# Tag Info

15

The generalized coordinates of a system of $N$ particles apply to the system as a whole, not the individual particles, and accordingly they can (and often do) combine the coordinates of multiple particles. One common example is that of two-body orbital motion: one generalized coordinate is the position of the center of mass of the system, $$\mathbf{q}_1 = ... 8 You seem to be talking about the "old covariant quantization" in which L_n for positive n and (L_0-a) annihilate physical ket states |\psi\rangle, right? It's analogous to the Gupta-Bleuler quantization http://en.wikipedia.org/wiki/Gupta-Bleuler_quantization which was a standard procedure used already in electromagnetism. The idea is that the ... 7 Given a system of N point-particles with positions {\bf r}_1, \ldots , {\bf r}_N; with corresponding virtual displacements \delta{\bf r}_1, \ldots , \delta{\bf r}_N; with momenta {\bf p}_1, \ldots , {\bf p}_N; and with applied forces {\bf F}_1^{(a)}, \ldots , {\bf F}_N^{(a)}. Then D'Alembert's principle states that$$\tag{1} \sum_{j=1}^N ( ...

6

1) According to usual terminology we wouldn't call a sliding friction force a constraint force as it doesn't enforce any constraint. (No pun intended.) In other words, a sliding friction does not by itself constrain the particles to some constraint subsurface, i.e., the particles can still move around everywhere. On the other hand, rolling friction and ...

5

You've duplicated constraints because if any one particle is constrainined in all three dimensions with all the other particles this constrains all the particles. The number of constraints is 3(N - 1). To give an example, take three particles a, b and c. If a is fixed relative to b and is also fixed relative to c, then b and c are fixed relative to each ...

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A modern treatment of this subject can be found in Segreev's book on the Kahler geometry of loop spaces also available online. This line of research started with the seminal work of Bowick and Rajeev: The holomorphic geometry of the closed bosonic string theory and $Diff S^1/S^1$ (Spires) (and independently Kirillov and Yuriev (Please see the reference in ...

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An equation of motion is a (system of) equation for the basic observables of a system involving a time derivative, for which some initial-value problem is well-posed. Thus a continuity equation is normally not an equation of motion, though it can be part of one, if currents are basic fields.

