For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.
2
votes
2answers
100 views
Principle of Least Action via Finite-Difference Method
I have to be honest, the principle of least action seems to me more of a religious claim one takes on complete faith, though of course I'm hoping this is just because I don't understand it. I tried to ...
-2
votes
0answers
20 views
Place in Europe where there are research groups with anti-de-Sitter geometry or minimal Lagrangian mappings [closed]
I am not sure whether this question is appropriate for this forum, but I am a mathematician with Ph.D. looking to apply to physics departments in Europe as a postdoctoral researcher where there are ...
1
vote
1answer
58 views
Is this a valid derivation of the Legendre transformation from the Euler-Lagrange condition
E-L condition:
$$\frac{d p}{dt}=\frac{\partial L}{\partial q}$$
Where $p=\frac{\partial L}{\partial \dot{q}}$
Are the following steps valid:
$$\frac{\partial q}{dt} dp=\partial L$$
$$\dot{q} \: ...
1
vote
1answer
73 views
Question about Lagrangian mechanics
I have a question about the Lagrangian-formalism; they state that the holonomic constraints are expressed as:
$$f_\alpha(x_1,y_1,z_1,...,x_n,y_n,z_n,t) = 0 $$ where $\alpha = 1,...,L$
They state ...
0
votes
3answers
100 views
How is a Hamiltonian constructed from a Lagrangian with a Legendre transform
many textbooks tell me that Hamiltonians are constructed from Lagrangians like
$$L=L(q,\dot{q})$$
with a Legendre transformation to obtain the Hamiltonian as
$$H=\dot{q}\frac{\partial L}{\partial ...
3
votes
1answer
60 views
Lagrangian contact interaction
Conside a contact interaction given by a delta function on their worldlines. Use a gauge fixed Lagrangian for two point particles in terms of their proper times $t$ and $t'$. Is it possible to find ...
-1
votes
1answer
60 views
Solving differential Equation for the Two-Body Problem
So, I'm following the derivation in D. Morin, Introduction to Classical Mechanics, of the equations for a two-body system. I understand all of it, aside from this one step.
When he's talking about ...
0
votes
1answer
76 views
Proper time of circular motion under Schwarzschild metric
I'm trying to calculate the proper time of a massive particle circulating Schwarzschild black hole, using EL equation of the following Lagrangian:
...
1
vote
1answer
84 views
Why do Lagrangians and Hamiltonians give the equations of motion? [duplicate]
I remember asking my second year Mechanics teacher about why do the Lagrangians give the equations of motion. His answer was that there is no answer to that, it is an empirical fact, and that asking ...
0
votes
0answers
18 views
Stability of trajectory of disc which moves along a straight curve
Let's have a disc which moves along a straight curve on a plane in a uniform gravitational field. There need to discover the stability of it's trajectory.
I represented the possible deviation of the ...
8
votes
2answers
139 views
Why are they called “cyclic” coordinates?
In Lagrangian formalism, when $\frac{\partial L}{\partial q} = 0$, the coordinate $q$ is called cyclic and a corresponding conserved quantity exists. But why is it called cyclic?
1
vote
0answers
44 views
Trouble with calculating Christoffel symbols of FLRW metric using Lagrangian method
The FLRW metric which I am using is $$ds^2 = dt^2 - \frac{a(t)^2}{c^2} \left( dx^2 + dy^2 + dz^2 \right)$$
where $a(t)$ is the so-called 'scale factor'. I did not want to calculate the Christoffel ...
7
votes
2answers
170 views
No closed orbits for a Newtonian gravitational field in 4 spatial dimensions
We are supposed to show that orbits in 4D are not closed.
Therefore I derived a Lagrangian in hyperspherical coordinates
$$L=\frac{m}{2}(\dot{r}^2+\sin^2(\gamma)(\sin^2(\theta)r^2 \dot{\phi}^2+r^2 ...
1
vote
0answers
26 views
Proving conservation of angular momentum in an elliptic billiard problem
This is for a course focusing on the connections between Newtonian, Lagrangian and Hamiltonian formalisms.
We're given an elliptic billiard table with foci 1 and 2, where $L_1$ and $L_2$ are the ...
0
votes
1answer
36 views
Kinetic rotational energy of a bar hooked to a coil
I have solved an exercise and I'd like to know if my proceeding about finding kinetic energy is correct or not, because this is the first time that I "meet" a situation like this.
