Questions tagged [lagrangian-formalism]
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.
5,793 questions
0
votes
0
answers
36
views
The Equations of Motion for the Spring-Mass Model with Three Masses, Two Springs [closed]
I am considering a three mass (central mass $m_b$ and two equivalent side masses $m_r$) connected by two springs. The system is only allowed to move vertically.
The associated potential energy of the ...
2
votes
1
answer
269
views
On the generalisation of Lagrangian systems to manifolds
I'm reading V.I. Arnold's Mathematical Methods of Classical Mechanics Second Edition and in part 4: "Lagrangian mechanics on Manifolds" in page 83 for a differentable function $L:TM\to\...
- 613
3
votes
0
answers
148
views
Spontaneous symmetry breaking in classical mechanics
Consider the following Lagrangian $L: \mathbb{R}^2\times T(\mathbb{R}^2)\times \mathbb{R}\rightarrow \mathbb{R}$,
$$L = \underbrace{\frac{1}{2}(\dot{x}^2+\dot{y}^2)}_{T\left(\dot{x}, \dot{y}\right)}-\...
- 337
3
votes
4
answers
706
views
Struggling with understanding generalised coordinates
I'm finishing up my first semester of Classical Mechanics. My professor wrote something to the effect of
The main idea behind the Calculus of Variations is given a set of generalised coordinates $\{...
- 31
-1
votes
1
answer
71
views
What is the expectation value of the action functional? Is it zero?
I am trying to compute
$$\langle S\rangle = \frac{1}{Z}\int [\mathcal{D}\phi]~S[\phi]~e^{iS[\phi]/\hbar}$$
Can one answer this question via some symmetry or formal arguments?
- 1,895
3
votes
2
answers
111
views
Proof that Higgs mechanism Lagrangian is invariant under the unbroken gauge group
When a gauge symmetry with gauge group $G$ is spontaneously broken by the Higgs mechanism, there may be a subgroup $H \subset G$ which leaves the ground state invariant.
I believe it is always true ...
- 391
2
votes
1
answer
64
views
Higgs Mechanism: Can different VEVs produce different residual gauge groups?
In the theory of the electroweak force, a classic example of the Higgs mechanism, the Lagrangian is given by
$$\mathcal{L} = \frac{1}{2g^2}F_{\mu\nu}F^{\mu\nu} + (D_\mu\phi)^\dagger D^\mu \phi+\mu^2\...
- 391
5
votes
3
answers
285
views
Stress tensor (density?) in curved space
The stress tensor is defined as
$$ T^{\mu\nu} = \frac{2}{\sqrt{-g}} \frac{\delta S}{\delta g_{\mu\nu}} $$
where $S$ is the action depending on some fields on a curved space with metric $g$.
I'm ...
- 3,468
3
votes
1
answer
117
views
Constraints on symmetries for Lagrangian formulation of Noether's theorem
I have an issue with the classical Lagrangian derivation of Noether's theorem, it seems to that there are infinitely many symmetries of a system. My issue stems from defining $\delta q = F(q, \dot{q}, ...
- 33
0
votes
0
answers
45
views
Derivation of stress-energy tensor with generic metric for a perfect fluid
I'm trying to understand the derivation of the stress-energy tensor of a perfect fluid with a generic metric, namely, $$T_{\mu\nu} = (\rho + P) u_{\mu}u_{\nu} + Pg_{\mu\nu}.$$
I've seen derivations ...
- 1,147
5
votes
1
answer
351
views
What form does kinetic energy take in the classical (non-relativistic) Lagrangian?
In classical (non-relativistic) mechanics, the Lagrangian often takes the form
$$L(\boldsymbol{q},\dot{\boldsymbol{q}},t) = K(\boldsymbol{q},\dot{\boldsymbol{q}},t) - U(\boldsymbol{q},t),$$
assuming ...
7
votes
3
answers
616
views
Why does gauge symmetries cause differential operator in action to be singular?
In order to get a generating functional for electromagnetism, we use integration by parts to obtain:
$$S = -\frac14 \int d^4 x \ (\partial_\mu A_\nu - \partial_\nu A_\mu)(\partial^\mu A^\nu - \partial^...
- 145
6
votes
2
answers
404
views
D'Alembert's Principle and its relation to the derivation of Lagrangian mechanics
In this YouTube video "Lagrangian Mechanics: when theoretical physics got real" by Dr. Jorge S. Diaz showing the derivation of Lagrangian mechanics, d'Alembert's principle, which generalizes ...
- 490
1
vote
1
answer
311
views
Doubt in Lagrangian formalism
I have a the solution to my system as $\psi(x,z)$ and $\psi^*(x,z)$, while doing some Lagrangian related calculations (while referring to 'Numerical approaches for solving the nonlinear Schrödinger ...
- 121
0
votes
0
answers
48
views
Lagrangian Mechanics Normal Modes Substitution [duplicate]
I have the following Lagrangian, which I am supposed to put in normal mode form:
$$
L = \frac{1}{2} m\ell^2 \left( 2\dot{\theta}^2 + (\dot{\theta} + \dot{\phi})^2 + (\dot{\theta} + \dot{\chi})^2 \...
