The constrained-dynamics tag has no wiki summary.
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Lagrangian with a general constraint [closed]
Can any body help me out to solve this problem?
I am familiar with mechanism of Lagrangian and I can solve some problems with constraints but this one is really hard to solve.
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1answer
266 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 ...
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2answers
208 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
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2answers
157 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 ...
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1answer
104 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 ...
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1answer
167 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 ...
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2answers
147 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 ...
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1answer
149 views
Significance of the the Lagrange multipliers in statistical mechanics
In classic thermodynamics one can derive the Maxwell Boltzmann statistics by solving a Lagrange multipliers equation. In this process a new parameter $\beta$ is introduced to take account of the total ...
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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 ...
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2answers
110 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 ...
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2answers
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Hamiltonian constraint in spherical Friedmann cosmology
I'm taking a GR course, in which the instructor discussed the 'Hamiltonian constraint' of spherical Friedmann cosmology action. I'm not quite clear about the definition of 'Hamiltonian constraint' ...
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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 $$\mathbf{r}_1 = \mathbf{r}_1(q_1, q_2,..., q_{3N-k}, ...
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2answers
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How is the physical Lagrangian related to the constrained minimization Lagrangian?
If we're minimizing an energy $V(q)$ subject to constraints $C(q) = 0$, the Lagrangian is
$$L = V(q) + \lambda C(q).$$
I have fairly solid intuition for this Lagrangian, namely that the energy ...
9
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2answers
168 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 ...
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1answer
112 views
Question about non-holonomic geometric constraints
Suppose a point particle is constrained to move on the curve $y=x^2$. This would then be a non-holonomic geometric constraint since the particle has one degree of freedom and requires two coordinates ...
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6answers
589 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, ...
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5answers
322 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 ...
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3answers
202 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 ...
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3answers
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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 ...
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2answers
84 views
Commutation for constraints
Suppose from the Hamiltonian I got the Primary constraints $$(\Phi_m,\Phi)$$ And $\dot \Phi_m$ , $\dot \Phi$ leads to secondary constraints $$(\gamma_m,\gamma)$$ respectively. Now if the commutation ...
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2answers
449 views
Why does tension not do work in this pulley system? etc
I have a slight difficulty understanding the solution to the following problem:
A light inextensible string with a mass $M$ at one end passes over a pulley at a distance $a$ from a vertically fixed ...
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4answers
400 views
First class and second class constraints
Hello I am working on a project that involves the constraints. I checkout the paper of Dirac about the constraints as well as some other resources. But still confuse about the first class and second ...
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4answers
231 views
What makes an equation an 'equation of motion'?
Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint.
For example, in the ...
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1answer
150 views
A particular case when Lagrange equation is equivalent to equation of motion on a Riemannian manifold
Suppose a particle is moving on a surface of a sphere,then it contains a holonomic constraint and so the three Cartesian co-ordinates are available with a constraint equation(equation of surface in ...
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1answer
92 views
Rotating sphere and circular trajectory: minimum speed
I have a sphere (mass = 3 kg), constrained to a fixed length rope, rotating (radius = 5 m) on a vertical plane.
My textbook ask me about the minimum speed in the highest point in order to keep the ...
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0answers
59 views
Secondary constraints leads to the value of lagrange multiplier
From Lagrangian I got two primary constraint $\phi_i$ and $\phi$. And my Hamiltonian in presence of the constraints becomes- $$H_p=p\dot q-L+\lambda_i\phi_i+\lambda\phi$$ here the $\lambda_i$ and ...
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3answers
129 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 ...
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1answer
250 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 ...
3
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1answer
254 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 ...
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2answers
423 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 ...
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2answers
216 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 ...
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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 ...
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1answer
388 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 ...
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1answer
128 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 - ...