"A bar has mass $M$ ...
0
votes
1answer
23 views
Doubt about coordinates and point of equilibrium
I'm solving an exercise about small oscillations and I have a doubt about coordinates that I have to use.
This is the text of the exercise:
"A bar has mass M and lenght l. Its extremity A is hooked ...
0
votes
0answers
39 views
Simple Calculus angular velocity problem [closed]
I have a Lagrange equation and I have no idea why the solution note I looked at have got
$d \over {dt}$$(mr\omega) = Q$,
$d \over {dt}$$(mr\ddot\omega) = Q$.
I had
$d \over {dt}$$(mr\dot\omega) = ...
1
vote
2answers
52 views
Relativistic Lagrangian transformations
I need to study the relativistic lagrangian of a free particle.
It's
$\ L= - m c^2 \sqrt[2]{1- \frac{|u|^2}{c^2}} $ I need to study the translation, boost and rotation symmetry. I say it doesn't ...
0
votes
1answer
88 views
Derivation of Dirac equation using the Lagrangian density for Dirac field
How can I find Dirac equation using the Lagrangian density for Dirac field?
0
votes
0answers
35 views
Lagrangian with a general constraint [closed]
Can any body help me out to solve this problem?
I am familiar with mechanism of Lagrangian and I can solve some problems with constraints but this one is really hard to solve.
1
vote
1answer
99 views
Retrieving Maxwell's equations from the minimum action principle
I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps.
Starting with the action:
$$S = \int dt \int ...
0
votes
0answers
40 views
Lagrangian of electromagnetic tensor in light cone coordinates? [closed]
I have Lagrangian Density of Electromagnetic field Tensor in light cone coordinates using D'Alembertian operator and Lagrangian density in Cartesian coordinates. I couldn't figure out the way to ...
0
votes
1answer
67 views
Why does Lagrangian of free particle depend on the square of the velocity ?
Why does Lagrangian of free particle depend on the square of the velocity ?
For example, $L(v^4)$ also doesn't depend on direction of $v$.
5
votes
0answers
75 views
Optimal tunnel shape for travelling inside the earth [duplicate]
Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other ...
4
votes
2answers
88 views
Independent systems and Lagrangians
Definition 1:
The notion of independent systems has a precise meaning in probabilities. It states that the (joint) probability or finding the system ($S_1S_2$) in the configuration ($C_1C_2$) is ...
0
votes
0answers
34 views
2 Masses attached to the roof by strings [closed]
There are two masses $M>>m$. $M$ is attached from a fixed surface by a string and ;m'mass is attached to $M$ by a string. Lengths L1, L2, c/s area A1, A2 of cords are given.I have to find ...
0
votes
1answer
87 views
A Type of Pendulum
Is there any chance that $$rtl(\ddot\omega+\ddot\phi)\cdot\sin{(\phi+\omega t)}- gl\dot\phi \cdot \sin{\phi} + ltr\dot\omega(\dot\phi^2-\dot\omega)\cdot \cos{(\phi+\omega t)}-gtr\cos{(\omega ...
2
votes
1answer
78 views
Determinant for a coupled fluctuation Lagrangian
Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the ...
1
vote
1answer
52 views
Total energy is extremal for the static solutions of equation of motions
In physics total energy is extremal for the static solutions of equation of motions.
Can anyone explain this sentence to me?
2
votes
3answers
153 views
What is the mathematical justification for the quadratic approximation to the energy of a spring in a one-dimensional lattice?
It follows easily from this draw, the length $l$ of this spring as a function of the vertical distance $x$, as $l(x)=\sqrt{1+x^{2}}$
Now, $l$ can be expressed as a MacLaurin expansion:
$$l(x) = ...
2
votes
3answers
150 views
Lagrangian mechanics and time derivative on general coordinates
I am reading a book on analytical mechanics on Lagrangian. I get a bit idea on the method: we can use any coordinates and write down the kinetic energy $T$ and potential $V$ in terms of the general ...
1
vote
1answer
113 views
Potential in Relativistic Scalar Field Theory
My intention is to establish a Soliton equation. I have cropped a page from Mark Srednicki page no 576.
I have understand the equation (92.1) but don't understand that how they guessed the ...
2
votes
2answers
118 views
Different approaches to calculating the Christoffel symbols
I would be very grateful to whoever can debug the following calculations...
