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2
votes
1answer
23 views

Degrees of freedom of a point mass sliding on a rigid curved wire without friction

I am very new to the subject and am going through Structure and Interpretation of Classical Mechanics. One exercise asks to find the degrees of freedom of a number of systems, one of which is a ...
2
votes
1answer
163 views

Reduction of Nambu Goto action to true degrees of freedom

First consider the particle $$S=m\int\sqrt{-\dot{X}^2}d\tau$$ if you choose the static gauge $\tau=X^0$ and replace it in the action you get $$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau$$ So now, you ...
2
votes
2answers
138 views

How to find the rank of the matrix $\frac{\partial ^2\mathcal{L}}{\partial \dot{X^\mu} \partial \dot{X^\nu} }$ for the Nambu-Goto string Lagrangian?

In this case $$\mathcal{L}~=~-T\sqrt{-\dot{X^2}X'^2+(\dot{X}\cdot X')^2}.$$ I was reading some books and papers about the constraints in the Nambu-Goto action, and all of them say something like ...
1
vote
0answers
23 views

Advantages of having a first class system and possibility of transforming a system into a first class one

I have two questions regarding first class systems. What are the advantages of having a first class Hamiltonian (a Hamiltonian whose all constraints are first class) in a theory or having a first ...
-1
votes
1answer
36 views

Parabolic slide [closed]

Can you give a big hint for this problem please? A material point is released at the end of a track in the form of a vertical arc of parabola $y = x^2$ in $[-1,1]$, the meter is selected as a unit ...
1
vote
1answer
42 views

What is the position as a function of time for a mass falling down a cycloid curve?

In the brachistochrone problem and in the tautochrone problem it is easy to see that a cycloid is the curve that satisfies both problems. If we consider $x$ the horizontal axis and $y$ the vertical ...
0
votes
0answers
31 views

Can some one explain the types of constraints with easy and simple examples?

I have read different types of constraints like primary, secondary, 1st class and 2nd class. I have a little idea but not enough. wikipedia couldn't help here. It will be so nice if some one explains ...
0
votes
1answer
41 views

Infinite Energies of a particle in a rectangular box

For a particle trapped inside a rectangular box of side lengths $l_x$ $l_y$ and $l_z$, the energies are ...
3
votes
2answers
143 views

Non-relativistic QFT Lagrangian for fermions

Take the ordinary Hamiltonian from non-relativistic quantum mechanics expressed in terms of the fermi fields $\psi(\mathbf{x})$ and $\psi^\dagger(\mathbf{x})$ (as derived, for example, by A. L. Fetter ...
2
votes
1answer
87 views

Hamiltonian field equations constraints

Let's consider the Lagrangian $$\mathcal{L}~=~-\frac{1}{2}(\partial_\mu\phi^\nu)^2+\frac{1}{2}(\partial_\mu\phi^\mu)^2+\frac{1}{2}m^2\phi_\mu \phi^\mu,$$ with Minkowski metric $\eta_{\mu\nu}={\rm ...
1
vote
1answer
140 views

Calculus of variations and string theory

In Polchinski's String theory book, Vol 1., in chapter 1, p. 18, he is deriving the Lagrangian in the light cone gauge (that's not necessary to know in order to answer this question), and he gets ...
2
votes
1answer
72 views

Lapse and shift in ADM decomposition

Poisson in Relativist's Toolkit and also other authors in various papers state explicitly that after one does the 3+1 decomposition, the lapse and shift $N$ and $N^a$ are non-dynamical variables, and ...
2
votes
2answers
125 views

Hamiltonian from a Lagrangian with constraints?

Let's say I have the Lagrangian: $$L=T-V.$$ Along with the constraint that $$f\equiv f(\vec q,t)=0.$$ We can then write: $$L'=T-V+\lambda f. $$ What is my Hamiltonian now? Is it $$H'=\dot q_i p_i ...
0
votes
1answer
42 views

Degrees of freedom in double Atwood machine?

Why the degree of freedom in double Atwood machine (one block on one side and a pulley with one block in its each side on other side) is 2 and not 1? According to the formula $s=3*n-m$; where ...
0
votes
0answers
28 views

Unilateral Torque Constraint on the foot-ground interface

I was studying the basics of legged locomotion and came across the unilateral force and torque constraints at the foot-ground interface. I understood the implication of the unilateral constraint on ...
1
vote
2answers
140 views

Definition of generalised coordinates?

