# Tagged Questions

The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.

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### Lagrangian for relativistic massless point particle

For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~\sqrt{ g_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of ...
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### Physical motivation for Lagrangian formalism

This is more of a request for clarification of understanding and intuition rather than a question, but I hope people can help me with it. I have learned calculus of variations and have subsequently ...
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### Higher than Lagrangian/action?

When you begin learning physics, you start with equations of motion applied to various physics systems. In classical mechanics course you learn, that exists Lagrangian/action of a system, which gives ...
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### Why is a theory Lorentz invariant if the Lagrangian is Lorentz invariant?

For if I started by trying to make the Hamiltonian Lorentz invariant, I would have failed. Indeed, the Hamiltonian is part of a covariant tensor. But how do I know that the Lagrangian is not a part of ...
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### Must there exist a Lagrangian for any 2nd order ordinary derivative equation?

We know if there exist a Lagrangian of some ODE, then it must exist many equivalent Lagrangian. My question: Then must there exist a Lagrangian for any 2nd order ODE? If not, do we have some ...
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### The cosmological constant as a Lagrange multiplier?

The cosmological constant $\Lambda$ can be introduced into the gravitational action like this : S = \frac{1}{2 \kappa} \int_{\Omega} (R - 2 \Lambda) \sqrt{-g} \; d^4 x + \text{matter ...
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### Is there an action for every physical law?

Given an action, I can get the differential equation governing the evolution of the system by applying the principle of least action. Does it work the other way around? Given any differential ...
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### Why does the non-linearity of the string action prohibit stretching due to strong excitations?

From 't Hooft's String Theory lecture notes on page 8 (paraphrased): To understand hadronic particles as excited states of strings, we have to study the dynamical properties of these strings, and ...
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### How to calculate Lagrangian density function in classical field theory

In Lagrangian mechanics observing the possible degrees of freedom we first write down our Lagrangian. Then we use E-L equation to determine equation of motion and using sufficient boundary condition ...
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### Coordinates from action-angle variables

I'm interested into getting the original coordinates, $q(t)$ and $p(t)$, from the action, $J=\oint p dq$, and angle, $w(t)=\frac{dH}{dJ}t+\beta$, variables for a 1-D, one particle system. I know that ...
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### General Relativity as a Special Relativistic Field Theory

In this question, I want to consider only the classical case. I have seen the statement that general relativity can be considered as a spin-2 field living on a Minkowski background. In that case, you ...
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### Are cumulus clouds analogous to airships flying downwards?

First of all, cumulus clouds are amazing. Big puffy white clouds floating on the air. Some of them produce updrafts of over 100 Km per hour. Now, if an airship had its engine pointed towards the ...
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### Reality of the action in QFT [duplicate]

Following Ramond, 1.5 Field Theory, it is mentioned that the classical Lagrangian density in (workable for HEP) QFT theories has to be Real, otherwise total probability is not conserved. Can someone ...
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### Why does the action have to be hermitian?

The hermiticity of operators of observables, e.g. the Hamiltonian, in QM is usually justified by saying that the eigenvalues must be real valued. I know that the Lagrangian is just a Legendre ...
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### Are there (interesting) Poincare-invariant QFTs with non-invariant Lagrangian densities?

In all QFTs I know, the Lagrangian density is completely invariant under the Poincare group, $$\mathcal L \to \mathcal L.$$ On the other hand, the action would be invariant even if the Lagrangian ...
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### Bulk action - Can a brane velocity be defined?

a) If a brane action in a bulk is defined, in that case, that a brane is modelwise moving through a bulk, how is this ratio defined? Is this a regular "velocity" in that meaning, that space is being ...
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### Action with self-dual field strength

It is said that writing down an action in presence of a self-dual field strength is subtle and not known till date. The familiar example people give is that of type IIB super-gravity which has a ...
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### Euclidean classical action

