Questions tagged [constrained-dynamics]

A constraint is a condition on the variables of a dynamical problem that the variables (or the physical solution for them) must satisfy. Normally, it amounts to restrictions of such variables to a lower-dimensional hypersurface embedded in the higher-dimensional full space of (unconstrained) variables.

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6
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4answers
230 views

Hamiltonian of non-regular Lagrangian is well-defined on phase space

In section 1.1.3 of Quantization of Gauge Systems by Henneaux and Teitelboim, it is stated that the Hamiltonian $$H=\dot{q}^np_n-L,\tag{1.8}$$ although trivially a function of $q$ and $\dot{q}$, can ...
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1answer
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Applied dynamics (intermediate frames) [closed]

It is a question involving a rigid body and the use of intermediate frames. (Sorry for inconvenience, as i was unable to upload pictures on the original website due to unknown reasons.) Here's the ...
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3answers
435 views

Why is the d'Alembert's Principle formulated in terms of virtual displacements rather than real displacements in time? [duplicate]

Why is the d'Alembert's Principle formulated in terms of virtual displacements rather than real displacements in time? EDIT In other words, which step of the derivation of D'Alembert's principle (or ...
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1answer
67 views

Constructing Lagrangian from Hamiltonian for Majorana fermions

The text gives the Hamiltonian density as \begin{equation}{\cal H}=\frac{v}{2}\Big(\psi^\dagger\frac{\partial\psi^\dagger}{\partial x}-\psi\frac{\partial\psi}{\partial x}\Big)+\Delta\Psi^\dagger\Psi \...
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6answers
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What are holonomic and non-holonomic constraints?

I was reading Herbert Goldstein's Classical Mechanics. Its first chapter explains holonomic and non-holonomic constraints, but I still don’t understand the underlying concept. Can anyone explain it to ...
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4answers
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D'Alembert's Principle: Necessity 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 ...
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0answers
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D'Alembers Principle - further explanation [duplicate]

In question : Why is the d'Alembert's Principle formulated in terms of virtual displacements rather than real displacements in time? there is a response : ...
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4answers
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Derivation of Lagrange Equations from Newton's Second Law for a Non-holonomic System of Particles

I am interested to write down a derivation of Lagrange equations from Newton's second law for a non-holonomic system of particles. Here, I mention my derivation where I am stuck right at the last step....
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1answer
387 views

Proof of Thomson's theorem on electrostatics using variational calculus

I'm following a proof of Thomson's theorem but I'm a bit confused when they use a lagrange multiplier to include the charge conservation constraint, could someone explain to me how is this multiplier ...
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1answer
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Can you help me understand how constraints are mathematically expressed? [duplicate]

Self studying Lagrangian mechanics using Goldstein. Holonomic constraints, an example being the distance between two particles of a rigid body, can be expressed as $(r_i - r_j)^2 - c_{ij}^2 =0$ and ...
3
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1answer
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How can we know that changing variables to conjugate momenta is possible?

I am reviewing the derivation of Hamiltonian mechanics from Lagrangian mechanics, but I simply cannot understand how we can 'change variables' from $\dot q$ to $p$. Even on a very simple level, how ...
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2answers
343 views

Why is this a non-holonomic constraint?

Wikipedia states: holonomic constraints are relations between the position variables (and possibly time1) which can be expressed in the following form: $$f(q_{1},q_{2},q_{3},\ldots ,q_{n},t)=0$$ ...
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1answer
65 views

Deriving Euler-Lagrange equations for generalized coordinates without “virtual work”?

I have been reading "Classical mechanics" by Goldstein, Poole, and Safko. In particular, the section on "D'alembert's principle and lagrange's equations", in which the principle of virtual work is ...
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2answers
972 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 ...
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2answers
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What is difference between variations of the work and virtual work?

