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4
votes
2answers
77 views

Are Lagrange's equations physical laws?

Well, I studied that a physical law is an equation between tensors that are function of position and time because when the frame is changed tensors change in order to leave the equation true: $$T_1(\...
1
vote
3answers
146 views

Origin of $\sqrt{-g}$ in the integral of action $S$

I have a question that might (and probably will) be stupid: I do not understand where does the factor $\sqrt{-g}$ (i.e. $\sqrt{-\det\left(g_{\mu\nu}\right)}$) come from in the action integral S when ...
1
vote
0answers
26 views

Help with the proof of the independence of the form of Lagrange equation wrt. choice of coordinates

I am reading about Lagrange-Euler equation here. When they prove that the formula is independent of the choice of coordinate, there is this reasoning, but I could not understand (probably my calculus ...
1
vote
2answers
127 views

Varying a scalar field Lagrangian density

I was varying a scalar field density and I look at this term $${\cal L}~=~-\frac{1}{2}\partial _\mu\phi\partial^\mu\phi.$$ The result that I need to come is $$-\frac{1}{2}\delta(\partial _\mu\phi\...
3
votes
2answers
55 views

Effect of Co-ordinate Change on Euler-Lagrange Equations for Scalar Fields

Consider a single scalar field $\phi$ on a manifold $\mathcal{M}$. Suppose in $\{x^\mu\}$ co-ordinates, the Lagrangian density is $\mathcal{L}(\phi, \frac{\partial \phi}{\partial x^\mu})$. This means ...
7
votes
2answers
236 views

Varying the Einstein-Hilbert action without reference to a chart

In most treatments of General Relativity, when the the Einstein-Hilbert action over some manifold $\mathcal{M}$ (plus Gibbons-Hawking-York term if $\mathcal{M}$ has a boundary), given by $$S=\frac{1}{...
1
vote
2answers
44 views

Different weights for time and spatial derivative in Lagrangian Density

I'm new to QFT and trying to understand the form of the Lagrangian densitys used. As a simple model you often see a Lagrangian density of the form $${\mathcal L} = \frac{1}{2} \partial_j \phi_n \...
3
votes
1answer
102 views

What's the Lagrangian density means?

I remember from my classical mechanics class the Lagrangian function, but I don´t understand what's the meanning of the lagrangian density in General Relativity. What's the difference between one and ...
3
votes
0answers
84 views

Showing path integral formalism is Lorentz-invariant without resorting to Hamiltonian formalism

I think people typically say that path integral formalism is manifestly Lorentz-invariant, because Lagrangian density is Lorentz-invariant. However, path formalism is typically defined with time ...
0
votes
1answer
63 views

How does effective potential transform under coordinate transformation?

Let us say we have an equation of motion of the following form, $$\ddot{x}=g\tag{1}$$ For this system an effective potential can be defined as, $$\ddot{x}=-\dfrac{d}{dx}U_\text{eff}$$ $$U_\text{...
2
votes
1answer
84 views

Action in curved space

I am reading Carroll's book on GR and am confused about the generalization of the action principle to curved space. Please refer to the snippet from the book below. After writing equation 4.47, we ...
2
votes
1answer
59 views

Computing the derivatives of a Lagrangian on a Riemannian manifold

Consider a $Riemannian\space n-dimensional\space manifold$ with coordinates {$x^i$} $(i=1,...,n)$. Let the arc length parameter be $t$. So that $\frac{d}{dt}x^i(t)\equiv\dot{x}^i(t)$ is the usual ...
1
vote
2answers
115 views

Geometrical picture of change of coordinates in case of Lagrangian

I was reading the part that Euler-Lagrange equation holds even on changing the coordinates. In the book by David Morin, the author talks of geometrical picture of the change of coordinates. He makes ...
2
votes
0answers
141 views

Lagrangian transformation [closed]

I want to prove that the Lagrangian transformation is covariant for $x_{i}\rightarrow q_{i}(x)$ and $x_{i}\rightarrow q_{i}(x,t)$. So far I've proven that it holds for the first transformation as ...
0
votes
1answer
149 views

Confusion regarding the $\partial_{\mu}$ operator

I've been confused about the $\partial_{\mu}$ operator. Peskin and Schroeder defines it as $\partial_{\mu} = \frac{\partial}{\partial x^{\mu}}$ For example, the Euler Lagrange equation of motion is ...
1
vote
1answer
458 views

Transformation of generalized coordinates

One of the advantages of Lagrangian formulation is that the equations of motion have the same form regardless of the choice of generalized coordinates. Suppose that a system has $s$ degrees of freedom,...
6
votes
3answers
340 views

What is the purpose of emphasizing that an action is invariant under diffeomorphism?

When learning field theory and string theory, I always see physicists stress the fact that the action, which is an integral of the Lagrangian density $S(x)=\int L(x,\dot{x})dt$, is invariant under ...
5
votes
1answer
575 views

Explicitly show covariance of Euler Lagrange equations

I know that the Euler Lagrange equation (here only in 1D) $$ \left(\frac{d}{dt}\frac{\partial}{\partial\dot{x}}-\frac{\partial}{\partial x}\right)L\left(x,\dot{x},t\right)=0 $$ is invariant under (...
10
votes
1answer
1k views

Invariance of action $\Rightarrow$ covariance of field equations?

Invariance of action $\Rightarrow$ covariance of field equations? Is this statement true? I have only seen examples of this, like the invariance of Electromagnetic action under Lorentz ...
2
votes
1answer
274 views

How can we derive the gauge field Lagrangian?

I learned the gauge field Lagrangian is given in this form: $$\mathcal{L} = -\frac{1}{4} \mathrm{Tr}(F_{\mu \nu}F^{\mu \nu}).$$ But how one can derive this equation starting from defining the ...
3
votes
2answers
415 views

Momentum vector transformation

I am confused about the way momentum vector transforms in the following case: $$q_k \to q_k'= q_k + \epsilon f_k(q)$$ The Jacobian is thus $\Lambda_{ij} = \frac{\partial q'_i}{\partial q_j} \approx \...
5
votes
2answers
1k views

Covariance of Euler-Lagrange equations under change of generalized coordinates

Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways: ...
2
votes
2answers
189 views

With respect to what quantities do I vary Lagrangians in field theory?

I have recently been wondering, with respect to which quantities (covariant or contravariant) one should vary QFT Lagrangians and whether there is some rule regarding this. Let me give an example ...