# Tagged Questions

The tag has no usage guidance.

107 views
+50

### Is boundary well defined if variation of metric don't vanish on the boundary?

Suppose that you want to calculate the variation $\delta S$ of an action induced by some arbitrary variation $\delta g_{\mu \nu}$ of the spacetime metric : S = \int_{\Omega} ...
207 views

### What are the boundary conditions associated to this lagrangian?

Suppose that $L(q^i, \dot{q}^i)$ is a standard and well behaved lagrangian associated to some Dirichlet boundary conditions : $q^i(t_1) = q_1^i$ and $q^i(t_2) = q_2^i$. Now I have this new lagrangian ...
28 views

### Gibbons-Hawking Variation

I know there already exist some questions about this and some very good answers. However, I am still having trouble understanding one part of the calculation. The GHY term is given by ...
89 views

### Boundary conditions of fields from the stationary action principle

First principle of stationary action Consider a real Klein-Gordon scalar field $\phi$ living in a $D$ dimensional flat spacetime. The field is considered off shell (the on shell condition is defined ...
15 views

### Commutativity Variation and derivative

When in general is true that $\nabla_{\mu}\delta=\delta\nabla_{\mu}$ where $\nabla_{\mu}$ is covariant derivative? I was thinking this when one has to find the equations of motion for example in ...
99 views

### Why can't we fix the metric and its derivatives at boundary, with the variational method?

In general relativity and for its Einstein-Hilbert action, we usually ask that the metric variations $\delta g_{\mu \nu}$ cancel on the boundary $\partial \, \Omega$ of some region $\Omega$ of the ...
159 views

### Variation of the Einstein-Hilbert action in D dimensions without the Gibbons-Hawking-York term

Consider the standard Einstein-Hilbert action in $D \ne 2$ dimensions spacetimes : $$S_{EH} = \frac{1}{2 \kappa} \int_{\Omega} R \; \sqrt{- g} \; d^D x,$$ where $\Omega$ is ...
29 views

### Proof of Hamilton's principle [duplicate]

Is there a anything like a proof of Hamilton's principle? Where would I find it?
25 views

### Function for which action is the minimum

On page 2 of "Mechanics" Landau & Lifshitz say that $q=q(t)$ is a function for which action is a minimum. Before this they say that at times $t_1$ and $t_2$, the system occupies coordinates ...
87 views

43 views

### Variational calculus, bending a stick and stationary states

We have a horizontal stick, one of its ends is on the wall, and we can apply a force to the other end. We assume that anything that we can do will leave this in the same plane. Our question is to ...
140 views

### Time dependence of the Lagrangian of a free particle?

I am working through Landau's book on Classical Mechanics. I understand the logic and physics of isotropy and homogeneity of space-time behind the derivation of the Lagrangian for a free particle, but ...
66 views

121 views

### Question on Einstein's derivation of the equation of the geodesic line?

While reading one of the original paper on general relativity written by Albert Einstein, titled the foundations of general relativity, I came across the following passage in pages 167-168, or pages ...
135 views

### How to use Euler-Lagrange when Lagrangian is $L=\sqrt{t}\sqrt{1+(dy/dt)^2}$

In this Lagrangian problem, action is $$S = \int_{t_1}^{t_2} \sqrt{t}\sqrt{1+\dot{y}^2} \,\,dt$$ where $\dot{y} = dy/dt$ and $t_1$ and $t_2$ are some fixed points. I tried to solve this problem using ...
52 views

### What is a sufficient condition to determine if the action integral reaches an extremal through the trajectory described by $q(t)$?

