The tag has no wiki summary.

learn more… | top users | synonyms

11
votes
2answers
2k views

Are there examples in classical mechanics where D'Alembert's principle fails?

D'Alembert's principle suggests that the work done by the internal forces for a virtual displacement of a mechanical system in harmony with the constraints is zero. This is obviously true for the ...
10
votes
2answers
364 views

Virasoro constraints in quantization of the Polyakov action

The generators of the Virasoro algebra (actually two copies thereof) appear as constraints in the classical theory of the Polyakov action (after gauge fixing). However, when quantizing only "half" of ...
10
votes
3answers
433 views

What exactly is a virtual displacement in classical mechanics?

I'm reading Goldstein's Classical Mechanics and he says the following: A virtual (infinitesimal) displacement of a system refers to a change in the configuration of the system as the result of any ...
9
votes
4answers
298 views

What makes an equation an 'equation of motion'?

Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint. For example, in the ...
9
votes
1answer
391 views

How do I find constraints on the Nambu-Goto Action?

Let $X^\mu (t,\sigma ^1,\ldots ,\sigma ^p)$ be a $p$-brane in space-time and let $g$ be the metric on $X^\mu$ induced from the ambient space-time metric. Then, the Nambu-Goto action on $X^\mu$ is ...
8
votes
1answer
388 views

Relation between Dirac's generalized Hamiltonian dynamics method and path integral method to deal with constraints

What is the relation between path integral methods for dealing with constraints (constrained Hamiltonian dynamics involving non-singular Lagrangian) and Dirac's method of dealing with such systems ...
8
votes
4answers
534 views

D'Alembert's Principle: Necesssity of virtual displacements

Why is the D'Alembert's Principle $$\sum_{i} ( {F}_{i} - m_i \bf{a}_i )\cdot \delta \bf r_i = 0$$ stated in terms of "virtual" displacements instead of actual displacements? Why is it so necessary ...
7
votes
2answers
243 views

Why so many arguments for the transformation equations of generalized coordinates?

For a system of $N$ particles with $k$ holonomic constraints, their Cartesian coordinates are expressed in terms of generalized coordinates as $$\mathbf{r}_1 = \mathbf{r}_1(q_1, q_2,..., q_{3N-k}, ...
7
votes
1answer
138 views

Request for Reference: BRST formalism/transformations

Could anyone please suggest a very basic paper/reference/literature on BRST symmetry/formalism that requires rudimentary knowledge of Dirac's method for dealing with constrained systems and generation ...
6
votes
1answer
779 views

When is the principle of virtual work valid?

The principle of virtual work says that forces of constraint don't do net work under virtual displacements that are consistent with constraints. Goldstein says something I don't understand. He says ...
6
votes
2answers
293 views

From Lagrangian to Hamiltonian in Fermionic Model

While going from a given Lagrangian to Hamiltonian for a fermionic field, we use the following formula. $$ H = \Sigma_{i} \pi_i \dot{\phi_i} - L$$ where $\pi_i = \dfrac{\partial L}{\partial ...
6
votes
2answers
198 views

Are Poisson brackets of second-class constraints independent of the canonical coordinates?

Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\{\chi_N(q,p),\chi_M(q,p)\}_P$ . Is ...
6
votes
1answer
57 views

Dirac bracket and second class constraints in first-order gravity formalism

In the first order formulation of general relativity, the frame field $e_{\mu}^a$ and $\mathrm{SO}(3,1)$ spin connection $\omega_{\mu c}^b$ are independent variables. In the Hamiltonian formulation of ...
5
votes
1answer
150 views

Why are D'Alembert's Principle and the Principle of Least Action Related?

Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common. $$\sum _i ( ...
5
votes
1answer
155 views

Is there a systematic way to derive constraint equations?