We have the metric for static spacetime: $$ds^2 = -\exp(2U(\vec x))dt^2+h_{ij}(\vec x) d x^i d x^j$$
I want to find the ...
2
votes
2answers
98 views
Higher order covariant Lagrangian
I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at ...
1
vote
0answers
44 views
Lagrangians for non-local equations of motion
Say I have a multicomponent field $X_a(x,t)$ such that I know it Fourier modes satisfy the following equation of motion,
$(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(k,t) = e^t \int \frac{d^3p ...
2
votes
2answers
112 views
Lorentz invariance of the action for free relativistic particle
I tried to check the Lorentz invariance of the standard special relativity action for free particle directly: ($c=1$)
$$
S=\int L dt=-m\int\sqrt{1-v^{2}}dt
$$
Lorentz boost:
$$ ...
2
votes
2answers
63 views
Where is the magnetic self energy term in $L$ for a charged particle in an electromagnetic field?
In the Lagrangian for a charged particle in an electromagnetic field
$$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$
the energy of the particle is contained in the kinetic term, ...
1
vote
0answers
115 views
Deriving torque from Euler-Lagrange equation
How could you derive an equation for the torque on a rotating (but not translating) rigid body from the Euler-Lagrange equation? As far as I know from my first class in Classical Mechanics, there is ...
0
votes
2answers
118 views
A small oscillations of a rod on the cylinder
Let's have the next case.
A rod (with mass $m$, length $L$ and a momentum of inertia $I$) at the initial time is located on a cylinder (with radius $R$) surface so that it's (rod's) center of mass ...
-1
votes
1answer
116 views
Lagrangian formulation for relativistic case
Lagrangian for a real scalar field:
$$\mathcal{L}=\frac{1}{2}\eta^{\mu \nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^2\phi^2 $$
Can someone simply drive me how can I write it from ...
1
vote
2answers
128 views
How the boundary term in the variation of the action vanishes
Can someone explain a little more that why the last term in equation (1.5) vanishes?
Reference:
David Tong, Quantum Field Theory: University of Cambridge Part III Mathematical Tripos, Lecture ...
1
vote
3answers
179 views
Must the Lagrangian always be known for the Euler-Lagrange equations to be of any use?
When studying classical mechanics using the Euler-Lagrange equations for the first time, my initial impression was that the Lagrangian was something that needed to be determined through integration of ...
5
votes
1answer
88 views
Does anybody know of any good sources that explain (generically) how we form Lagrangians/Actions/Superpotentials for different field content?
I regularly find that I'll understand where the field content in a particular physics paper comes from, but then a Lagrangian or action or superpotential is stated and I don't know how it's derived. ...
2
votes
1answer
82 views
Varying an action (cosmological perturbation theory)
I am stuck varying an action, trying to get an equation of motion. (Going from eq. 91 to eq. 92 in the image.)
This is the action
$$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$
...
0
votes
0answers
97 views
Small oscillations: diagonal matrix [closed]
I'm solving an exercise about small oscillations.
I name $T$ the kinetic matrix and $H$ the hessian matrix of potential.
The matrix $\omega^2 T- H$ is diagonal and so find the auto-frequencies is ...
0
votes
1answer
45 views
if i want action to be positive number then it require that $\tau_i$ be bigger than $\tau_f$, isn't it true? [closed]
the action is the length of the geodesic
$S=-E_o\int_i^f d\tau$
we get an action that is minimised for the correct path.
if i want action to be positive number then it require that $\tau_i$ be ...
4
votes
2answers
181 views
Why is the Lagrangian quadratic in $\dot{q}$? [duplicate]
My teacher said we only consider Lagrangians which are quadratic in $\dot{q}$, and we don't take other Lagrangians. I couldn't understand why. Can anyone please explain this?
6
votes
5answers
275 views
Form of the Classical EM Lagrangian
So I know that for an electromagnetic field in a vacuum the Lagrangian is $\mathcal L=-\frac 1 4 F^{\mu\nu} F_{\mu\nu}$, the standard model tells me this. What I want to know is if there is an ...
3
votes
3answers
141 views
Virtual differentials approach to Euler-Lagrange equation - necessary?
I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for the Euler-Lagrange equation. The whole notion of, and ...
2
votes
2answers
187 views
Null geodesic given metric
I (desperately) need help with the following:
What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$
I don't know how to transform a metric into a geodesic...! There is no need to ...