I really want to know whether or not both equations are the same mathematically. I think that they are the same, I just want to be sure. (Reference: this website.)
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1answer
140 views

Locally accessible dimensions of configuration space

I am reading a book called "Structure and Interpretation of Classical Mechanics" by MIT Press.While discussing configuration space and degrees of freedom,the authors remark the following: Strictly ...
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2answers
83 views

Conversion of non-holonomic constraints to holonomic

In the case of a disc rolling without slipping, we have a constraint $\dot{x}=a\dot{\theta}$ where $a$ is the radius of the disc. Note that I have considered $x$ and $\theta$ as the generalized ...
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1answer
195 views

Physical Constraints

In physics, what does one mathematically mean by constraint in classical mechanics? What are the the different types/cases and how do people deal with them?
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Hamilton Constraint of the WdW equation

Can someone explain specifically what the surface term of the hamilton constraint in quantum cosmology actually describes and how it creates time even though we start with a timeless universe? And why ...
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1answer
193 views

Motion of a particle constrained on a rotating rod

A sphere of neglectable radius is placed on a very long and frictionless rod (which we can approximate to a straight line) on which it is able to move. The rod rotates around one of its end points ...
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1answer
174 views

Rigorous definition of generalized coordinates

In Goldstein's classical mechanics and in many other books I haven't seen a rigorous definition of generalized coordinates. In a system of $N$ particles described by $\textbf{r}_1, \dots, \textbf{r}...
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1answer
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In 2D machines, why does higher pair joints deduct 1 D.O.F.?

I have been taught that higher pair joints (e.g. Gears, cams, rollers) deduct 1 d.o.f. Due to the fact that they still allow 2 motions translation along the tangent surface rotation around the ...
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1answer
59 views

Understanding the equations of motion for the Polyakov action in string theory

I want to make a small numerical simulation of how strings in theory move under their equations of motion but I'm getting stuck at implementing the constraints. The Polyakov action reads $$S=-\frac 1{...
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0answers
28 views

Is Lagrange multiplier an arbitrary gauge?

A related post might be found here: Is the gauge transform field in electromagnetism a Lagrange multiplier? Consider the case of Lagrange multiplier $L=f(x)+\lambda g(x)$ where $g(x)=0$ was a ...
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Poisson bracket of momentum constraints in general relativity

I wish to compute the Poisson bracket of the momentum constraints in general relativity. Unfortunately, I am not able to do it correctly and the answer I am getting is not a linear combination of the ...
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1answer
263 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 ...
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1answer
40 views

Example of a single constraint force doing virtual work despite the sum of work done by constraints being zero

When deriving d'Alembert's Principle it must be assumed, that the total virtual work done by constraint forces vanishes. $$\sum_{j=1}^N\mathbf{C}_j\cdot\delta \mathbf{r}_j=0.$$ In the books I've read, ...
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0answers
34 views

How to solve this one-sided constraint problem?

As shown in the figure below, ball 1 and ball 2 are connected by a rigid rod without mass. The wall and floor are absolutely smooth, and the mass of ball 1 and ball 2 are all $m$. The other geometric ...
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1answer
86 views

How to find an underlying holonomic constraints from a differential constraint?

I have been reading through "The Variational Principles of Mechanics" by Lanczos (if anyone is familiar with this text), and I am currently reading through the section discussing holonomic constraints....
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0answers
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Find the time period for pulley spring system [closed]

What I tried to do was first consider this system at equilibrium, assuming the pulley to be a disc of radius $'r'$ and I calculated the extensions of the 2 springs as $e(=mg/k)$ for lower spring and $...
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2answers
236 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). ...
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0answers
41 views

Constrained Hamiltonian dynamics [closed]

Suppose that I have a state $q$ that is constrained to satisfy $\mu(q)=0$. Assuming that $\nabla_q \mu(q)$ is a matrix of full-rank, I can use the Jacobian of the constraint function to define tangent ...
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3answers
817 views

Hamilton's principle and virtual work by constraint forces

I have a question about the following pages(pg 47 and 48) from Goldstein's "Classical Mechanics" I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. How ...
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1answer
36 views

Equations of motion for a certain constrained system

As an exercise in Lagrangian and Hamiltonian mechanics, I am looking at a system with the following Lagrangian: $$L=\dot R \cdot\dot R-\theta\dot R\cdot (SR)+\lambda(R\cdot R-1) $$ $R$ is a vector ...
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0answers
24 views

Performing Legendre transformation when conjugate momentum is independent of time derivative of generalised co-ordinates [duplicate]

Suppose there is a Lagrangean $L = \frac{1}{2} m c \dot x - \frac{1}{2}kx^2$ where $c$ is a constant to keep the dimensions right. The conjugate momentum is then $ p_x = \frac{\partial L} {\...
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1answer
2k views

Why friction force is force of constraint?

