The tag has no wiki summary.

learn more… | top users | synonyms

-2
votes
0answers
34 views

How to prove that generalized coordinates are independent?

I know that the generalized coordinates obtained after taking into account the constraint forces are independent. How can I prove this?
5
votes
1answer
135 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 ( ...
1
vote
3answers
76 views

About constraints of the first class and electrodynamics

Let's have some theory in hamilton formalism and let's assume that it has the constraints between canonical variables $Q, \pi$. By the Dirac terminology, the set of constraints $F_{a}(Q, \pi) \approx ...
8
votes
4answers
499 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 ...
6
votes
1answer
53 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 ...
2
votes
3answers
75 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 ...
10
votes
3answers
376 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 ...
1
vote
1answer
265 views

Virtual displacement and generalized coordinates

I have a doubt regarding the expression of a virtual displacement using generalized coordinates. I will state the definitions I'm taking and the problem. The system is composed by $n$ points with ...
5
votes
1answer
168 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 ...
0
votes
1answer
40 views

How do I find the generalized coordinates in a certain system?

I'm learning about constraints and I know the following: If there are $N$ particles in 3 dimensional space, I have $3N$ degrees of freedom. If I have $n_b$ holonomic constraints and I switch over to ...
4
votes
2answers
134 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 ...
0
votes
0answers
47 views

Question about an example of non-integrable constraints

The example is a thin disk rotating of an inclined plane. The disk can roll not only down the plane, but also "sideways". Let $(x,y)$ be the position of the CM, where the $y$ axis is down the slope ...
1
vote
0answers
70 views

Restrained double pendulum

The equations of motion of a double pendulum are well-known. Usually you'd have the them expressed in the rotations $\theta_1(t)$ and $\theta_2(t)$. There are two degrees of freedom. Now consider the ...
5
votes
1answer
126 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 ...
2
votes
1answer
74 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 ...
0
votes
0answers
49 views

Classical rod-wall-floor system

I have a homogeneous thin rod $AB$. $A$ can slide along $z$-axis, $B$ along $xy$ plane. There is no friction. We have any initial conditions. Now the question is to write down the equation of motion ...
1
vote
3answers
213 views

Why does diamond have lower tensile strength than Iron?

Let me first give you the tensile strength of both substance: Diamond: 16000 MPa Steel : 2617 MPa As you guys should know, tensile strength is how much a ...
2
votes
0answers
46 views

A question about the constraints in BRST-Fock theories

In BRST Symmetry in the Classical and Quantum Theories of Gauge Systems, Henneaux says the Fock representation is not applicable to an odd number of constraints. Then he goes on to say that the ...
3
votes
1answer
146 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 ...
2
votes
2answers
563 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
1answer
148 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
532 views

What are the forces of constraint if there are multiple equivalent constraints?

Suppose a large (rigid) block is sitting on top of two smaller blocks of equal height $1$, both of which rest on the ground. We wish to find the position of the block (easy) and the forces of ...
0
votes
0answers
56 views

Dirac Parenthesis and redefinition of constraints

As I know in the analyzing a constrained system always we have freedom to change a constraint by adding another constraint to it or by multiplying it by anything except zero. Because this modification ...
6
votes
2answers
197 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 ...
3
votes
1answer
176 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 ...
1
vote
1answer
177 views

Calculate Hamiltonian from Lagrangian for electromagnetic field

I am unable to derive the Hamiltonian for the electromagnetic field, starting out with the Lagrangian $$ \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\partial_\nu A^\nu \partial_\mu A^\mu ...
4
votes
1answer
222 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: ...
3
votes
2answers
274 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
222 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 ...
3
votes
1answer
236 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 ...
1
vote
1answer
199 views

Constrained motion in a parabolic tube [closed]

A smooth parabolic tube is placed with vertex downwards in a vertical plane. A particle slides down the tube from rest under the influence of gravity. Prove that in any position, the reaction of ...
3
votes
1answer
143 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 ...
9
votes
1answer
390 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 ...
3
votes
0answers
82 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
1answer
268 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
1answer
312 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 ...
2
votes
6answers
1k 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 ...
3
votes
0answers
75 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 ...
1
vote
1answer
89 views

Lagrange multiplier dependent on time

At the moment I am following a course on variational methods for mathematicians. Last week we derived the Euler-Lagrange equations for a functional under a constraint. In this derivation we found that ...
1
vote
3answers
1k views

How is a Hamiltonian constructed from a Lagrangian with a Legendre transform

many textbooks tell me that Hamiltonians are constructed from Lagrangians like $$L=L(q,\dot{q})$$ with a Legendre transformation to obtain the Hamiltonian as $$H=\dot{q}\frac{\partial L}{\partial ...
2
votes
1answer
217 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 ...
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 ...
1
vote
3answers
727 views

Frequency of small oscillation of particle under gravity constrained to move in curve $y=ax^4$

How to find the frequency of small oscillation of a particle under gravity that moves along curve $y = a x^4$ where $y$ is vertical height and $(a>0)$ is constant? I tried comparing $V(x) = \frac ...
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 ...
1
vote
1answer
51 views

How to see timelike excitation has a negative norm from the “old covariant quantization”

I have a question in reading Polchinski's string theory vol I p 123, about the "old covariant quantization". It is said ... $\langle 0;k | 0; k' \rangle = ( 2\pi)^D \delta^D (k-k') \tag{4.1.15}$ ...
4
votes
1answer
173 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 ...
3
votes
2answers
350 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 ...
2
votes
1answer
216 views

Euler-Lagrange for constrained system

Suppose we have Euler-Lagrange system with generalized coordinate $r_1$ and $r_2$, and input $u_1$ and $u_2$. I know how to prove this system is indeed Euler-Lagrange system. Suppose now if we have a ...
1
vote
1answer
194 views

Calculation of Commutation in constraint analysis

During analysis the constraint from a theory, suppose my canonical Hamiltonian is $$H_c=P^A\dot{A}+P^B\dot{B}-L$$ where $P^A=\frac{\partial L}{\partial \dot A}$ and $P^B=\frac{\partial L}{\partial ...
6
votes
2answers
285 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 ...