Questions tagged [variational-principle]
Any of several principles that find the physical trajectory of a system by minimizing or maximizing some value computed over the proposed path (for instance geometric optics can be reproduced by insisting on a minimum time principle).
1,228 questions
6
votes
4
answers
317
views
Is there a version of Hamilton's Principle of Stationary Action when only initial conditions are known and the final end state is unknown? [duplicate]
Consider a dynamical system with Lagrangian $L$ and configuration space $X$, we are interested in trajectories of this system over a time interval $q:[t_0,t_1]\rightarrow X$.
When one has the ...
- 153
0
votes
0
answers
52
views
EOM of Nambu-Goto in second fundamental form
I am computing the EOM of the Nambu-Goto action $$S[X] = -T\int d^2 \sigma \sqrt{-\det{(\partial_a X^\mu \partial_b X_\mu)}}$$ and I want to write this in a specific form using the second fundamental ...
- 688
0
votes
0
answers
75
views
Deriving equations of motion in Lagrangian mechanics with semi-holonomic constraints
I am trying to understand the derivation of the equations of motions in Lagrangian mechanics in the presence of constraints. I believe the idea is just to apply the Hamilton's principle (the actual ...
- 1,783
1
vote
1
answer
130
views
Classical Equivalent of time evolution given by on-shell action?
In Quantum Mechanics, the time evolution of an observable in the Heisenberg picture is determined by the Dirac bracket with the Hamiltonian operator
$$ i\hbar\frac{d}{dt}\hat{\mathcal{O}}(t)=[\hat{\...
- 4,936
2
votes
1
answer
93
views
Getting the symplectic form from a double variation of the action functional
I was trying to get the Hamilton-Jacobi equations starting from the phase-space version of the action and I run into some problems. Let's start from the beginning:
Action principle in coordinate space
...
- 4,936
-1
votes
1
answer
149
views
Why isn’t Lagrangian defined as only kinetic energy?
In classical physics, as per the principle of least action, if an object moves from $x=x_i$ at $t=t_i$ to $x=x_f$ at $t=t_f$, energy is conserved and all possible paths within the specified time $t_f-...
- 351
0
votes
0
answers
60
views
Proving that $L = T - U$ without using Newton's second law [duplicate]
I was recently reading Landau & Lifshitz's first book on mechanics and I'm having trouble trying to prove something that he gives for granted.
After stating that the Lagrangian of a system is a ...
- 1
2
votes
0
answers
110
views
Euler-Lagrange Equations under constraint in Keldysh formalism
I have 2 questions regarding the minimization procedure of an effective action in the Keldysh formalism. In the Keldysh path integral one deals (after Keldysh rotation) with a partition function of ...
- 169
0
votes
1
answer
111
views
Light path of least time paradox [duplicate]
Light takes the path of least time when refracting from one medium to another.
I understand this principle very well in this simple scenario.
Similarly it takes the path of least time, also a geodesic,...
- 85
1
vote
2
answers
226
views
What are the conceptual foundations of Lagrangian formulation? [duplicate]
When learning Lagrangian mechanics in a standard mechanics course, it is typically introduced as an alternative formulation of classical mechanics which can be derived from Newtonian mechanics. ...
- 304
3
votes
2
answers
183
views
The contribution of higher-order derivatives on the equations of motion in QFT [closed]
I am currently preparing for a final on QFT and one of the old exam questions involves calculating the equations of motion (EOM) of a Lagrangian. Now this should be an easy question, but the ...
- 41
1
vote
1
answer
147
views
Galilean Transforms on Lagrangian of a Free Particle [duplicate]
In writing the Lagrangian of for free motion of a particle in an inertial frame, a key assumption is that the equation of motion must have the same form in every inertial frame. For velocities $\...
- 240
1
vote
2
answers
202
views
Equivalence between Hamilton's equations for a free particle and the geodesic equation
For a free particle in curved spacetime with signature $(-,+,+,+)$, consider the Hamiltonian $$H = \frac{1}{2}\left(g^{\mu\nu}p_\mu p_\nu + m^2\right).\tag{0}$$ Hamilton's equations are given by:
$$\...
