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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Writing a non-minimally coupled Einstein-Maxwell action
Usually you study a GR system with an electromagnetic field using the standard action
\begin{equation}
S=\int{(R-\frac{1}{4}F^2)\sqrt{-g} d^4 x}
\end{equation}
(where $F_{\mu\nu}=A_{\mu,\nu}-A_{\nu,\...
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1answer
88 views
Is there anything natural about the principle of “stationary action”?
In Taylor's classical mechanics, he derived Lagrange equations and showed that they are equivalent to Newton's second law. Then, it was obvious that Lagrange equations are similar to the Euler-...
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1answer
57 views
Derivation of Euler-Lagrange Equations
I am studying the Euler Lagrange equations and have some problems understanding its derivation.
Consider a path $y(x)$ where a slight deviation from the path is given by $$Y(x,\epsilon) = y(x) + \...
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17 views
Higher-order variation of an action
In general relativity, the first-order variation of a point particle action gives the geodesic equation while a second-order variation gives the geodesic deviation equation. Similarly, is there any ...
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1answer
54 views
When can I treat derivative as a fraction? (Brachistochrone)
My teacher was solving the Brachistochrone problem in class. She parametrized the required path with $x(y)$, then said $T=\int_0^Tdt=\int_{y_1}^{y_2}\frac{dt}{dy}dy=\int_{y_1}^{y_2}\frac{dy}{dy/dt}$. ...
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1answer
59 views
Does modifying the geodesic Lagrangian $L$ with a smooth function $f(L)$ give the same geodesic curves as solutions?
Mathematical side of the problem
Given the metric
$$ds^2 = dr^2+r^2d\theta^2+r^2\sin^2\theta d\varphi^2$$
we can easily construct the action of a free particle
$$S=\alpha \int d\tau \underbrace{\sqrt{\...
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I have problem understanding something from the variational principle for free particle motion (James Hartes' book, chapter 5)
I am currently studying general relativity from James Hartle's book and I have trouble undestanding how he goes to equation (5.60) from equation (5.58). It's about the variational principle for free ...
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2answers
80 views
Lagrangian for scalar field in terms of klein Gordon equation
I am Studying Peskin and Schroeder, at page 287 , Lagrangian for scalar field is
$$L={1\over 2}(\partial _\mu \phi )^2-{1\over 2}m^2 \phi^2.$$
It can be rewritten as
$$L={1\over 2} \phi (-\...
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0answers
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Boundary terms in AdS space
Given the metric in AdS space
$$
ds^2=\frac{r^2}{L^2}(-dt^2+d\vec{x}^2)+\frac{L^2}{r^2}dr^2
$$
I am trying to calculate the variation of the action of the KG equation in this metric.
What would be ...
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2answers
50 views
Why is this integral that comes up during the derivation of the Euler-Lagrange equation equal to 0 only if the integrant is 0?
I just learned about the derivation of the Euler-Lagrange equation and I couldn't understand the last step which is looking at when this integral is equal to 0.
$$\int_{t_1}^{t_2}\left(\frac{\partial ...
2
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1answer
61 views
What actually IS action? [duplicate]
I am delighted that the the whole of physics derives from a single simple principle. Since the time of Lagrange, the principle of least action has not only become the founding principle of classical ...
0
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1answer
56 views
Issue with a derivation of the Hamilton-Jacobi equation
I'm trying to derive the HJ the easiest way I can but some issues come up.
$$\mathrm{dS}=\dfrac{\partial S}{\partial q}\mathrm{d}q+\dfrac{\partial S}{\partial t}\mathrm{d}t\Rightarrow\displaystyle{S=\...
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0answers
32 views
Legendre transform in classical mechanics and statistical entropy maximization
I am trying to get some understanding of convex optimization, and in particular of why the Legendre transform appears in certain optimization problems. I am particularly interested in two examples, ...
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1answer
51 views
Einstein equations in 2 dimensional dilaton-gravity theories
In Ref. as Jensen, the model Eq.(12) does not contain any kinetic term for the field $\varphi$:
\begin{equation}
S = \int d^2 x \sqrt{-g} \left( \varphi R + U[\varphi] \right)
\end{equation}
The ...
3
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4answers
133 views
Einstein-Hilbert action does not yield the same results as the Einstein field equations for a given metric
I am trying to derive the second-order equations of motion for a metric variable using two approaches: the formal vacuum Einstein field equations (with $T_{\mu\nu}=0$)
$$G_{\mu\nu}=R_{\mu\nu}-\frac{1}...
2
votes
1answer
32 views
How does derivation of Lagrange equation from d’Alembert principle differ from the derivation of it from principle of least action?
Using d’Alembert principle, one doesn't require any assumption like the one made in other case where particle has to follow the path of least action.
