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).
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Variational Principle for the $\rm H_2^+$ ion
I am following the book "Introduction to the Quantum Theory" by David Park, and I've run into a problem. The text calculates the ground state potential energy of the $\rm H_2^+$ ion using ...
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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 ...
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How does the relativistic action logically follow from the nonrelativistic action, and why is proper time involved? [closed]
In nonrelativistic mechanics, the action for a particle of mass $m$ moving in a potential $V(\mathbf{x})$ is
$$
S_{\text{classical}} = \int \left(\frac{1}{2} m \mathbf{v}^2 - V(\mathbf{x}) \right) dt.\...
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EOM of Nambu-Goto in second fundamental form
I am computing the EOM of the Nambu-Goto action $$S[X] = -T\int d^2 \sigma \sqrt{-\det{(\partial_a X^\mu \partial_b X_\mu)}}$$ and I want to write this in a specific form using the second fundamental ...
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Reasoning for fixed endpoints when constructing the action from the EL equations
Given a particle in space, the EL equations give us a differential equation for determining how the partilce will move as time evolves.
I am comfortable deriving a least action principle which ...
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Deriving equations of motion in Lagrangian mechanics with semi-holonomic constraints
I am trying to understand the derivation of the equations of motions in Lagrangian mechanics in the presence of constraints. I believe the idea is just to apply the Hamilton's principle (the actual ...
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How to derive the field equations for cubic gravity in arXiv:1704.01590?
For the cubic gravity with Lagrangian
$$
\sqrt{-g} \left(R+\frac{1}{\Lambda^4}R_{\mu \nu}{ }^{\alpha \beta} R_{\alpha \beta}{ }^{\gamma \sigma} R_{\gamma \sigma}{ }^{\mu \nu}\right),
$$
where $\Lambda$...
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Classical Equivalent of time evolution given by on-shell action?
In Quantum Mechanics, the time evolution of an observable in the Heisenberg picture is determined by the Dirac bracket with the Hamiltonian operator
$$ i\hbar\frac{d}{dt}\hat{\mathcal{O}}(t)=[\hat{\...
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Hamilton-Jacobi confusion: What is the coordinate dependence of the Hamilton principal function?
The way I usually see the Hamilton-Jacobi equation derived goes like this. Start with an action principle
$$ \mathcal{S}[q]=\int L(q,\dot{q})dt.$$
Evaluate this action on a solution to the Euler-...
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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
...
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In general relativity can the Einstein-Hilbert action ever be stationary but *not* least, i.e. a saddle?
In some cases of motion, the action is not minimized, but only stationary.
Is there an example of a system described by general relativity - thus by the Einstein-Hilbert action and thus the Einstein ...
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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-...
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Does Fermat’s principle require equal optical path lengths for all rays or just minimization for each individual ray? How this applies in a GRIN lens?
My problem is about rays traveling through a GRIN medium, where the refractive index is maximum at the center and decreases quadratically outward.
Intuitively, I would expect rays starting above or ...
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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 ...
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Equations of motion from (TQFT) actions with discrete-valued cochains
I was wondering how the equations of motion (EOM) for a TQFT defined with discrete cochains actually follow from the action.
Consider for example a four dimensional space $X$ and $\mathbb{Z}_2$-valued ...
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