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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How to implement constraints via delta functions?
I have a question regarding the implementation of constraint equations as delta functions in integrals.
My confusion can best be illustrated with a quick example:
Consider a Gaussian integral of the ...
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Constraints in Dirac's Lectures
Currently, I am reading section 3 of a 1950 lecture/paper (PDF) by Dirac, about general hamiltonians and dynamics in the formalism. He defines
$$H= \mathfrak{H(q,p)},\tag{7}$$
weakly (as in only holds ...
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Application of Dirac brackets to Dirac fermions in a tetrad gravity background
I have been studying the Dirac equation in curved spacetime, with the Lagrangian
$$L=\Psi^{\dagger} \gamma^{0}(i\gamma^{\mu} D_{\mu} -m) \Psi$$
(I think, however, I have seen it without the $\Psi^{\...
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Equation of motion of a point sliding down a parabola [closed]
I have a frictionless parabola $ (t,t^2) $ on the $x,y$ plane. I was having difficulties deriving the equations of motion for a point P placed at a height h on the parabola and let go of without any ...
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Deriving equations of motion in Lagrangian mechanics with semi-holonomic constraints
I am trying to understand the derivation of the equations of motions in Lagrangian mechanics in the presence of constraints. I believe the idea is just to apply the Hamilton's principle (the actual ...
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How to deal with Equation of Motion with singularities?
I was trying to describe the movement of a ball rolling on bowl.
The degrees of freedom of the system are the following:
The Position of the Center Of Mass (where $r$ is the distance from the origin ...
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Position function on curved path [closed]
could anyone tell me how to derive the position function, of a body that slide down on frictionless circle fragment. given by the equation $x^2+y^2=r^2$ where $r$ is a radius, in 3rd quarter of $xy$ ...
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Why the election of the reference frame make constraints appear or not?
Maybe this is a dumb question, but imagine we have the following system:
If we work on the red reference frame (the inclined plane frame, denoted by $I$), this system is easily solvable:
$$
\mathbf ...
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Lagrangian Equations for systems with holonomic constraints
The auxiliary conditions, due to the $𝑚$ holonomic algebraic constraints for the $𝑛$ variables $𝑞_𝑖$, can be expressed by the $𝑚$ equations with $n$ variables:
$$f_a(\mathbf{q})=0$$
The ...
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Finite Symmetry Transformations in Classical Mechanics
This is not a homework exercise. I graduated from univerisity more than 10 years ago. I ask questions from my self-study.
There're two types of symmetry transformations in classical mechanics. One is ...
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Gauge Transformation in One Dimension Revisited
Yesterday, I asked this question about gauge transformations in classical mechanics, and received an answer from Qmechanic. However, I genuinely don't understand his/her answer, and so I tried to ...
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Variations with Non-Holonomic Non-Integrable Constraints
I am reading this paper by Flannery. He considers a non-holonomic non-integrable set of constraints
$$g_k=g_k(q_i,\dot q_i,t)=0, \quad k=1,\ldots,c,\tag{3.1}$$
where the $i$ index runs to $n$. He ...
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Gauge Transformation in Classical Mechanics
A few days ago, I asked this question about computing the Noether charges associated with gauge redundancies in classical mechanics, and received an answer from Qmechanic. However, in my previous ...
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The Grassmann-Odd Part of the Lagrangian of a Spinning Particle
I have the following series of questions from the lecture notes "Constrained Hamiltonian Systems and Relativistic Particles" by Fiorenzo Bastianelli. On page 15, section 2.2 the Lagrangian ...
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Intuition behind Dirac’s Quantization of the Hamiltonian
In Lectures on Quantum Mechanics by Paul A.M. Dirac, he works through the quantization of the Hamiltonian on flat and curved spacetime. However before he ensures it is relativistic he has a ...
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