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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.

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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Calculus of variations -- how does it make sense to vary the position and the velocity independently?

In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat them as independent variables?
grizzly adam
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Why treat complex scalar field and its complex conjugate as two different fields?

I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar field: $$\mathcal{L}_\text{compl…
BMS
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Rigorous underpinnings of infinitesimals in physics

Just as background, I should say I am a mathematics grad student who is trying to learn some physics. I've been reading "The Theoretical Minimum" by Susskind and Hrabovsky and on page 134, they introduce infinitesimal transformations. Here's the…
Brian Klatt
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Why is the shape of a hanging chain not a "V"?

From Wikipedia: To answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy. The gravitational potential energy is highest at…
Jossie Calderon
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Dynamics of a rotating coin

Sometimes when I'm bored in a waiting area, I would take a coin out of my pocket and spin it on a table. I never really tried to figure out what was going on. But, recently I wondered about two things: Can I determine the critical rotational speed…
user29305
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Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are the same is presented in the book I am reading as a…
user37155
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Derivation of Euler-Lagrange equations for Lagrangian with dependence on second-order derivatives

Suppose we have a Lagrangian that depends on second-order derivatives: $$L = L(q, \dot{q}, \ddot{q},t).\tag{1}$$ If we're working on the variational problem for this Lagrangian, then I know that we'll wind up with the following Euler-Lagrange…
anygivenpoint
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Variation of the Christoffel symbols with respect to $g^{\mu\nu}$

I'm trying to find the field equations for some particular Lagrangian. In the middle I faced the term $$\frac{\delta \Gamma_{\beta\gamma}^{\alpha}}{\delta g^{\mu\nu}} \, .$$ I know that $$\delta \Gamma_{\beta\gamma}^{\alpha} =…
Alejandro Guarnizo
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Is it possible to prove that planets should be approximately spherical using the calculus of variations?

Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical? Let's assume that a planet is 'born' when lots of particles of uniform density are piled together, and…
Jose Javier Garcia
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Where can I find the full derivation of Helfrich's shape equation for closed membranes?

I have approximately 10 papers that claim that, from the equation for shape energy: $$ F = \frac{1}{2}k_c \int (c_1+c_2-c_0)^2 dA + \Delta p \int dV + \lambda \int dA$$ one can use "methods of variational calculus" to derive the following: $$\Delta…
Samuel Reid
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What's the variation of the Christoffel symbols with respect to the metric?

By the Leibniz rule, I expected it to be $$\delta \Gamma^\sigma_{\mu\nu} = \frac 12 (\delta g)^{\sigma\lambda}(g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu}-g_{\mu\nu,\lambda}) + \frac 12 g^{\sigma\lambda}(\partial_\nu (\delta g)_{\mu\lambda}+\partial_\mu…
Rodrigo
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Why boundary terms make the variational principle ill-defined?

Let me start with the definitions I'm used to. Let $I[\Phi^i]$ be the action for some collection of fields. A variation of the fields about the field configuration $\Phi^i_0(x)$ is a one-parameter family of field configurations $\Phi^i(\lambda,x)$…
Gold
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What is the relation between (physicists) functional derivatives and Fréchet derivatives

I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books: $$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\epsilon \delta(x-y)]-F[f(x)]}{\epsilon}$$ from the…
Forever_a_Newcomer
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Introductory texts for functionals and calculus of variation

I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good introductory text for this topic. Any idea will be…
skywaddler
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Anticommutation of variation $\delta$ and differential $d$

In Quantum Fields and Strings: A Course for Mathematicians, it is said that variation $\delta$ and differential $d$ anticommute (this is only classical mechanics), which is very strange to me. This is in page 143-144: If we deform $x$ we…
Zihni Kaan Baykara
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