My understanding about constraint force is that it is a force which limits the geometry of particle's motion. For example, situations such as the particle trapped in a track or limited in domain can ...
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1answer
279 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 ...
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0answers
43 views

Difference between finite and infinitesimal motion

I am studying Arnold Sommerfeld mechanics. Here they talk about finite and infinitesimal motion. Quoted from the text: The simplest example of a non-holonomic condition is furnished by a sharp ...
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2answers
341 views

Confusion with Virtual Displacement

I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
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0answers
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Can constraint forces be parallel to virtual displacements?

It is often stated that virtual displacements must always be orthogonal to constraint forces. However, an example where this seems to fall apart is that of the Atwood machine, where the virtual ...
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5answers
150 views

D'Alembert's principle and the work done by constraint forces in Atwood's machine

From what I understand, constraint forces do no work because they are perpendicular to the allowed virtual displacements of the system. However, if you consider an unbalanced Atwood machine, in which ...
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2answers
171 views

Ambiguity in d'Alembert's principle

It seems to me that many different momenta $\dot{\bf p}_j $ can satisfy d'Alembert's principle: $$\tag{1} \sum_{j=1}^N ( {\bf F}_j^{(a)} - \dot{\bf p}_j ) \cdot \delta {\bf r}_j~=~0 $$ in a ...
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1answer
88 views

Why is it important that there is no variation of time $\delta t=0$ in the definition of virtual displacement?

In Goldstein's Classical mechanics I found a proposition that I don't understand: Similarly, the arbitrary virtual displacement $\delta \mathbf{r}_i$ can be connected with the virtual displacement $...
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2answers
83 views

Determine force of particle along trajectory [closed]

The problem: I have a trajectory in cartesian plane defined by points. A particle with mass $m$ runs through this trajectory, with a defined initial velocity. At every moment, there is a force $F$ ...
0
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1answer
354 views

Lagrange multipliers - isothermal-isobaric ensemble

I know that the entropy of isothermal-isobaric ensemble is given by: $$S = -k \sum_{i=1}^M p_i \ln p_i \quad \textrm{where $p_i$ must be normalized} \quad \sum_{i=1}^M p_i = 1 \, .$$ The average ...
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0answers
18 views

During BRST quantization of non Abelian gauge fields, is it necessary for the quantized particles to be transverse? [duplicate]

While covariantly quantizing non-Abelian gauge theories, we first impose the condition that the action of the BRST charge on physical states must yield zero. Then we further demand that such states ...
2
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4answers
96 views

Why the total virtual work done by forces from constraints vanishes? (Perpendicularity of two or more particles)

My mechanics book claims that the total force on the $i$-th particle is $$ F_i=K_i+Z_i \tag{2.5} $$where $Z_i$ is the force due to constraints and $K_i$ the real, dynamic force. Then, the book states ...
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1answer
82 views

Is a brachistochrone a straight line in curved space?

Please bear with me, and don't get upset if i have lack in knowledge about spacetime. Brachistochrone: Given two points A and B in a vertical plane, what is the curve traced out by a point acted ...
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1answer
214 views

Quantizing one real fermion

It is well-known how to canonically quantize the Lagrangian $$L = i \bar{\psi} \dot{\psi} - \omega \bar\psi \psi$$ I now wonder how one quantizes the Lagrangian with one real fermion $$L = i \psi \...
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0answers
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Automatically embedded in Lagrangian formulation

Since bond forces are automatically embedded in Lagrangian formulation, while in Newton they are not, and you have to operate directly with them [you have freedom from generalized coordinates]. That's ...

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