There's this problem in Goldstein's (Classical Mechanics) derivations section: 5. Two wheels of radius $a$ are mounted on the ends of a common axle of length $b$ such that the wheels rotate ...
5
votes
0answers
71 views

The consistency conditions of constrained Hamiltonian systems

I am studying the Hamiltonian description of a constrained system. There are some questions puzzled me for days, which I have been stuck on it. From the lagrangian, we can obtain the primary ...
5
votes
1answer
177 views

Secondary constraints leads to the value of lagrange multiplier

From Lagrangian I got two primary constraint $\phi_i$ and $\phi$. And my Hamiltonian in presence of the constraints becomes- $$H_p=p\dot q-L+\lambda_i\phi_i+\lambda\phi$$ here the $\lambda_i$ and ...
4
votes
3answers
325 views

Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation

In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
4
votes
1answer
260 views

Is there a Hamiltonian for the (classical) electromagnetic field? If so, how can it be derived from the Lagrangian?

The classical Lagrangian for the electromagnetic field is $$\mathcal{L} = -\frac{1}{4\mu_0} F^{\mu \nu} F_{\mu \nu} - J^\mu A_\mu.$$ Is there also a Hamiltonian? If so, how to derive it? I know how ...
4
votes
1answer
179 views

Lagrange Multipliers Versus Generalized Coordinates

When forced to explain to someone why one could either set up a general Lagrangian & then incorporate constraints using Lagrange multipliers, as opposed to just setting up a Lagrangian with ...
4
votes
3answers
214 views

Odd number of second class constraints (!)

For my thesis, I have calculated the constraints for a system using Dirac method of constraint analysis. The problem is I got odd number of second class constraints (!), which gives me unusual numbers ...
4
votes
2answers
571 views

Counting degrees of freedom in presence of constraints

In a $N$ dimensional phase space if I have $M$ 1st class and $S$ 2nd class constraints, then I have $N-2M-S$ degrees of freedom in phase space. How can I calculate the degrees of freedom in ...
4
votes
2answers
142 views

Lagrange multiplier and constraint force

The Lagrangian with Lagrange multiplier in the form $$L= T- V + \lambda f(q, \dot{q},t).$$ But there are different ways of writing the constraint $f = 0$. Will that lead to different EOMs? Let me ...
4
votes
1answer
226 views

Constraints of massive relativistic point particle in hamiltonian mechanics

I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: ...
4
votes
1answer
122 views

Is it possible to project a problem of mechanics in a lower dimensionality?

I had the intuition that, in classical mechanics, when the trajectory of a body is known, then analysis of its motion can be done in the linear space of that trajectory, if all forces are projected on ...
3
votes
6answers
1k views

Degree of freedom paradox for a rigid body

Suppose we consider a rigid body, which has $N$ particles. Then the number of degrees of freedom is $3N - (\mbox{# of constraints})$. As the distance between any two points in a rigid body is fixed, ...
3
votes
4answers
751 views

First class and second class constraints

Hello I am working on a project that involves the constraints. I checkout the paper of Dirac about the constraints as well as some other resources. But still confuse about the first class and second ...
3
votes
1answer
177 views

Missing terms in Hamiltonian after Legendre transformation of Lagrangian

Short question Given any Lagrangian density of fields one could possibly conceive, is it the case that after one has performed a Legendre transformation, if the Hamiltonian is then expressed in terms ...
3
votes
2answers
351 views

primary constraints for constrained Hamiltonian systems

I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum dynamics of constrained Hamiltonian ...
3
votes
1answer
247 views

Point of Lagrange multipliers

I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
3
votes
1answer
358 views

What is a bilateral constraint?

In the realm of mechanics/rigid body dynamics, can anyone tell me what a bilateral constraint is? Can't seem to find any information on the exact definition, just uses of it such as "considering only ...
3
votes
1answer
521 views

Gauss law in classical U(1) gauge theory

I can see that $a_{0}$ is not an independent field and Gauss law is a constraint on the theory arising from field equations. But, I don't get the geometrical picture. Let $A$ be the space of all ...
3
votes
1answer
153 views

Clarifying constraint forces in Lagrangian dynamics

In the Lagrangian formulation, the addition of constraint forces that are unknown can be done with Lagrange multipliers, which allows for the forces to be found. Taking $k$ constraints of the form ...
3
votes
1answer
149 views

Why can we assume independent variables when using Lagrange multipliers in nonholonomic systems?