This is the Euclidean classical action $S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))$. It would be nice if somebody could explain the structure of the potential. I don't ...
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### Conceptual problem with action considered as function of endpoints

I am having some trouble with understanding why it makes sense to consider action in classical mechanics as function of endpoints $q_{initial}, \ q_{final}$ and endtimes $t_{initial}, \ t_{final}$. ...
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### Functional Derivative of action

Consider the action of free Klein-Gordon theory $S[\phi]=\frac{1}{2}\displaystyle\int d^4y(\partial_\mu\phi(y)\partial^\mu\phi(y)-m^2\phi^2(y))$ Integrating by parts in the first term gives me ...
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### When the equations of motion are not unique (eg. when they are given by eigenvectors), which will the free particle adhere to?

For this question I think it will be easier to express the usual equation describing the motion of a "free particle,"--viz. $g_{ij}\dot{x}^i\dot{x}^j$--in matrix form as follows: ...
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### How does satisfying the Euler-Lagrange equation put a Classical Path on-shell?

I am thinking of what the Euler-Lagrange equation, $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0$$ specifically represents in satisfying the ...
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### Principle of least action: $\frac{d S_{cl}}{dt_b} = \frac{\partial S_{cl}}{\partial t_b} + \frac{\partial S_{cl}}{\partial x_b}\dot{x}_b$

Question I cannot see how I can obtain the yellow highlighted section on the RHS from that of the LHS. The following equation can be found in both my lecture notes(*1) (page 9, equation 2.7) and is ...
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In Goldstein's Classical Mechanics, Chapter 2.3: Derivation of Lagrange's Equations From Hamilton's Principle part of the derivation involves each of the generalized coordinates being independent. $$... 2answers 84 views ### How to describe time-shifts in Noether's theorem in Hamiltonian formalism As was described in, for example, this post, one can formulate Noether's Theorem also in Hamiltonian Mechanics. Symmetries are then represented by vector fields generated by observables whose Poisson ... 1answer 46 views ### With radian as a unit, should action and angular momentum have the different units? If one accepts radian as a fundamental unit, does it make sense that action and angular momentum have units differing in radian to the power of one? The same question applies for energy and torque. ... 1answer 84 views ### Variations of actions of (lie algebra valued) differential forms I have always found it a bit difficult to understand the variation of an action written in differential form language. For example, take the action$$\int tr A\wedge A\wedge A$$where A=A_\mu ... 3answers 3k views ### Noether's current expression in Peskin and Schroeder In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence. But if we ... 1answer 153 views ### Is there a Maupertuis principle for General Relativity? The motion of a point particle in classical mechanics is given by Newton's equation, \mathbf{F}=m\mathbf{a}. Suppose all forces considered are conservative and we have a constant total energy h. ... 3answers 216 views ### Why is the action dimensionless in natural units? As I understand it, a natural system of units is one in which the numerical values of c and \hbar are unity, i.e. c=\hbar =1. What I find confusing is that they are still dimensionful, i.e. ... 1answer 85 views ### How to proceed (Tough Problem) [closed] The problem that I am considering is to find the shortest path (or geodesic) on a surface with the equation z=f(x,y). The path is parameterized by s so that the path goes from ... 1answer 54 views ### How to derive the true spatial paths (orbits) from the Jacobi-Maupertuis condition How can differential equations describing a physical object's true spatial paths (orbits) be derived from the time-independent Jacobi-Maupertuis principle of least action? According to this, it is ... 0answers 34 views ### Why do we sum relativistic intervals in relativistic action of a massive point-particle, and not a function from it? Relativistic action as follows (which should explain relativistic motion of a classical particle):$$ S = C \Delta s=C\int ds  Where $C$ is some constant and $\Delta s$ is relativistic interval. ...
Suppose that you want to calculate the variation $\delta S$ of an action induced by some arbitrary variation $\delta g_{\mu \nu}$ of the spacetime metric : S = \int_{\Omega} ...