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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Field degrees of freedom from equations of motion and higher spin

It is my understanding that we compute the number of degrees of freedom of a quantum field as the number of its components minus the number of non trivial equations we get by taking the divergence of ...
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Equation of constraint - Falling disc unrolling from an attached string

Where does the equation of constraint below come from? I've tried to rationalize it, but the angle will be 0 more than one time as the string unrolls, even though y will keep going down (right?), not ...
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Concerning Constraint Equations for Lagrangian Formalism

I was working on a problem studying for a classical mechanics class and came across an idea I'm not sure about concerning the formalism of Lagrangian mechanics concerning constraint problems. https://...
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Confusion about valid set of virtual displacement

The answer of the problem below makes me confused about my thinking about virtual displacement... Consider the system in the figure below, similar to mechanisms used for aerial filming of sporting ...
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Consider the Kalb-Ramond field $B_{\mu\nu}$ which is basically a massless $2$-form field with the Lagrangian $$\mathcal L = \frac{1}{2}P_{\alpha\mu\nu}P^{\alpha\mu\nu}\,,$$ where $P_{\alpha\mu\nu} \... 1answer 35 views Meaning and Origin of an Expression which Involves Virtual Displacement As an additional point of confusion related to the answer given here: Confusion with Virtual Displacement I have encountered the following expression in my study of virtual displacements. $$\delta{... 2answers 259 views Confusion with Virtual Displacement I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ... 1answer 166 views How to deal with no-slip non-holonomic constraints in Lagrangian? I'm solving a dynamical system of a ball of mass m and radius R rolling on a rotating platform ("turntable") for which I found the Lagrangian to be:$$L=\frac{1}{2} m (\dot{x} - \Omega y)^2 + \... 1answer 213 views D'Alembert's principle and equation of motion Is obtaining proper equation of motion from D'Alembert's principle a mere coincidence or there is some logic behind this? This is asked because while we are finding the equations in a regime where ... 0answers 67 views Block sliding down hemisphere problem (harder) (need pleb Newtonian explanation) So this is from a pretty recent (freshman-level) physics university exam. Part (a) is relatively simple and standard for a freshman physics course. What I am finding much more difficult is part (b). ... 1answer 177 views Constrained Hamiltonian systems: spin 1/2 particle I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action S=-m\int ... 2answers 124 views Geodesic equations from action with auxiliary field A textbook says that the geodesic equations (for both massive and massless) can be derived from the following action: $$S = -\frac{1}{2} \int d\tau \:\eta \: (\eta^{-2} \dot{x}^\mu \dot{x}^\nu g_{\... 5answers 534 views Hamiltonian for relativistic free particle is zero One possible Lagrangian for a point particle moving in (possibly curved) spacetime is$$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$where a dot is a derivative with respect to a parameter \... 3answers 113 views Does Noether's theorem apply to constrained system? The Lagrangian of a constrained system will be$$L-\lambda_1f_1-\lambda_2f_2-...\lambda_kf_k.$$If a transformation will not affect the constrained Lagrangian, the there is some corresponding ... 0answers 30 views If two surfaces slip on each other then their velocities along the common normal are equal? If two surfaces slip on each other then their velocities along the common normal are equal? 1answer 128 views Rigorous definition of generalized coordinates In Goldstein's classical mechanics and in many other books I haven't seen a rigorous definition of generalized coordinates. In a system of N particles described by \textbf{r}_1, \dots, \textbf{r}... 1answer 134 views Dirac Brackets in General Relativity I want to calculate Dirac brackets of different phase space variables in gravity. In case of electrodynamics, one does the same using the following steps: Looking at the momenta to find that \Pi^... 1answer 151 views Constraints in general relativity In this review on inflation, on Pg. 135, Baumann talks about the energy and the momentum constraints for gravity. Are these equations the G_{00} = T_{00} and G_{0i} = T_{0i} components of the ... 1answer 131 views Physical Constraints In physics, what does one mathematically mean by constraint in classical mechanics? What are the the different types/cases and how do people deal with them? 2answers 145 views When the constraints are not holonomic, why is it not possible to find such q_is that \delta q_is are independent of each other? In the book of Classical Mechanics by Goldstein, at page 20, it is given that However, I cannoot understand from what has been presented so far that when the constraints are not holonomics, why is it ... 1answer 75 views D'Alembert's principle when the mass of the particles are changing In the book of Classical Mechanics by Goldstein, at page 19, while deriving D'Alembert's principle, the author assumes that$$\dot p = m \ddot r.$$However, when the mass of the bodies also changes, ... 2answers 97 views Finding the value of the holonomic constraint forces So let's say I have a Lagrangian augmented with some holonomic constraints.$$L' = L + \sum_i \lambda_i(t) f_i(q,t).\tag{i}$$The solutions is the system of differential equations:$$\frac{\partial ... 2answers 139 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 ... 0answers 79 views Going downhill - what's the constraint force? [closed] We are going downhill on a path $$y=a(1-x^2),$$ and I need to calculate the constraint force-position function. What I've done is this: The lagrangian of the system is $$L=\frac{1}{2}(\dot{x}^2+\dot{... 0answers 49 views Lapse and shift inside or outside the Poisson bracket? For general relativity in the 3+1 ADM formulation, one has H=\int dx [N{\cal H}+N^a{\cal H}_a] with N and N^a the lapse and shift which are undetermined Lagrange multipliers. The dynamical ... 1answer 65 views Restriction Forces in Lagrangian Mechanics I was recently preparing for a test on Classical Mechanics and a friend of mine started wondering if there was any method through which we could obtain the restriction forces acting on a certain ... 0answers 97 views How to derive the Hamiltonian of general relativity (ADM formalism without surface terms)? Given that$$ds^{2} =−N^{2}dt^{2} +h_{ij}(dx^{i} +N^{i}dt)(dx^{j}+N^{j}dt)S=\int dt d^{3} x\sqrt{h} N(^{3}R+K_{ij}+K^{ij}-K^{2})$$where ^{3}R is the Ricci scalar of hij, h the ... 1answer 151 views Deriving the geodesic equation using a Lagrange multiplier to fix affine parametrisation The geodesic equation can be derived using the action$$S_0 ~=~ \int d\tau \sqrt{-\dot{x}_\mu\cdot \dot{x}^\mu}\tag{1}$$(I am using the (-+++) convention and \dot{x} = \frac{dx}{d\tau}). To ... 1answer 121 views Polyakov Lagrangian and Lagrange multipliers I'm reading Polchinski's Introduction to String Theory (volume I) and something got me quite puzzled in the beginning (At the top of page 19 to be precise). This part is about the open string and the ... 1answer 247 views Lagrangian of point mass on rod inside ring with holonomic constraint I am given a massless ring of radius R that is rolling along a flat plane without slipping. There is friction. A massless rod of length \frac {R} {2} is attached to the inside edge of the ring ... 1answer 202 views Hamiltonian Structure of Chern Simons Electrodynamics I am reading the review paper "Aspects of Chern-Simons Theory" by Gerald Dunne https://arxiv.org/abs/hep-th/9902115 Starting from p. 17, Dunne works on the Hamiltonian structure of the CS ... 0answers 91 views Variation with respect to a traceless symmetric tensor Suppose we have an action variation like$$\delta S[G]=\int \mathfrak{H}^{\mu\nu}\delta G_{\mu\nu} \,\, d^Nx,$$where$\mathfrak{H}^{\mu\nu}$is a tensor density. If the variation with respect to$...
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Consider a particle constrained to a manifold $\mathbb Q$ embedded within standard Euclidean 3-dimensional space that experiences no forces other than constraint forces keeping it within $\mathbb Q$. ...
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Hamiltonian for a variable length pendulum

