# 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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### Problems based on constrained motion and energy conservation

Referring to qs no.47(Fig.20), we are asked to find the M/m ratio for the situation stated.Here's the solution to it- (This is actually a question from the renowned S.S.Krotov problems book!).The ...
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### Lagrangian Mechanics - Bead sliding on a rotating rod

Say i have a bead of mass $m$ sliding on a friction-less rode (or wire) that is rotating with a permanent angular velocity $ω$. The whole system is under the influence of a uniform gravitational field ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### Write Equations of motion of a point particle that moves on the surface of a paraboloid [closed]

I'm not a physics major student and I call myself a total noob in physics, however, I have this issue to solve. The problem is that I don't even know where to start from. One little thing to add: this ...
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### Infinity problem in basic classical mechanics problem [closed]

My question arises from a classical mechanics problem from a Hong Kong physics training programme: This is not a homework question as I am not asking about how to solve the problems in the image. As ...
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### Conservation of total energy for a system with holonomic constraints

Consider a system with generalized coordinates $u_1, u_2$ and $u_3$ such that $u_1$ and $u_2$ are dependent through the following holonomic constraint \begin{equation} G(u_1, u_2)=0. \end{equation} It ...
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### Particle sliding on a sphere with friction

This is a generalization of the question Particle sliding on a sphere when we also have friction given by $F_f = \mu N$. See the following figure: Before doing anything, we can imagine what friction ...
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### Arnold's holonomic constraints being limits of potential energy

The following quote comes from Arnold's "Mathematical methods in mechanics" book: "We consider potential energy $U_N = Nq_2^2 + U_0(q_1, q_2)$, depending on parameter $N$ (which we will tend to ...
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### Non-holonomic constraints, degree of freedom and generalized coordinates

If a system has $N$ coordinates and $M$ number of holonomic constraints then number of degree of freedom $=N-M$ and generalized coordinates $=N-M$ too. But if there are $k$ non-holonomic constraints ...