All Questions
Tagged with boundary-conditions lagrangian-formalism
98 questions
10
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5
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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.
$$\delta S = \int_{t_1}^{t_2}\left[m\frac{d\underline{x}}{dt}\...
- 333
0
votes
2
answers
84
views
Transversality condition for Euler-Lagrange Equations with 1 variable end point
I am able to follow the derivation of the Euler-Lagrange equations, for 1 variable end point, but cannot make the final step regarding the additive term.
Specifically, I arrive at the path minima ...
4
votes
2
answers
230
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Deriving Hamilton's Principle from Maupertuis' Principle
I'm trying to understand the derivation of Hamilton's Principle from Maupertuis' Principle:
In this answer, they say "$E$ is a constant". Which is consistent with:
Maupertuis's principle ...
1
vote
2
answers
70
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Independence of Hamiltonian formulation from Lagrangian formulation
The Hamiltonian of a system is defined as the Legendre transform of the Lagrangian and I was wondering if there is a way to construct the Hamiltonian formalism without ever using a Lagrangian. This ...
- 109
1
vote
1
answer
54
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Variational approach to problem $1.14$ from Goldstein
To summarize, the problem states that we prove
$$\mathcal L'=\mathcal L+\frac{\mathrm d}{\mathrm dt}F(\mathbf q,t)$$
is also a valid Lagrangian, that is, it satisfies the Euler-Lagrange equations for ...
- 450
1
vote
1
answer
84
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Invariance of the action under a symmetry of 2D isotropic harmonic oscillator
I have a question on the invariance of the action under symmetry transformation.
As the simplest example, here I consider two dimensional Harmonic oscillator. After some rescaling, the Hamiltonian can ...
- 155
6
votes
1
answer
209
views
Ambiguity for boundary conditions after conformal transformation
Abbreviations
EOM = Equations of motion
BCs = Boundary conditions
CT = conformal transformation
Intro
I was playing around a bit with EOMs, action principle, CTs and BCs. There, I met a problem. ...
- 813
4
votes
1
answer
146
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Boundary conditions on Schwarzschild event horizon
Consider the variational problem for a scalar field in Schwarzschild spacetime $M$ with respect to Eddington-Finkelstein coordinates $(v,r, \theta, \varphi)$, i.e.
$$\delta I(\phi) = \int_M dV \ \Big(...
- 813
1
vote
0
answers
38
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How does $\dot{q}_i p_i - H = \dot{Q}_i P_i - K + \frac{d}{dt}F$ will give the same EL and EoM for corresponding coords? [duplicate]
How does $\dot{q}_i p_i - H = \dot{Q}_i P_i - K + \frac{d}{dt}F$ give the same Euler-Lagrange equations and Equations of motion (EoM) for corresponding coordinates and allow us to determine a ...
- 415
2
votes
0
answers
60
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Robin conditions from action principle
Consider the Lagrangian density
$$L(\tilde{\phi}, \nabla \tilde{\phi}, \tilde{g}) = \tilde{g}^{\mu \nu} \nabla_{\mu} \tilde{\phi} \nabla_{\nu} \tilde{\phi} + \xi \tilde{R} \tilde{\phi}^2$$
with $\...
- 813
1
vote
1
answer
90
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Possible boundary conditions in derivation of Euler-Lagrange equations
Given a Lagrange density
$$\mathcal{L} = g^{ij} \phi_{,i} \phi_{,j} - V(\phi)\tag{1}$$
I have read (e.g. here) that the boundary term that occurs through variation of the action
$$ \delta I = \int_V ...
- 813
1
vote
1
answer
123
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Classical open string in Polchinski -- consistency of Neumann boundary conditions with gauge choice
In Section 1.3 of String Theory, Volume 1, Polchinski derives the open string spectrum from the Polyakov action with Neumann boundary conditions, by first considering the classical open string in ...
- 23
2
votes
2
answers
136
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Variation of the Lagrangian expressed as a time derivative of a function
In chapter 4.5 of Jakob Schwichtenberg's Physics from Symmetry, he expresses the variation of the Lagrangian $L = L\left ( q, \dot{q}, t \right )$ with respect to the generalized coordinate $q$ as
$$\...
- 45
0
votes
1
answer
132
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Boundary conditions in $\delta I=0$ to derive Einstein's equations -- why the derivatives of $g_{\mu\nu}$ are held constant?
Dirac derives Einstein's field equations from the action principle $\delta I=0$ where $$I=\int R\sqrt{-g} \, d^4x$$ ($R$ is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-...
- 524
6
votes
3
answers
2k
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Something fishy with canonical momentum fixed at boundary in classical action
There's something fishy that I don't get clearly with the action principle of classical mechanics, and the endpoints that need to be fixed (boundary conditions). Please, take note that I'm not ...
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