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1answer
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How do we get Maupertuis Principle from Hamilton's Principle?
Maupertuis principle says that if we know the initial and final coordinates but not time, the total energy and the fact that energy is conserved, we can choose the "right" path from all mathematically ...
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1answer
46 views
Evaluation of an integral in the variational treatment of the ground state of $\rm He$ atom
The hamiltonian of the electrons in an He atom, in CGS units, is
$$
\hat{H} = -\frac{\hslash^2}{2m}\left(\nabla_1^2 + \nabla_2^2\right) - Ze^2\left(\frac{1}{r_1} + \frac{1}{r_2}\right) + \frac{e^2}{r_{...
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4answers
40 views
Is there any meaning for the path found by Hamilton's Principle to an impossible state?
Hamilton's principle is written as a statement about the path taken between two states of the system which occur. Is there any meaning found in solving the variational problem for points which can ...
3
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2answers
80 views
Assumptions reg. Kinetic energy and Potential energy in the Lagrangian formulation
I have recently been introduced to Lagrangian mechanics. My previous exposure to Lagrangian math has been in the form of optimizing constrained functions using Lagrange multipliers.
I get the math ...
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1answer
87 views
Legendre transformation in QFT
I know that given the Hamiltonian of a theory, there can be many different associated Lagrangians, or even none at all, but why is that so? In classical mechanics the Hamiltonian and Lagrangian ...
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1answer
82 views
QFTs without Lagrangian
I have been reading other questions in this site, but I have not found answers to all my questions about theories without Lagrangians.
What do we mean exactly when we say that they do not have a ...
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How to prove that a sphere has the minimum gravitational potential energy for a given volume of uniformly dense material? [duplicate]
Planets are spherical (roughly) in shape because a spherical configuration gives the minimum gravitational potential energy.
How does one prove that the sphere is the shape that gives minimum ...
2
votes
1answer
60 views
The gothicized metric and the Palatini formalism
In the Palatini formalism of GR, we had two results treating the metric $g_{\mu\nu}$ and the connection $\Gamma^\alpha_{\mu\nu}$ separately as dynamical variables, which are
The vaccum field ...
3
votes
1answer
30 views
Can we construct a system in which two distinct paths give the same actions? If so, how does the system evolve? [duplicate]
Say we construct the Lagrangian for a system and minimise the action, what happens if this is not unique? In other words the action is minimised by two distinct (not infinitesimally separated) paths. ...
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88 views
Variational principle with $\delta I \neq 0$
In Covariant Phase Space with Boundaries D. Harlow allows boundary terms in the variation of the action. If we have some action $I[\Phi]$ on some spacetime $M$ with boundary $\partial M = \Gamma \cup \...
4
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1answer
71 views
Can you numerically compute a trajectory by direct minimization of the action functional?
Is there a numerical approach to compute simple projectile motion by directly minimizing the action functional?
I was thinking that the trajectory is essentially a least cost path through phase ...
2
votes
1answer
107 views
Is there anything wrong with this modified Einstein-Hilbert action of first order?
The Einstein-Hilbert Action:
$$S_{EH}=\frac{1}{2\kappa}\int \sqrt{-g}g^{ab} \left({\Gamma^c}_{ab,c} - {\Gamma^c}_{ac,b} + {\Gamma^d}_{ab}{\Gamma^c}_{cd} - {\Gamma^d}_{ac}{\Gamma^c}_{bd}\right) d^4x$$
...
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2answers
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Principle of least action with non-conservative forces?
See this excerpt from Kinematic and Dynamic Simulation of Multibody Systems page 122-123:
Consider a system characterized by a set of $n$ independent coordiunates $q_i$. Let $L=T-V$ be the system ...
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Deduction of the action of $f(R)$ gravity
From $$S=\int{d^4}x\sqrt{-g}f(R)$$
I want to deduce the $f(R)$ field equations which are:
$$f'(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}-[\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box]f'(R)=\kappa T_{\mu\nu} ,$...
1
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1answer
74 views
Deriving Euler-Lagrange equations for generalized coordinates without “virtual work”?
I have been reading "Classical mechanics" by Goldstein, Poole, and Safko. In particular, the section on "D'alembert's principle and lagrange's equations", in which the principle of virtual work is ...
5
votes
4answers
184 views
Deeper Meaning to the Nature of Lagrangian
Is there a more fundamental reason for the Classical Lagrangian to be $T-V$ and Electromagnetic Lagrangian to be $T-V+ qA.v$ or is it simply because we can derive Newton's Second Law and Lorentz Force ...
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1answer
56 views
Guessing eigenvalue solution
I am reading The Theory of Magnetism I, by Mattis. In Chapter 2, he proposes the following eigenproblem:
$$ \left ( \begin{matrix} V & U \\ U^\dagger& V \end{matrix} \right ) \left ( \begin{...
