For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.

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224 views

Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
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1answer
52 views

Reduction of Nambu Goto action to true degrees of freedom

First consider the particle $$S=m\int\sqrt{-\dot{X}^2}d\tau$$ if you choose the static gauge $\tau=X^0$ and replace it in the action you get $$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau$$ So now, you ...
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2answers
144 views

Lagrange's equation implying Newton's 2nd law?

The typical first application of Lagrange's equation is showing that it implies Newton's law for a particle whose Lagrangian is $L=\frac{1}{2}mv^2-V(x)$. Plugging this Lagrangian into Lagrange's ...
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1answer
21 views

Degrees of freedom of a point mass sliding on a rigid curved wire without friction

I am very new to the subject and am going through Structure and Interpretation of Classical Mechanics. One exercise asks to find the degrees of freedom of a number of systems, one of which is a ...
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1answer
43 views

Classical spin viewed as SU(2)

In which sense is the configuration variable of a classical spin $SU(2)$? I can view a classical spin as a unit vector in $\mathbb{S}^2$ (2-dim. sphere), but it seems it is really given by a matrix ...
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2answers
79 views

How to find the rank of the matrix $\frac{\partial ^2\mathcal{L}}{\partial \dot{X^\mu} \partial \dot{X^\nu} }$ for the Nambu-Goto string Lagrangian?

In this case $$\mathcal{L}~=~-T\sqrt{-\dot{X^2}X'^2+(\dot{X}\cdot X')^2}.$$ I was reading some books and papers about the constraints in the Nambu-Goto action, and all of them say something like ...
2
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1answer
48 views

Proving independence of the lagrangian on position of a free particle using the euler-lagrange equation

I asked a similar question some time back but am trying to work this from another angle. In deriving the lagrangian of a free particle, we use the homogeneity of space to conclude that the lagrangian ...
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1answer
658 views

Conversion of the Nambo-Goto action into the Polyakov action?

I`ve read that the Nambo-Goto action containing the induced metric $\gamma_{\alpha\beta}$ $$\tag{1} S_{NG} ~=~ -T\int_{\tau_i}^{\tau_f} d\tau \int_0^{\ell} d\sigma \sqrt{-\gamma}$$ can be converted ...
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1answer
54 views

Is it possible to formulate a Hamiltonian for a damped system?

I recently found out that it is possible to formulate a Hamiltonian for a system with time-dependent coordinates such that the Hamiltonian is not the same as the energy When is the Hamiltonian of a ...
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1answer
47 views

Equation of motion of an auxiliary field

I'm a newbie in the field of QFT and SUSY, so I'm warning you: this might be a stupid question. I'm working with auxiliary fields to describe supersymmetric models and I understand that upon ...
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2answers
512 views

Can Lagrangian mechanics be justified without referring to Newtonian mechanics?

Are there any ways of justifying Lagrangian mechanics as a foundation of classical physics, without referring to Newtonian mechanics? In other words, what is the deeper reason or intuition why ...
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2answers
83 views
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3k views

Deriving the Lagrangian for a free particle

I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away. Proving that a free particle ...
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1answer
70 views

Lagrangian for free particle in special relativity

From definition of Lagrangian: $L = T - U$. As I understand for free particle ($U = 0$) one should write $L = T$. In special relativity we want Lorentz-invariant action thus we define free-particle ...
2
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1answer
48 views

Poincare non-invariance in real world and field theory

This may be a very blunt question but I wonder why we always use Poincare invariant Lagrangians in field theory. After all, the entire world around us is by no means homogeneous, isotropic and so on. ...
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1answer
38 views

How to calculate the period of the movement from a potential?