- 729
1
vote
0
answers
52
views
How is the Lagrangian extracted from a given line element? [duplicate]
Given a spacetime with the line element
$$ds^2 = g_{\mu\nu}(x) \, dx^\mu dx^\nu$$
How can one extract the lagrangian?
I have seen the following formula used:
$$\mathcal{L} = \frac{1}{2} g_{\mu\nu}(x) \...
- 522
3
votes
2
answers
236
views
Why can you not take $u \cdot u = c^2$ in the relativistic free massive particle Lagrangian?
In my classical electrodynamics class, we use the Lagrangian of the relativistic free massive particle as $$L = - mc\sqrt{\dot{r}\cdot\dot{r}}.$$ Where $\dot{r}^\mu = u^\mu = \frac{dr^\mu}{d \tau}$; $...
- 1,267
6
votes
5
answers
897
views
Formulating the catenary problem as a variational problem
I am interested in studying the curve a string follows when fixed by two points and subject to a (uniform) gravitational field. Say the string has a constant length $L$ and is fixed on two points A, B ...
- 2,638
2
votes
2
answers
196
views
Understanding Hamilton's principle with constraints in section 2.4 of Goldstein, 3rd edition
I'm working through Goldstein's Classical Mechanics, 3rd edition. In section 2.4, we are extending Hamilton's Principle to a system with constraints. In the beginning of the section he makes a couple ...
- 119
1
vote
1
answer
185
views
Why is integral of relativistic action $-\alpha \int_{a}^{b} \, \mathrm ds$ minimised with respect to $\mathrm ds$?
While reading some aspects concerning the conclusion that bodies follow geodesics of spacetime, I ran into relativistic action on p.24 in chapter 2 $\S8$ of the 2nd volume of Landau & Lifshitz:
$$...
- 2,069
10
votes
5
answers
2k
views
Feynman's derivation of Euler-Lagrange equations
From Chapter 19 of Volume 2 of The Feynman Lectures on Physics, the following integral is supposed to be zero for any $\eta(t)$ I choose.
$$\delta S = \int_{t_1}^{t_2}\left[m\frac{d\underline{x}}{dt}\...
- 343
6
votes
2
answers
410
views
Levi-Civita Christoffel symbol in geodesic
I was trying to solving the geodesic equation using calculus of variations, which generally is
$$ \delta L = \delta\int \mathrm{d}t \sqrt{g_{mn}(\gamma(t))\frac{\mathrm{d}\gamma^m(t)}{\mathrm{d}t}\...
- 61
2
votes
0
answers
105
views
Is the d'Alembert's principle arbitrary and redundant? [duplicate]
D'Alembert principle states that:
$$
\sum_i \left (\mathbf F_i^{(a)}- \mathbf{\dot p}_i \right) \cdot \delta \mathbf r_i = 0
$$
but I'm not quite getting it. The derivation seems trivial for me, ...
- 2,069
5
votes
1
answer
363
views
Brachistochrone problem with initial velocity
The Brachistochrone problem is usually presented with the having a ball dropped into the slide with initially zero velocity and at position $(x, y)=(0, 0)$.
I would like to know the more general ...
- 3,075
0
votes
1
answer
103
views
Equilibrium Configuration of $n$ Positive Charges Constrained to Move on a Spherical Surface [duplicate]
I'm considering a system of $n$ positive charges that are constrained to move on a spherical surface, meaning their distances from a central point are fixed. I'm trying to determine the equilibrium ...
- 123
0
votes
0
answers
36
views
What is the simplest Lagrangian for the 1D Schroedinger equation? [duplicate]
I need the simplest 1D Lagrangian for the 1D Schroedinger equation. If you know one please
share it with me.
1
vote
2
answers
240
views
Transversality condition for Euler-Lagrange Equations with 1 variable end point
I am able to follow the derivation of the Euler-Lagrange equations, for 1 variable end point, but cannot make the final step regarding the additive term.
Specifically, I arrive at the path minima ...
