Linked Questions
10 questions linked to/from Theory invariance after substitution of theory's field equations back into theory's action functional?
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Using EOM in QED Lagrangian [duplicate]
Let's have the QED Lagrangian.
$$\mathcal{L} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} + \bar{\Psi}(i\partial_\mu \gamma^\mu - m)\Psi + g\bar{\Psi}A_\mu \gamma^\mu \Psi.\tag{1}$$
The equations of motion are:...
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Can the Lagrangian be written as a function of ONLY time?
The lagrangian is always phrased as $L(t,q,\dot{q})$.
If you magically knew the equations $q(t)$ and $\dot{q}(t)$, could the Lagrangian ever be written only as a function of time?
Take freefall for ...
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Lagrangian of an effective potential
If there is a system, described by an Lagrangian $\mathcal{L}$ of the form
$$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$
where $T$ is the kinetic ...
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In a Lagrangian, why can't we replace kinetic energy by total energy minus potential energy?
TL;DR: Why can't we write $\mathcal{L} = E - 2V$ where $E = T + V = $ Total Energy?
Let us consider the case of a particle in a gravitational field starting from rest.
Initially, Kinetic energy $T$ is ...
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How can you solve this "paradox"? Central potential
A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one.
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When is numerical value of Lagrangian evaluated on-shell a full differential?
I noticed recently that for many field equations, Lagrangian evaluated on-shell (i.e. using equations of motions) is a full derivative- a divergence or something, or in other words a boundary term. ...
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Solving 3D Kepler Problem substitution goes wrong
I'm trying to arrive at the effective potential equation in Kepler Problem using Routh reduction method. We can procede in two ways, either using polar coordinates in the plane where the orbit happens ...
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Inconsistency? Lagrangian with its Euler–Lagrange equation as condition
Consider the action
$$A_{1} = \int{L(q, \dot{q})}{dt}\tag{1}$$
and the corresponding Euler–Lagrange equation
$$\frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{q}...
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Consistency of substitution of a canonical variable from EoM back into (momentum-less) action
I was reading this answer, where the issue of substituting equations of motion (eoms) into the action is addressed. I am fine with the basic idea that the action principle is destroyed when the eoms ...
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Weird sign in EOM: Centripetal vs. centrifugal term [duplicate]
Something goes wrong when I was deriving the equation of motion in Kepler's probelm, as below,
Angular momentum conservation $L = Mr^2\dot{\theta}^2$.
And Lagrangian is $L = \frac{1}{2}M(\dot{r}^2 + ...
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