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2answers
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Work done by internal forces of a rigid body

I am reading Goldstein's Classical Mechanics book, and I came across that: In a rigid body the internal forces do no work Is this statement based on the assumption that the internal forces are ...
0
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0answers
26 views

In d'Alembert's principle, how that the Reverse effective force and Force of constraints are different?

In d'Alembert's principle, how that the Reverse effective force and Force of constraints are different? Both are opposition or restriction on the body but how both be different in real?
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4answers
106 views

Question on the constraint force of a string

I would like to know if the following statement is true or false even if I expect that it is true. Notation: I will consider a string that has no mass and that can not be extended. Saying that a ...
0
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2answers
55 views

Virtual Work- Is the presentation in Cornelius Lanczos wrong?

Book: The Variational Principles of Mechanics by Cornelius Lanczos Edition: 4th Chapter: 3, The Principle of Virtual Work I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1). ...
3
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1answer
56 views

Why does the 'Jacobian of at least one combination of $n$ functions shall be different from zero'?

I've started reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt from p. 11: The generalized coordinates $q_1,q_2,\ldots, q_n$ may or may not have a ...
3
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1answer
53 views

A pendulum attached to a spring and all the system is rotating with angular velocity

Find the all the constraints and a set of generalized coordinates A pendulum attached to a spring and all the system is rotating with angular velocity $\omega$. this is what I have done, I do not ...
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0answers
12 views

Computing the dynamics with a bilateral constraint in position

Let's says I have a dynamical system of generalized coordinates $x$, subject to a bilateral constraint $g(x(t),t)=x_1(t)-d(t)=0$ where $d$ is a prescribed displacement. I am looking for the ...
3
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2answers
92 views

Lagrange Multipliers and Virtual Work: Are Joos & Freeman wrong?

I have come to suspect that the treatment of virtual work in configuration space using Lagrange multipliers given here "Theoretical Physics, by Georg Joos & Ira M. Freeman, pg 114" is not correct. ...
1
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1answer
100 views

Hamilton's equations of motion on Dirac's formalism

I'm having several doubts about the procedure proposed by the Dirac-Bergmann algorithm in order to get the correct equations of motion of electrodynamics (Maxwell's equations). Suppose I've already ...
0
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1answer
68 views

Find the acceleration of the bead [closed]

Two identical, uniform large rings, each of mass $\text{m}$ are connected through a bead of same mass, which can move freely. When bead is released, it starts sliding down. The large rings roll over a ...
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1answer
54 views

Membrane Theory

I'm not a physicist or educated mathematician, so please excuse me if my question is scientifically rudimentary. It concerns Membrane Theory. If all open strings are attached to the surface of the D ...
1
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0answers
57 views

Non-null hessian condition for regular dynamical systems

I'm "researching" on unquantised Yang-Mills theory. For that I'm studying the Dirac's method for singular constrained systems and having problems to follow the first considerations on that matter. I ...
0
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1answer
56 views

Build rotational Hamiltonian based on Lagrangian of general form

I've been told that one could build rotational Hamiltonian based on Lagrangian of general form: $\mathcal{L} = \mathcal{L} (\vec{\Omega})$. By introducing Euler angles one could rewrite Lagrangian in ...
2
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1answer
83 views

Non-holonomic constraints in Dirac-Bergmann theory

The Dirac-Bergmann algorithm effectively isolates the physical degrees of freedom of a system, by changing from Poisson brackets $\{\cdot,\cdot\}_\mathrm{PB}$ to Dirac brackets $\{\cdot,\cdot\}_\...
0
votes
1answer
42 views

Conservation of energy in the “ideal physics world”

Let us assume we are in the textbook ideal physics world, without any friction. Now, if I place a ball at the top of a parabola U and let it go under gravity, the ball should roll up and down forever ...
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1answer
73 views

Is air friction active force or constraint force?

