Summary: In QFT, if we add a gauge breaking term to the Lagrangian, do we still need to introduce Faddeev-Popov ghost particles?
Ghosts seems to be introduced to maintain gauge invariance. But suppose we have eliminated the gauge invariance, from the start, by explicitly introducing a gauge...
On page 224 of the 5th edition of Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion, the authors introduced the ##δ## notation (in section 6.7). This notation is given by Equations (6.88) which are as follows:
$$\delta J = \frac{\partial J}{\partial...
1. Homework Statement
A rigid cylinder of radius ##R## and mass ##\mu## has a moment of inertia ##I## around an axis going through the center of mass and parallel to the central axis of the cylinder. The cylinder is homogeneous along its central axis, but not in the radial and angular...
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Description of the system:
The masses m1 and m2 lie on a smooth surface. The masses are attached with a spring of non stretched length l0 and spring constant k. A constant force F is being applied to m2.
My coordinates:
Left of...
1. Homework Statement
A circular hoop of radius R rotates with angular frequency ω about a vertical axis coincident with its diameter. A bead of mass m slides frictionlessly under gravity on the hoop. Let θ be the bead’s angular position relative to the vertical (so that θ = 0 corresponds to...
In Lagrangian mechanics, both q(t) and dq/dt are treated as independent parameters. Similarly, in Hamiltonian mechanics q and p are treated as independent. How is this justified, considering you can derive the generalized velocity from the q(t) by just taking a time derivative. Does it have...
1. Homework Statement
A bead of mass m slides in a frictionless hollow open-ended tube of length L which is held at an angle of β to the vertical and rotated by a motor at an angular velocity ω. The apparatus is in a vertical gravitational field.
a) Find the bead's equations of motion
b)...
In field theory a typical Lagrangian (density) for a "free (scalar) field" ##\phi(x)## is of the form $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi -\frac{1}{2}m^{2}\phi^{2}$$ where ##m## is a parameter that we identify with the mass of the field ##\phi(x)##.
My question is...
In 3rd edition of Goldstein's "Classical Mechanics" book, page 335, section 8.1, it is mentioned that :
In Hamiltonian formulation, there can be no constraint equations among the co-ordinates.
Why is this necessary ? Any simple example which will elaborate this fact ?
But in Lagrangian...
1. Homework Statement
Two blocks of equal mass, m, are connected by a light string that passes over a massless pulley. One block hangs below the pulley, while the other sits on a frictionless horizontal table and is attached to a spring of constant k. Let x=0 be the equilibrium position of the...