This question is taken from the book "Classical Dynamics of Particles and Systems" - Marion, problem 7.24. The problem is about a pendulum that is set into motion, it's length varies at a constant ...
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Holonomic and non-holonomic constraints

Is it possible for a system to have holonomic and non-holonomic constraints at the same time? If so, in this scenario does it make sense to talk about a set of independent 'generalized coordinates'?
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Constraint forces do no virtual work: does this always apply?

The thread title is my main question, but to give some context, I'll include a particular example that made me ask the question in the first place. In Hand and Finch, a small block is on a ...
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Constraint equation for Einstein-Hilbert action in Light-cone gauge

In Light cone coordinate system $(+,-,i)$, where $i=1,2$, the light cone coordinates are defined as $x^{\pm}=\frac{x^0 \pm x^3}{\sqrt{2}}$, if we consider the $+$ coordinate to be our "timelike" ...
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Question on the no-slipping constraint of a cylinder rolling off another cylinder

Why is the constraint not $R*\theta_1 = a*\theta_2$? Why isn't the arc length cut out by the smaller cylinder proportional to its angle of rotation?
I'm faceing a problem of a thin wheel of radius R rolling without slipping on a track (y = f(x); on xy-plan). The wheel plane stays vertical and tangent to the track at the contact point P. $\alpha$ ...