Can air be regarded as a constraint body when a rigid body is moving? Or is the moving itself cause the friction so it is an active force?
1
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1answer
72 views

Rigid body motion degrees of freedom

A rigid body moving in $\mathbb{R^2}$ has 3 degrees of freedom and in $\mathbb{R^3}$ has 6 degrees of freedom. Could you please help me show that a rigid body moving in $\mathbb{R^n}$ has $\frac{n+n^2}...
2
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1answer
116 views

Classical particle in a box [closed]

I'm trying to work out some of the details for this system. A particle with mass $\mu$, initial velocity $v_0$ at $x_0$ and moving freely between two walls located at $\pm L/2$, with which it bounces ...
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0answers
36 views

Finding the constraint equation

I am trying to solve a problem on Constraint equations for a triple pendulum model, but was not able to derive a constraint equation for the last mass. I solved constraint equations for Masses 1 ...
0
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1answer
69 views

How did he find the “lambda” value in this question? [closed]

There is a pdf i found when searching about Lagrangian Multpliers, but i was not able to understand how he derived lambda from two differential equations. If anyone can walk me through it, i would be ...
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6answers
176 views

People went down two different length slides end up at the bottom at the same time [closed]

Discussion I am having: If there are two slides that are at the same height. One slide is 100m long and the other slide is 200m long. The endpoint and start point are the same displacement. In ...
0
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1answer
50 views

Counting number of degrees of freedom in constrained system

Following Counting degrees of freedom in presence of constraints, we know that there would be N-2M-S dofs if we have M 1st-class constraints and S 2nd-class constraints in N-dim phase space. I don't ...
1
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1answer
112 views

Differential holonomic constraints

Differential holonomic constraint is an integrable homogeneous first order differential equation: $$\sum_{i}\mathcal{E}_{i}(q)\frac{dq_{i}}{d\tau}=0;$$ in which $\sum_{i}\mathcal{E}_{i}(q)dq_{i}$ is ...
1
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0answers
35 views

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: $$g_{ij}\dot{x}^i\...
1
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2answers
95 views

Are the generalized coordinates in Lagrangian mechanics really independent?

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. $$ ...
0
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0answers
55 views

Relation between interaction Lagrangian and interaction Hamiltonian

I work with this interaction Lagrangian density $$\mathcal{L}_{int} = ia\bar{\Psi}\gamma^\mu\Psi Z_\mu +ib(\phi^\dagger\partial_\mu \phi - \partial_\mu\phi^\dagger \phi)Z^\mu,$$ where $Z^\mu$ is an ...
0
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1answer
25 views

A function of a constraint in a mechanics question not equaling zero?

Is this an error (I wrote what I think should be written), or is it okay that the constraint functions do not equal zero? I thought constraint functions are of the form $g(x,\dot {x},t)=0$, so in ...
0
votes
1answer
106 views

Problem in constraint equations

In this, if I want the acceleration constraint between $M$ and $2M$, I write $AM+2AB$=LENGTH OF STRING, which on differentiating twice gives $a_{m}=2a_{2m}$(which turns out to be correct). However, if ...
2
votes
1answer
90 views

Dirac bracket for a constrained particle

I am trying to work through a simple example of how to use the Dirac bracket from the following paper. In particular section 4 where the authors consider a constrained particle with the following ...
1
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0answers
85 views

Why don´t we just do a Legendre transform for a GR hamiltonian?

In general, if one has a well defined lagrangian for a field theory, which depends on a field, say $A_{\mu}$ and on its first spatial and temporal derivatives, we can simply define the canonical ...
0
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0answers
26 views

Is static gauge quantization of the particle equivalent to covariant quantization?

In the covariant quantization one is able to get directly (from the constraint $p^\mu_\mu+m^2=0$) the Klein-Gordon equation. But if one uses the parametrization $\tau=X^0$ then the Schrodinger ...
1
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1answer
118 views

Quantum field theory with constraint: energy-momentum conservation?

Suppose I have a 2-form field $B$ and a Lagrange multiplier field $\lambda$, then the Lagrangian $S = \int (B \wedge \delta B + \lambda \delta B \wedge \delta B)$ with a Lie derivative operator $\...
2
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0answers
102 views

Why do Lagrange multipliers work in mechanics?