3
votes
1answer
51 views
How does a falling rock minimize action?
Consider a single two dimensional system with a rock that is influenced by gravity. The Action of this system is defined as $\int_0^\infty [T(\dot x(t))-V(x(t))]dt$, where $T$ is the kinetic and $V$ ...
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56 views
The nature of the variation of the metric tensor and the Christoffel symbols
We needed to define the covariant derivative of tensors to preserve its nature as tensors after differentiating, I mean we are so careful when we apply anything to geometric objects. Now I don't get ...
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2answers
134 views
Why boundary terms make the variational principle ill-defined?
Let me start with the definitions I'm used to. Let $I[\Phi^i]$ be the action for some collection of fields. A variation of the fields about the field configuration $\Phi^i_0(x)$ is a one-parameter ...
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1answer
45 views
What path will the system take if there are more than one path for which the action integral takes a stationary value?
Hamilton's principle states that "The true evolution $q(t)$ of a system described by $N$ generalized coordinates $q = (q_1, q_2, ..., q_N)$ between two specified states $q_1 = q(t_1)$ and $q_2 = q(t_2)...
0
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1answer
58 views
Problem in deriving Electromagnetic tensor
I'm having troubles in understanding a mathematical step in the derivation of the electromagnetic tensor. In Landau&Lifshitz's book I found that the action of a particle in an electromagnetic ...
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0answers
67 views
Is there a minimum principle which governs neuronal activity in brain function? [closed]
Reinforcement learning is a well established field which attempts to explain the behavior and decision-making of toddlers, among other agents. The core principle of this theory is based on dynamic ...
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votes
4answers
122 views
Problem with Klein-Gordon equation derivation
In Notes for a course on Classical Fields by R. ALdrovandi, one the the exercises in page 94 is to derive the klein Gordon equation $(\Box + m²)\phi = 0$ from the following lagrangian density
\begin{...
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1answer
58 views
How do the Euler-Lagrange equations generalise to an arbitrary manifold?
So every formalism for the EL equations I have seen relies on choosing a coordinate chart. However, if we had say, a field on a sphere, then we can’t have global coordinates.
How, in principle, ...
3
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1answer
46 views
Given equations of motion, how can we check if there is a lagrangian from which we can derive them? [duplicate]
Suppose we are given a set of equations of motion for $N$ bodies, which generically will go like this
\begin{equation}
\frac{d^2\mathbf{r}_i}{dt^2}=\mathbf{F}_i \left((\mathbf{r}_i)_{1 \leq i \leq N},...
1
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1answer
70 views
Why do we get this formula for the action for Maxwell theory? [duplicate]
I have a question on the variational formulation of electromagnetism.
I read that the action is given by
$\mathcal{S}_{EM} = \int d^4x\ F_{\alpha \beta} F^{\alpha \beta}$ (1)
and the source term ...
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0answers
37 views
Lagrangian via equation of the motion [duplicate]
If I have a lagrangian, then I can get the equations of the motion using the Euler-Lagrange equations. My question is about the converse of this statement: If one knows the equations of motion, like $�...
3
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1answer
58 views
Solving projectile motion using least action principle and level sets
I'm trying to compute 1D projectile motion -- basically throwing a ball up and catching it in the same hand. I want to use Lagrangian dynamics and find a numerical solution out of interest.
I ...
0
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1answer
89 views
Why do we integrate over the Lagrangian to get action?
As action is defined as
$$S = \int_{t_1}^{t_2}{\mathcal{L}(q,\dot{q},t)}dt $$
For any time interval $(t_1, t_2)$.
As $t_1$ and $t_2$ are arbitrary $t_2$ can be taken arbitrarily close to $t_1$ and ...
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1answer
54 views
Adding a term to the Lagrangian Density without changing Lagrange's Equations
I'm stuck on a problem in the last chapter of Hamill's Student's Guide to Lagrangians and Hamiltonians. It asks why adding:
$$
\frac {\partial \mathscr{L}} {\partial t} + \frac {\partial \mathscr{L}} ...
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0answers
49 views
Why Principle of least action is true? [duplicate]
Why the principle of least action is true? I mean it is equivalent with Newton's laws(Lagrangian and Newtonian equations are equivalent) but is that the way it was formulated?
I read somewhere the ...
0
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1answer
28 views
Smaller elapsed time with higher velocity, but free fall maximize elapsed time, who clearly have a velocity compare to a stationary object
I am a little confused because an object with velocity would experience smaller elapsed time compared to an object that is not. But in GR elapsed (proper) time is maximized by free fall who have ...
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51 views
Why is tensor networks applicable only to ground states?
When one uses tensor network as a wave function ansatz for a variational method, we usually use this scheme to find a ground state. Why can’t we apply the tensor network formalism to find any excited ...