I think the definition of generalised coordinates is something along the following lines: A set of parameters that discribe the configuration of a system with respect to some refrence ...
0
votes
1answer
67 views

Simple explanation of first and second class constraints with an example

Can someone give a simple physical example of first class and second class constraints? I mean, if you were giving a classical mechanics lecture for undergraduates, how would you explain this concept ...
0
votes
1answer
86 views

Motion in the gravitational field along a trajectory [closed]

A point particle of mass $m$ is moving in the gravitational field along some trajectory which is described by the function $y=y(x)$, which is in a vertical plane and is continuous. Initial ...
0
votes
1answer
31 views

Locally accessible dimensions of configuration space

I am reading a book called "Structure and Interpretation of Classical Mechanics" by MIT Press.While discussing configuration space and degrees of freedom,the authors remark the following: Strictly ...
4
votes
2answers
57 views

Confusion with potential in simple pendulum

I'm a maths student taking a course in classical mechanics and I'm having some confusion with the definition of a potential. If we consider a simple pendulum then the forces acting on the end are ...
0
votes
1answer
70 views

Normal force on object on a curved 3D surface

Let's imagine we are on the top of some axisymmetric surface. Let it be $r = r(z)$ in cylindrical coordinates $\left(r,\varphi, z\right)$. It can be a sphere, which we have discussed in my previous ...
1
vote
1answer
97 views

Coupled wheel and rod (analytical mechanics) [closed]

I am struggling with formulating the equations of motion. Consider a coordinate system with origin in $O$ ($y$ upwards and $x$ to the right), label the center of mass of rod $AB$ with $G$ then: ...
3
votes
2answers
113 views

Hamiltonian for a Lagrangian with coupling

I am dealing with the following Lagrangian density $$\mathscr{L}_{em}= -\frac{1}{2}\rho\omega^2 u^2 +\frac{1}{2}\nabla u:\Sigma :\nabla ...
6
votes
2answers
213 views

Mass particle trajectory on a sphere

So, I am trying to simulate mass particle motion on the outer surface of sphere using cartesian coordinates. Let's conclude just a gravity and frictionless movement. Sphere $x^2 + y^2 + z^2 = 1$, ...
0
votes
1answer
34 views

Is the movement of a projectile in 2D a Holonomic system? [closed]

Is the well known problem of the movement of a projectile, no friction, in two dimensions a holonomic system? If yes.. Why? If Not.. Why?
0
votes
1answer
66 views

Finding Lagrangian with Non-holonomic constraints

I am stuck working on a problem that involves finding the Lagrangian for a free particle constrained to move on the surface of a disk of radius $a$. The particle collides elastically with the edge of ...
1
vote
1answer
99 views

Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I ...
1
vote
0answers
50 views

Calculation of the Poisson bracket of a (Classical) Yang-Mills generator

This question might be too technical or minute, but I believe someone can give me the right advise. What I want to calculate is a Poisson bracket algebra of classical YM gauge generators, ...
0
votes
2answers
153 views

A question regarding 3 bodies connected as a system

Let us consider three bodies of equal mass connected to each other with 2 ideal strings of length l. The three bodies are placed in a straight line In this arrangement there is 1 body connected to 2 ...
2
votes
1answer
99 views

Poisson brackets for constrained system

Let's have some Hamiltonian which involves the set of first class constraints $\varphi$ and set of constraints $\kappa $, which play role of canonical conjugated momentums for $\varphi$,. They're ...
3
votes
1answer
116 views

Yang-Mills constraints and Poisson brackets

Let's have constraints for Yang-Mills theory: $$ \varphi_{a} = \partial_{i}\pi^{i}_{a} - f_{abc}\pi^{b}_{i}A^{c}_{i}. $$ I have read the statement that $$ \tag 1 [\varphi_{a}(\mathbf x), ...
0
votes
0answers
44 views

Question on a variant of Atwood machine with strings unwinding in one side

The figure is standard Atwood machine except that for the right object string tension acts on the left side of the object, and strings unwind from the object. No slip condition on a rolling ...
0
votes
1answer
198 views

Position based dynamics constraint scaling factor

Reading through Müller et al., Position Based Dynamics, 2007 I got lost when passing from equation (5) $$\Delta p = \frac{C(p)}{|\nabla_pC(p)|^2}\nabla_pC(p)$$ to equation (6) (and applying the ...
1
vote
1answer
130 views

Variational principle for a point particle (massive or massless) in curved space

We know that for a point particle, the action is $$ S[x,e] ~=~ \frac{1}{2}\int_{\lambda_A}^{\lambda_B} d\lambda\left[e^{-1}(\lambda)~g_{\mu\nu}(x(\lambda))~\dot{x}^\mu(\lambda)~\dot{x}^\nu(\lambda) ...
3
votes
2answers
168 views

Ball Bearing Inside a Hollow, Spinning Rod: where is the logical flaw?

