Physics Stack Exchange
Physics Stack Exchange

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.

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.

726 questions
39
votes
6 answers

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 me in detail and in simple language?
Akash Shandilya
  • 401
36
votes
4 answers

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 arbitrary infinitesimal change of the coordinates…
Gold
  • 38,087
  • 19
  • 112
  • 289
32
votes
3 answers

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 constraint of a rigid body where all the particles…
Larry Harson
  • 5,456
28
votes
1 answer

Lagrangian of Schrödinger field

The usual Schrödinger Lagrangian is $$ \tag 1 i(\psi^{*}\partial_{t}\psi ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi, $$ which gives the correct equations of motion, with conjugate momentum for $\psi^{*}$ vanishing. This Lagrangian density is not real…
user5468
  • 381
25
votes
1 answer

Why can't d'Alembert's Principle be derived from Newton's laws alone?

The wiki article states that D'Alembert's Principle cannot derived from Newton's Laws alone and must stated as a postulate. Can someone explain why this is? It seems to me a rather obvious principle.
CrazyDolphin
  • 415
25
votes
6 answers

Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\lambda$. This Lagrangian gives an action…
Javier
  • 28,811
21
votes
3 answers

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 $$H=\dot{q}\frac{\partial L}{\partial \dot{q}}-L$$ but none of the textbooks explain how…
0range
  • 313
20
votes
2 answers

Can one write down a Hamiltonian in the absence of a Lagrangian?

How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to construct the Hamiltonian of a such system starting…
John Doe
  • 764
19
votes
2 answers

Constraints of 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: $$S=-m\int d\tau \sqrt{\dot x_{\nu}\dot x^{\nu}}.$$ Here…
xxxxx
  • 1,595
  • 1
  • 16
  • 24
19
votes
1 answer

Why are Hamiltonian Mechanics well-defined?

I have encountered a problem while re-reading the formalism of Hamiltonian mechanics, and it lies in a very simple remark. Indeed, if I am not mistaken, when we want to do mechanics using the Hamiltonian instead of the Lagrangian, we perform a…
Frotaur
  • 2,361
  • 1
  • 17
  • 41
18
votes
4 answers

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 to "freeze" time in displacements? Also, what…
Sandesh Kalantre
  • 1,497
17
votes
1 answer

What is the full algebra of BRST-invariant observables for general relativity?

The Hamiltonian formulation of general relativity - either in the ADM formalism or in Ashtekar variables - is straightforwardly a gauge theory. While the BRST formalism has primarily been developed to quantize such theories, it can be applied to…
ACuriousMind
  • 132,081
16
votes
2 answers

The role of the virtual work principle

Lanczos' masterpiece "The Variational Principle of Mechanics" has, on page 76, the following statement: Postulate A (virtual work): The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with…
QuantumBrick
  • 4,183
16
votes
6 answers

Why does diamond have lower tensile strength than Iron?

Let me first give you the tensile strength of both substances: Diamond: 1600 MPa Steel : 2617 MPa As you guys should know, tensile strength is how much a material can be stressed or pulled before it breaks. The data above shows that diamond is…
Vaishnavi
  • 1,107
15
votes
1 answer

Does the inverse of the Dirac conjecture hold?

In the theory of constrained Hamiltonian systems, one differentiates between primary and secondary constraints, where the former are constraints derived directly from the Hamiltonian in question and the latter are only realized 'on-shell', i.e. once…
luksen
  • 6,245
1
2 3
…
48 49