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If you work with a smaller number of coordinates (usually "curved ones" in a sense) and no Lagrangian multipliers, you are simply considering a configuration space that is a submanifold of the full configuration space in the calculation that does include Lagrange multipliers. Extremizing the action $S_{full}$ with Lagrange multipliers $$\delta S_{full} = ... 4 (1) You have a set of irreducible constraints, \lbrace \phi_j\rbrace, both primary and secondary This set of constraints defines a submanifold M within the "full" (unconstrained) phase space. (2) A function on the phase space is set to be weakly zero if it vanishes when restricted to the constrained submanifold M. A function is called strongly zero if ... 4 Well, when canonically quantizing a system with constraints, you have two methods: Dirac's approach "Quantize, then Constrain"; Reduced Phase Space approach "Constrain, then Quantize". Although these two approaches have analogs with path integral quantization, the Path integral approach sweeps a lot of problems under the rug when you pick a particular ... 4 Every rigid body has 3 translational dof. In addition, there are 0, 2, or 3 rotational dof, depending on the geometry, giving a total of 3, 5, or 6 dof. A spherically symmetric rigid body has no other dof. A rigid body with rotational symmetry around an axis has 2 rotational dof, namely two angles for orienting the symmetry axis along a direction. All ... 4 The canonical momenta don't change if you add a total derivative to the Lagrangian. The particular total derivative you wanted to add to the Lagrangian as well as the Lagrangian itself has free i,j indices. You surely meant something else because the Lagrangian should have no free indices like that. Let me assume that you meant both expressions to be ... 3 In field theory, a conservation law just states that some quantity is conserved: if \partial_\mu \, \star = 0 where \star is a vector or a tensor, you can associate a conserved charge etc. - you know the spiel I guess. Constraints are something you impose by hand (or by experiment). Finally, equations of motion are dynamical equations that follow from ... 3 There is a simple reason why we can consider variations on the whole A rather than the quotient C=A/G and the reason is following: all configurations that are G-equivalent have the same value of the action S. That's what we mean by the statement that the theory has the symmetry G. So the variation of the action S in the directions that are ... 3 OP wrote(v2): What makes an equation an 'equation of motion'? As David Zaslavsky mentions in a comment, in full generality, there isn't an exact definition. Loosely speaking, equations of motion are evolution equations, with which the dynamical variables future (and past) behavior can be determined. However, if a theory has an action principle, then ... 3 Constraints are handled in Lagranian mechanics through either of two approaches: 1) The constraint equation is used to reduce the degrees of freedom of the system. For example, if a particle is constrained to the surface of a sphere, then the Lagrangian can be written entirely in terms of two generalized coordinates and their associated momenta (typically, ... 3 Yes, of course that the p-v relationship may be transcendental so that it cannot be inverted in terms of elementary functions. That doesn't mean that the inverse function doesn't exist, however. Even functions that can't be written down in terms of elementary functions may exist. For example, consider the Lagrangian$$ L =\exp(bv^2)\cdot mv^2 $$It ... 3 Adding to Lubos Motl's correct answer, it should be stressed that one may not always invert the relation p_i=f_i(q,\dot{q},t) to isolate \dot{q}^j, not even in principle, because of constraints. Such cases are known as singular Legendre transformations, and they are the starting point of the topic of constrained dynamics. Example. Consider e.g. the ... 3 I) Let us suppress position dependence q^i and explicit time dependence t in the following, and also assume that the Lagrangian L=L(v) is a smooth function of the velocities v^i, where i=1, \ldots, n. The Hessian matrix is defined as$$\tag{1} H_{ij}~:=~\frac{\partial^2 L}{\partial v^i \partial v^j}.$$Let us consider an open neighborhood^1 ... 3 I managed to find a quasi-systematic way to do this. The idea that allowed me to do this was inspired by Noether's Theorem. Re-parameterization invariance is a symmetry of the system, a symmetry much stronger than an ordinary global symmetry. Similarly, however, a constraint is also a conserved quantity, but it is something much stronger than that. ... 3 We note$$ \left[ \alpha_m^0, \alpha_n^0 \right] = \eta^{00} \delta_{m+n,0} = - \delta_{m,-n} $$A timelike excitation is \alpha_{-n}^0 \left| 0; k \right>. The norm of this state is \begin{split} \left<0;k'\right| \alpha_{m}^0 \alpha_{-n}^0 \left|0;k\right> &= \left<0;k'\right| \left( \left[ \alpha_{m}^0 , \alpha_{-n}^0 ... 3 Basically, the multiplier method is a way to encode the constraint information of the system directly into the Lagrangian so that you don't have to worry about screwing up the physical requirements of the problem when you solve the equations of motion. In other words, instead of solving the equations of motion and constraining the results, you're ... 2 The problem is that you are double counting a lot of your constraints. If the (vector) displacements between particles A and B, and between B and C is fixed, then the displacement between A and C is fixed. Therefore the constraint on distance between A and C is redundant, and you can't count it separately. 2 The k holonomic constraints are used to eliminate k qs, so reducing their number from 3N to 3N-k. This then introduces the dependence of some of the transformation equations on t and other qs. You have k holonomic constraints of the form$$\mathbf{f}_1(q_1, q_2,..., q_{3N},t) = 0...\mathbf{f}_k(q_1, q_2,...,q_{3N}, t) = 0$$3N q ... 2 The first step is the completion of the whole constraint list (primary, secondary , ternary etc.), and checking that no secondary constraint leads to a contradiction (i.e., empty constraint surface). Remark: The time evolution of the constraints is performed with the "total" Hamiltonian: H_T(p,q) = p\dot{q}-L+\sum_{\alpha}\lambda_{\alpha} ... 2 The number of physical degrees of freedom (DOF) or dynamical variables is simply the number of generalized positions whose evolution is given by a second order in time differential equation. Using the OP's notation, the number of DOF is$${1\over 2}(N-2M-S) For instance, in electrodynamics the phase-space is six-dimensional $\{A_i,F_{0i}\}_{i=1}^3$ and ...

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In general, a dynamical equation of motion or evolution equation is a (hyperbolic) second order in time differential equation. They determine the evolution of the system. $\partial_{\mu}F^{i\mu}$ is a dynamical equation. However, a constraint is a condition that must be verified at every time and, in particular, the initial conditions have to verify the ...

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Here tension is zero for very sort span of time (infinitesimally sort time), or for an instant only. When ever it moves away from the vertically top position it will fill tension again. So it will move in circular trajectory. And if we consider instantaneous velocity it is tangential. I think this clarify your doubt.

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If the constrained Hamiltonian system has a finite number of real degrees of freedom$^1$, and if all the constraints are regular, then it is mathematically impossible to have an odd number of second-class constraints. (The proof is very similar to the reason why a symplectic manifold or vector space must be even-dimensional.) Perhaps OP is actually ...

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