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### Functional derivative for the action $S$

From Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur, p. 15: Example 1.3 The Lagrangian $L$ can be written as a function of both position and velocity. Quite generally, one can ...
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### How to determine $n(x)$ when the functional depends exclusively on $n(x)$ and $x$? (Fermat's principle)

Recently I was taught an introduction to calculus of variations in reference to a course on analytical mechanics, where one problem involved Fermat's principle, stating that the path taken by a light ...
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### If the Lagrangian included acceleration [duplicate]

If the Lagrangian included acceleration, then how many conditions would be needed to unambiguously define the trajectory, on what variations would the minimum action problem be considered, what would ...
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### Functional Calculus in QFT

Does anybody know some good sources with detailed derivations of the main results we need to compute generating functionals in QFT (and functional calculus used in the subject in general). I find that ...
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### Search for maximum and minimum functionality

How can I prove that the trajectory I found after applying the principle of least action corresponds to the minimum of action, and not to the maximum?
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### Christoffel symbols for diagonal metric with the Lagrangian [closed]

I have been trying to find an explicit formula for the Christoffel symbols of an arbritrary diagonal metric $g_{\mu\nu}$. I do not wish to expand the symbols with the formula which contains the metric ...
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### How can the Euler-Lagrangian equation be applied to the Schroedinger Lagrangian of several fields?

In a Lagrangian like the one that follows: $$\mathcal{L} = {i} \Psi^*\dot{\Psi} - \frac{1}{2m} \nabla{\Psi} ^* \nabla \Psi.\tag{1}$$ How can I apply the Euler-Lagrangian equation, shown in $(2)$, to ...
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### Action in Lagrangian Mechanics [duplicate]

I editted this question since it was closed because it is a duplicate. However, answers in the referenced question didn't solve my question, so I am writing it again. Lagrangian mechanics is built ...
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