As described in the title, suppose we have a frictionless, hollow rod that is rotating in the $xy$-plane with some fixed angular velocity $\omega$. The rod is pivoting around its midpoint. Suppose we ...
3
votes
0answers
199 views

How is the method of Lagrange multipliers used for multiple constraints of multiple variables? [closed]

Let's say for example that I have two constraints $f(x,\dot{x},y,\dot{y})$ and $g(x,\dot{x},y,\dot{y})$ and a Lagrangian $L(x,\dot{x},y,\dot{y})$. What are the Euler-Lagrange equations of the first ...
0
votes
1answer
62 views

Why is the slippage constraint for one moving cylinder and one fixed cylinder $r(\phi - \theta)=R \theta$? [closed]

Why is the slippage constraint for one moving cylinder and one fixed cylinder $r(\phi - \theta)=R \theta$? Every time I write it down on paper I get the result $r\phi = R \theta$. I am not sure if I ...
2
votes
1answer
178 views

Example of Hamilton's Principle to Systems with Constraints (Goldstein)

I'm currently studying Goldstein's Classical Mechanics book and I can't get my head around his reasoning in section 2.4. (Extending Hamilton's principle to systems with constraints). I'd like to ...
2
votes
3answers
159 views

Virtual Work: How is the applied force related to the coordinates chosen?

I have a question after reading a section from Goldstein's Classical Mechanics. The question deals with equation 1.43 in the text (given below): $$ \tag{1.43} \sum\limits_{i} {\bf F}_i^{(a)}\cdot ...
2
votes
0answers
104 views

Intuition behind the principle of virtual work

To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset ...
5
votes
1answer
358 views

Why are D'Alembert's Principle and the Principle of Least Action Related?

Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common. $$\sum _i ( ...
6
votes
2answers
136 views

Dirac bracket and second class constraints in first-order gravity formalism

In the first order formulation of general relativity, the frame field $e_{\mu}^a$ and $\mathrm{SO}(3,1)$ spin connection $\omega_{\mu c}^b$ are independent variables. In the Hamiltonian formulation of ...
2
votes
3answers
193 views

Configuration manifolds and constraints

In Classical Mechanics there's this notion of configuration manifold. Although I've heard about that a lot and although I often use that concept, I'm not sure I really understand them well because ...
11
votes
3answers
2k views

What exactly is a virtual displacement in classical mechanics?

I'm reading Goldstein's Classical Mechanics and he says the following: A virtual (infinitesimal) displacement of a system refers to a change in the configuration of the system as the result of any ...
4
votes
3answers
172 views

About constraints of the first class and electrodynamics

Consider a theory in the Hamiltonian formalism and assume that it has constraints between canonical variables $Q, \pi$. By the Dirac terminology, the set of constraints $F_{a}(Q, \pi) \approx 0$ of ...
0
votes
1answer
126 views

How do I find the generalized coordinates in a certain system?

I'm learning about constraints and I know the following: If there are $N$ particles in 3 dimensional space, I have $3N$ degrees of freedom. If I have $n_b$ holonomic constraints and I switch over to ...
0
votes
0answers
67 views

Question about an example of non-integrable constraints

The example is a thin disk rotating of an inclined plane. The disk can roll not only down the plane, but also "sideways". Let $(x,y)$ be the position of the CM, where the $y$ axis is down the slope ...
4
votes
2answers
206 views

Lagrange multiplier and constraint force

The Lagrangian with Lagrange multiplier in the form $$L= T- V + \lambda f(q, \dot{q},t).$$ But there are different ways of writing the constraint $f = 0$. Will that lead to different EOMs? Let me ...
1
vote
0answers
96 views

Restrained double pendulum

The equations of motion of a double pendulum are well-known. Usually you'd have the them expressed in the rotations $\theta_1(t)$ and $\theta_2(t)$. There are two degrees of freedom. Now consider the ...
0
votes
0answers
72 views

Classical rod-wall-floor system

I have a homogeneous thin rod $AB$. $A$ can slide along $z$-axis, $B$ along $xy$ plane. There is no friction. We have any initial conditions. Now the question is to write down the equation of motion ...