I have an assignment, where I have an object moving in 1-D with a given mass and energy, and the potential V(x), and I'm supposed to calculate the period of the movement as a function of the energy ...
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1answer
107 views
0
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1answer
48 views

Deriving lagrangian of a free particle - How do you arrive at Lagrangian independency conclusions

I guess this question has been asked before, but I'm looking at a slightly different aspect. I'm reading Landau's book on classical mechanics. In deriving the lagrangian for a free particle, I ...
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0answers
53 views

Decoupling of generalized coordinates in lagrangian

Say you have a lagrangian $L$ for a system of 2 degrees of freedom. The action, S is: $S[y,z] = \int_{t_1}^{t_2} L(t,y,y',z,z')\,dt \tag{1}$ If $y$ and $z$ are associated with two parts of the ...
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1answer
37 views

Total time derivative of magnetic vector potential A

I am looking at this document, which tries to establish the Lagrangian of the Lorentz force. Everything is fine, but I don't see why: $$\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+\frac{\partial ...
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1answer
47 views

Calculating Christoffel symbols from Lagrangian

I was given the following metric for a sphere $$g_{\mu\nu} = diag(1, r^2, r^2\sin^2\theta)$$ and tasked to calculate the Christoffel symbols. There are 2 ways that I know of to calculate them. One ...
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1answer
97 views

Coupled wheel and rod (analytical mechanics) [closed]

I am struggling with formulating the equations of motion. Consider a coordinate system with origin in $O$ ($y$ upwards and $x$ to the right), label the center of mass of rod $AB$ with $G$ then: ...
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0answers
59 views

Error in Susskind's “The Theoretical Minimum”?

Susskind's "The Theoretical Minimum", p. 117: Why isn't the last formula $$ \mathcal{A} = \int_{t_0}^{t_1} \biggl[\frac{1}{2}m\Bigl(\dot{X}+\dot{f}\Bigr)^2-V\bigl(X \color{red}{+ f(t)}\bigr)\biggr] ...
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1answer
107 views

Differentiating the Lagrangian to find geodesic equations?

I'm stuck pretty much at the first hurdle trying to follow the derivation of the geodesic equations from the Lagrangian ...
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1answer
49 views

Why is the solution of the $\phi^6$ potential not a soliton?

Consider a theory with a $\phi^6$-scalar potential: $$ \mathcal{L} = \frac{1}{2}(\partial_\mu\phi)^2-\phi^2(\phi^2-1)^2. $$ I solved its equation of motion but found that the general form of its ...
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1answer
81 views

Invariance of Fermionic action under Lorentz transformations

Suppose I have an Lagrangian $$\mathcal{L} = \frac{1}{2}g_{ab} \bar{\psi}^a \Gamma^k \partial_k \psi^b $$ and I want to show it's invariance under the infinitesimal Lorentz transformations $$\delta ...
4
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1answer
364 views

Sufficient conditions for the energy to be not conserved?

I'm almost embarrased to ask this question because I thought I was by now very confident with classical mechanics. Someone has stated that given a mechanical system with a Lagrangian $L$ s.t. ...
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1answer
258 views

Calculating the moment of inertia in bifilar pendulums

I'm an A2 student, and I've been looking into how experimental and theoretical determined mass moments of inertia differ. I came across a method (search Youtube for Measuring Mass Moment of Inertia - ...
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1answer
294 views

Determinant for a coupled fluctuation Lagrangian

Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the ...
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3answers
93 views

Why are complex fields in the Lagrangian?

I know that a complex field has twice the number of degrees of freedom of a real field, and that fields (in QFT) aren't observables so we don't really care if they are real. But why the need for ...
2
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1answer
68 views

Null geodesic equation

For a null geodesic curve $X^i$, $$0=g_{ij}V^iV^j.$$ When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the ...
2
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1answer
200 views

Hookes Law and Objective Stress Rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
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2answers
73 views

Lagrangian $L' = L + \frac{df}{dt}$ gives the same equations of motion

It is well known that when a Lagrangian $L$ is incremented by the total time derivative of a function $f$ that does not depend on the time derivatives of the generalized coordinates, the same ...
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3answers
111 views

What is a “Reversed Effective Force”?

