1. ### Weird condition describing symmetry transformation

I'm a bit confused about the condition given in the description of the symmetry transformation of the filed. Usually, given any symmetry transformation ##x^\mu \mapsto \bar{x}^\mu##, we require $$\bar\phi (\bar x) = \phi(x),$$ i.e. we want the transformed field at the transformed coordinates to...

4. ### A Variation of scalar field action

I am reading Polchinski's review on AdS/CFT https://arxiv.org/abs/1010.6134. I have a very simple question, and please help me out. Thanks in advanced. The question abou formula (3.19) The scalar effective bulk action is given by  S_0=\frac{\eta}{2}\epsilon^{1-D}\int d^Dx \phi_{\rm cl}...
5. N

### Benefits of variation?

Homework Statement Variation during reproduction is beneficial to the species but not necessarily for the individual? Homework Equations Not any The Attempt at a Solution I only know about variation during reproduction is beneficial to the species but I don't know anything about how it is not...
6. ### Variations of Regular Curves problem

Homework Statement Let γs : I → Rn, s ∈ (−δ, δ), > 0, be a variation with compact support K ⊂ I' of a regular curve γ = γ0. Show that there exists some 0 < δ ≤ ε such that γs is a regular curve for all s ∈ (−δ, δ). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of...
7. ### Momentum not consistent with definition in Landau's book?

Homework Statement I am not sure whether the meaning of the equation ##(3)## which used for deriving momentum is as same as equation ##(4)##.I will make a detailed description below. The lagrangian function for a free particle is ##L=-mc^2\sqrt{1-\frac{v^2}{c^2}} \quad (1)## The action from...
8. ### I Extremal condition in calculus of variations, geometric

Hi folks, I am a bit confused with the extreme condition used in the calculus of variations: δ = 0 I don't understand this rule to find extreme solutions (maximum or minimum) If in normal differential calculus we have a function y = y(x) and represent it graphically, you see that at the...
9. ### Derivation of Lagrange's eqs

Homework Statement So I'm deriving Lagrange's equations using Hamilton's principle which states that the motion of a dynamical system follows the path, consistent with any constraints, that minimise the time integral over the lagrangian L = T-U, where T is the kinetic energy and U is the...
10. ### What is the total variation of sin(x) on [a,b]?

Homework Statement For a given function ##g:[a,b]→ℝ, 0 < a < b##, compute its total variation \underset{[a,b]}{\mathrm{Var}} (g) where ##g(x) = \sin(x), x\in[a,b].## Homework Equations The Attempt at a Solution I know that between odd multiples of ##\frac{\pi}{2}##, ##\sin(x)## is monotone...
11. ### Need help with counting problems

Homework Statement Counting problems are a very tough subject to me, so if someone could give me tips, examples explaining what's really happening, that would be great. Homework Equations I know what permutations, variations, combinations, ... are. The problems involving only one of those...
12. ### Permutation with exception/repetition

I need a formula to calculate permutation. For example I have a 5 numbers and I creating a 3 digit number from it. The numbers are: 1, 1, 1, 2, 3; I could write up 13 variations, but I couldn't work out the formula. If the numbers are: 1, 1, 2, 2, 3 the number of variations are 18 (if I wrote...
13. ### Domestic Water Pump (1 HP)

Hi there, I have spent quite some time on PF as an unregistered user reading through various stuff and have learned a lot. Just registered now to seek help on something that's been bothering me a lot and to which I haven't managed to find a solution yet. Please let me know if this is not the...