I'm studying from Goldstein's Classical Mechanics. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form $f_\alpha(q, \dot{q}, t) =0$, $\alpha = 1 ...
3
votes
2answers
283 views

Elimination of velocities from momenta equations for singular Lagrangian

this doubt is related to Generalized Hamiltonian Dynamics paper by Dirac. Consider the set of $n$ equations : $p_i$ = $∂L/∂v_i$, (where $v_i$ is $q_i$(dot) = $dq_i/dt$, or time derivative of ...
3
votes
1answer
248 views

Showing constraint is nonholonomic

One example of a nonholonomic constraint is a disk rolling around in the cartesian plane that is constrained to not be slipping. These leads to the constraint $dx - a \sin\theta d\phi = 0$ and $dy - ...
3
votes
1answer
277 views

D'Alembert's principle

Actually I have some troubles to understand what this principle is all about, so I want to use the simple pendulum in order to get the idea. Since I have read a few passages that dealt with this ...
3
votes
2answers
146 views

How is the physical Lagrangian related to the constrained minimization Lagrangian?

If we're minimizing an energy $V(q)$ subject to constraints $C(q) = 0$, the Lagrangian is $$L = V(q) + \lambda C(q).$$ I have fairly solid intuition for this Lagrangian, namely that the energy ...
3
votes
0answers
85 views

Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: ...
3
votes
0answers
78 views

Do primary first class constraints change the electric field in the Hamiltonian form of Maxwell's theory?

In my understanding of Dirac's theory of constrained Hamiltonians, the primary (and also the secondary) first class constraints are generators of canonical transformations that do not change the ...
2
votes
6answers
2k views

How are constraint forces represented in Lagrangian mechanics?

Suppose we try to obtain the movement equation for a particle sliding on a sphere (no friction, ideal bodies...). The only forces acting on the particle are its weight and - here's my problem - a ...
2
votes
2answers
645 views

Euler-Lagrange equations and friction forces

We can derive Lagrange equations supposing that the virtual work of a system is zero. $$\delta W=\sum_i (\mathbf{F}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=\sum_i ...
2
votes
2answers
2k views

degree of freedom of a rigid body 5 or 6?

I'm confused here. I have a three particle (rigid) system. What would be the degree of freedom? I found out five. 3 coordinates for center of mass and 2 for describing orientation. But we have only ...
2
votes
3answers
91 views

Configuration manifolds and constraints

In Classical Mechanics there's this notion of configuration manifold. Although I've heard about that a lot and although I often use that concept, I'm not sure I really understand them well because ...
2
votes
1answer
172 views

Can the Lagrange Multipliers depend on the coordinates?

When dealing with Lagrange multipliers to solve systems with constraints we usually have two ways if the constraints are holonomic: Differentiate the constraint and add the appropiate term to the ...
2
votes
1answer
68 views

Example of Hamilton's Principle to Systems with Constraints (Goldstein)

I'm currently studying Goldstein's Classical Mechanics book and I can't get my head around his reasoning in section 2.4. (Extending Hamilton's principle to systems with constraints). I'd like to ...
2
votes
3answers
102 views

Virtual Work: How is the applied force related to the coordinates chosen?

I have a question after reading a section from Goldstein's Classical Mechanics. The question deals with equation 1.43 in the text (given below): $$ \tag{1.43} \sum\limits_{i} {\bf F}_i^{(a)}\cdot ...
2
votes
1answer
80 views

Interpreting Lagrange Multipliers as forces

I am (still) working on getting a good understanding of Lagrange multipliers. I understand their function in an optimization problem that is subject to some constraint. For the specific case of ...
2
votes
1answer
218 views

Primary constraints for Hamiltonian field theories

I am currently trying to carry out the construction of the generalised Hamiltonian, constraints and constraint algebra, etc for a particular field theory following the procedure in Dirac's "Lectures ...
2
votes
3answers
178 views

Quantizing first-class constraints for open algebras: can Hermiticity and noncommutativity coexist?

An open algebra for a collection of first-class constraints, $G_a$, $a=1,\cdots, r$, is given by the Poisson bracket $\{ G_a, G_b \} = {f_{ab}}^c[\phi] G_c$ classically, where the structure constants ...