I have some confusion about the "Reversed effective force" as it appears in the derivation of D'Alembert's principle. In Goldstein d'Alembert's principle is given as: $(F-\dot{p}) \cdot \delta r = ...
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2answers
101 views

Do the same equations of motion imply the same Lagrangians? [duplicate]

If two Lagrangian (densities) $\mathcal{L}$ give the same equations of motion, are they equivalent?
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1answer
220 views

Pressure and density using a general Lagrangian

Given a lagrangian of a form: \begin{equation}\mathcal{L}=f(\phi,\partial_{\mu}\phi\partial^{\mu}\phi)\end{equation} where $f$ is a function, I need to derive pressure and density in a FLRW universe ...
2
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1answer
88 views

Symmetries of the action of the free classical Klein-Gordon field

I've read that the action for the free classical Klein-Gordon field $$S = \int \mathrm{d}^4x~ \mathcal{L} = \frac{1}{2} \int \mathrm{d}x^4 \left(\partial_\mu \phi(x) \, \partial^\mu \phi(x) - ...
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1answer
123 views

Why a timelike geodesic maximizes path length?

I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is ...
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0answers
70 views

Lagrangian, geodesics and relativity [closed]

My background is in maths, but I have been studying some basic physics with occasional input from a friend who is studying for a physics PhD. Due to my background, I am keen to visualize things ...
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1answer
58 views

Is angular momentum the conjugate momentum of an angle?

Lagrangian mechanics can be used to describe the double pendulum (see here, for example). In this development are the conjugate momenta $p_{\theta_i}$ exactly the angular momenta $m_i l_i \frac{d ...
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1answer
38 views

Holonomic constraints and degrees of freedom?

Can we see that a constraint can decrease the degrees of freedom of a system if and only if it is holonomic. Either way please can you explain why?
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2answers
183 views

Kepler's Laws to Determine Radius of Circular Orbit

"In nonrelativistic limit of general relativity there is a correction to the Newtonian gravitational potential energy $−h/r^3$ with $h = αL^2/(mc)^2$, where $c$ is the speed of light, $α = GMm$ ...
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1answer
89 views

Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I ...
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0answers
26 views

Equivalence between Chern-Simons action and first order formalism

I can not derive second line from Chern-Simons action \noindent $S_{cs}$=k$\int{Tr(A \wedge dA+\frac{2}{3}}A \wedge A\wedge A)$ \noindent =k$\int{\ e }^a\wedge $R[$\omega $] we have to use ...
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2answers
140 views

Why do the $1/2$ factor appear in the Majorana mass Lagrangian?

In case of Dirac neutrino there is no $1/2$ factor in the mass Lagrangian but for Majorana type neutrino there is a half factor in the mass Lagrangian.
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1answer
123 views

Are there other less famous yet accepted formalisms of Classical Mechanics?

I was lately studying about the Lagrange and Hamiltonian Mechanics. This gave me a perspective of looking at classical mechanics different from that of Newton's. I would like to know if there are ...
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0answers
94 views

Why don't we have logarithms or exponentials of the fields in the Lagrangians?

All tbe Lagrangian densities I have seen have always been polynomials of the fields. Is this a coincidence or is there a reason forbid, say, Lagrangians with logarithms or exponentials of the fields?
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3answers
96 views

Lagrangian/Hamiltonian mechanics at high school?

Has anyone developed an approach to teaching mechanics based on Lagrangian/Hamiltonian mechanics from the ground up. I mean from high school on up. This is akin to explicitly not talking about ...
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1answer
38 views

Which of the Physics textbooks would you recommend I read this quarter (Analytical Mechanics)? [duplicate]

My Analytical Mechanics class this quarter has one required textbook: "Classical Dynamics of Particles and Systems" by Thornton & Marion and three recommended readings: "Mechanics" by Landau ...
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1answer
41 views

Solving Lagrangian equations of motion for two point-bodies with gravitational interaction

I would like to solve the equations of motion with the Lagrangian function for two point-bodies that interact gravitationally via the potential $$V= {-Gm_1m_2 \over r_{12}} $$ where $$r_{12} = **r_1 ...