As I understand it, the Euler-Lagrange equation is a necessary but not a sufficient condition to determine if the action integral reaches an extremal through the trajectory described by $q(t)$. If ...
152 views

I'm self-studying Mechanics and I have a little problem: We can see that in Landau's book or in Wikipedia that when we inject the lagrangian in Euler Lagrange equation the term $\frac{\partial ... 2answers 1k views ### Why are generalized positions and generalized velocities considered as independent of each other? I'm confused how $$\dot{\mathbf{r}}_{j}=\sum_{k}\frac{\partial\mathbf{r}_{j}}{\partial q_{k}}\dot{q}_k+\frac{\partial\mathbf{r}_{j}}{\partial t}$$ leads to the relation, ... 0answers 44 views ### Interchanging of variation and integration operator for holonomic systems Meirovitch says in his "Principles and Techniques of Vibrations" (1997) on p.85: In the case of holonomic systems, the variation and integration processes are interchangeable (...) which means ... 0answers 55 views ### What is the functional shape assumed by a flexible rod? Be L a flexible rod. Say that it is very difficult to significantly stretch it, so that we can uniquely identify a point on it by a parameter$l \in [0, L]$where$L$is its length. Be$Ca set of ... 3answers 466 views ### Does the variation of the Lagrangian satisfy the product rule and chain rule of the derivative? I have seen wikipedia use the product rule and maybe the chain rule for the variation of the Langragin as follows: \begin{align} \dfrac{\delta [f(g(x,\dot{x}))h(x,\dot{x})] } {\delta x} = \left( ... 3answers 735 views ### What is the equation for the tension on the ends of a cable suspended at different heights? I'm looking for an equation to find the tension on the ends of a cable suspended between two poles (one higher than the other) with no load but the cable itself. I determined that the tension would ... 1answer 128 views ### Finding the magnetic vector potential by calculus of variations Given the functional $$F[A]=\int_{\mathbb{R}^3}\{\frac{1}{2\mu(x)}|\nabla\times\vec{A}|^2-\vec{J}\cdot\vec{A}\}d^3x$$ with\vec{A}$is a candidate vector potential for the field ... 0answers 56 views ### Calculus of variations applied to a rotating liquid I thought I knew how to use calculus of variations, but then I started thinking about the problem of a rotating liquid and it confused me a great deal. It would be nice to hear your thoughts on the ... 1answer 103 views ### The Euler-Poincare equation Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone ... 0answers 62 views ### Total Vs Partial in Lagrange density? I have a question regarding the red term below. This is the integration by parts during the derivation of the Euler-Lagrange equation for continuous systems. Why is this not the time derivative ... 1answer 121 views ### Variational derivatives of strongly connected diagrams functional in gauge theory Background In Jorge C. Romao's "Advanced Quantum Field Theory", at the end of page 218, Eq (6.266) reads: $$\tag{1} \left.\frac{\delta^{2}}{\delta \omega^{b}(y)\delta A_{\mu}^{c}(z)}\left[ ... 4answers 1k views ### What is the relation between (physicists) functional derivatives and Fréchet derivatives I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books:$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} ... 1answer 166 views ### Subtlety in derivation of Noether's theorem by Di Francesco In the book 'Conformal Field Theory' by Di Francesco et al, a derivation of Noether's theorem is demonstrated by imposing that, what I believe is said to be a more elegant approach, the parameter ... 0answers 264 views ### Question about an integration by parts in Feynman's Quantum Mechanics [closed] I have begun reading Feynman & Hibbs Quantum Mechanics and Path Integrals. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I ... 0answers 839 views ### Shape of a string/chain/cable/rope/wire? The height of a string in a gravitational field in 2-dimensions is bounded by$h(x_0)=h(x_l)=0$(nails in the wall) and also$\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take$h$as a function of arc ... 1answer 226 views ### Finding the Shape of a Hanging Massive Wire This came from a physics wave book. I have a static, massive wire on the x-axis, with a y-displacement due to a force per unit length$F_y$. I start with the equation$F_y = ...
The last time I took a plane the following problem crossed my mind. Setting: take the Earth and neglect its rotation around the Sun. It then only rotates on itself with angular velocity $\Omega$. ...