# 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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### 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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### 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 ...
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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### Constructing Lagrangian from Hamiltonian for Majorana fermions

The text gives the Hamiltonian density as {\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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### 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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### 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 ...
48 views

### 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 : ...
1k views

### 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....
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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### 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 ...
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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 ...
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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 ...