I understand that it is not always simple to find generalized coordinates that satisfy the constraint equations, so we try to find an alternative (more mechanical) method that yields curves that ...
0
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0answers
32 views

The Hamilton-Suslov principle

Building on the introduction given in the following paper, how does the Hamilton-Suslov principle generalise Hamilton's principle to consider contained mechanics? \begin{equation} \int ^{t_1}_{t_0}\...
2
votes
1answer
127 views

Why does a system have to be holonomic?

So I'm doing some work from Taylor's mechanics book. He says for the problems in the book, we require the system to be holonomic - that is the number of generalized coordinates = number of Deg. of ...
0
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1answer
93 views

Allowed Virtual Displacements

I am having trouble understanding the kinds of virtual displacements which are permitted for a given constrained system. I have a specific example in mind: A block of wood resting on a table parallel ...
3
votes
1answer
113 views

Fermionic Poisson bracket

I'd like to understand the Poisson bracket for fermions in classical field theory defined on a cylinder (with coordinates $(t,x)$, $x$ being the compact direction) and propagating on $T^n$ with ...
2
votes
1answer
113 views

Canonical commutation relations in Light-cone gauge

It seems that when trying to identify the physical degrees of freedom for the string some authors$^1$ use: $$ q^-=\frac{1}{\ell}\int_0^{\ell} X^-(\tau,\sigma)d\sigma$$ Then, the commutation relation ...
1
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1answer
127 views

Dirac bracket for the Majorana Lagrangian

Note: See update below. Consider the Majorana Lagrangian $$\mathcal{L}=-\psi ^{\mathrm{T}}\mathrm{i}% \gamma ^{0}\left( \gamma ^{\rho }\partial _{\rho }+m\right) \psi ,\tag{1}$$ where $% \psi \in ...
2
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1answer
97 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 ...
3
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1answer
384 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 ...
2
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2answers
227 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 ...
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0answers
72 views

Advantages of having a first class system and possibility of transforming a system into a first class one

I have two questions regarding first class systems. What are the advantages of having a first class Hamiltonian (a Hamiltonian whose all constraints are first class) in a theory or having a first ...
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1answer
57 views

Parabolic slide [closed]

Can you give a big hint for this problem please? A material point is released at the end of a track in the form of a vertical arc of parabola $y = x^2$ in $[-1,1]$, the meter is selected as a unit ...
2
votes
2answers
230 views

What is the position as a function of time for a mass falling down a cycloid curve?

In the brachistochrone problem and in the tautochrone problem it is easy to see that a cycloid is the curve that satisfies both problems. If we consider $x$ the horizontal axis and $y$ the vertical ...
0
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1answer
85 views

Infinite Energies of a particle in a rectangular box

For a particle trapped inside a rectangular box of side lengths $l_x$ $l_y$ and $l_z$, the energies are $E_{n_x,n_y,n_x}=\frac{\pi^2\hbar^2}{2m}(\frac{n_x^2}{l_x^2}+\frac{n_y^2}{l_y^2}+\frac{n_z^2}{...
3
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2answers
285 views

Non-relativistic QFT Lagrangian for fermions

Take the ordinary Hamiltonian from non-relativistic quantum mechanics expressed in terms of the fermi fields $\psi(\mathbf{x})$ and $\psi^\dagger(\mathbf{x})$ (as derived, for example, by A. L. Fetter ...
2
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1answer
134 views

Hamiltonian field equations constraints

Let's consider the Lagrangian $$\mathcal{L}~=~-\frac{1}{2}(\partial_\mu\phi^\nu)^2+\frac{1}{2}(\partial_\mu\phi^\mu)^2+\frac{1}{2}m^2\phi_\mu \phi^\mu,$$ with Minkowski metric $\eta_{\mu\nu}={\rm ...
1
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1answer
209 views

Calculus of variations and string theory

In Polchinski's String theory book, Vol 1., in chapter 1, p. 18, he is deriving the Lagrangian in the light cone gauge (that's not necessary to know in order to answer this question), and he gets $$L~...
3
votes
1answer
302 views

Lapse and shift in ADM decomposition

Poisson in Relativist's Toolkit and also other authors in various papers state explicitly that after one does the 3+1 decomposition, the lapse and shift $N$ and $N^a$ are non-dynamical variables, and ...