Lagrangian mechanics Definition and 188 Threads

Homework Statement The problem is the following: Using a Lagrangian, find the equations of motion of a mass hanging from a massless string, with the pendulum pivot moving in a horizontal circle at constant angular velocity. I take the mass to be m, the length of the string L, the radius of the...

Please help, I've put this is true on my personal statement, but I'm now fairly sure that it is not, but a more expert opinion would be extremely helpful.

In Lagrangian mechanics the energy E is given as : E = \frac{dL}{d\dot{q}}\dot{q} - L Now in the cases where L have explicit time dependence, E will not be conserved. The notes I am referring to provide these two examples to distinguish between the cases where E is energy and it is not...

Hello, I am stuck on the following problem. 1. Homework Statement Consider the continuous family of coordinate and time transformations (for small ##\epsilon##). Q^{\alpha}=q^{\alpha}+\epsilon f^{\alpha}(q,t) T= t+\epsilon \tau (q,t) Show that if this transformation preserves the action...

First, let me take as the definition of a Lagrangian the quantity that when put into the Euler Lagrange equations, it gives the correct equation of motion. It sounds like we need to know the equations of motion first. For example. the Lagrangian for a particle subject to a constant magnetic...

What exactly was the purpose for the development of Lagrangian mechanics? Does it describe physical systems and situations that Newtonian mechanics cannot? I would also like to know why the Hamiltonian reformulation of mechanics occurred after the development of Lagrangian mechanics.

Homework Statement a. Suppose two particles with mass $m$ and coordinates $x_1$, $x_2$ collides elastically, find the lagrangian and prove that the linear momentum is preserved. b. Find new coordiantes (and lagrangian) s.t. the linear momentum is conjugate to the cyclical coordinate. Homework...

Homework Statement Consider a theory with a \phi^6-scalar potential: \mathcal{L} = \frac{1}{2}(\partial_\mu\phi)^2-\phi^2(\phi^2-1)^2. Why is the solution to the equation of motion not a soliton? Homework Equations \phi''=\frac{\partial V}{\partial\phi} The Attempt at a Solution...

This is not a homework equation at all, however I have devised my own example problem in order to convey my misunderstanding. (My question is at the end of the problem) Question that I had come up with: A particle's motion is described in the x direction by the equation x = x(t). The particle's...

An example problem in Chapter 7 of "Classical Dynamics of Particles and Systems" by Marion, Thornton uses Lagrangian equations with undetermined multipliers to solve for the motion of a disc rolling down an incline. The resulting Lagrangian equations are: Mg sin α - M d2y/dt2 + λ = 0...

I've been asked to find the conserved quantities of the following potentials: i) U(r) = U(x^2), ii) U(r) = U(x^2 + y^2) and iii) U(r) = U(x^2 + y^2 + z^2). For the first one, there is no time dependence or dependence on the y or z coordinate therefore energy is conserved and linear momentum in...

Homework Statement (a)Find Christoffel symbols (b) Show the particles are at rest, hence ##t= \tau##. Find the Ricci tensors (c) Find zeroth component of Einstein Tensor Homework EquationsThe Attempt at a Solution Part (a)[/B] Let lagrangian be: -c^2 \left( \frac{dt}{d\tau}\right)^2 +...

Homework Statement An object of mass m, and constrained to the x-y plane, travels frictionlessly along a curve f(x), while experiencing a gravitational force, m*g. Starting with the Lagrangian for the system and using the method of Lagrange multipliers, derive the equations of motion for the...

This is a slightly physics oriented question, so apologies for that. Basically, having started studying differential geometry it has started to become a little clearer to me why one can consider the Lagrangian as a function of position and velocity, but I don't feel I'm quite there yet. My...

Homework Statement A particle slides on the outer surface of an inverted hemisphere. Using Lagrangian multipliers, determine the reaction force on the particle. Where does the particle leave the hemispherical surface? L - Lagrangian qi - Generalized ith coordinate f(r) - Holonomic constraint...

Homework Statement [/B] A uniform solid ball of mass m rolls without slipping down a right angled wedge of mass M and angle θ from the horizontal, which itself can slide without friction on a horizontal floor. Find the acceleration of the ball relative to the wedge. 2. The attempt at a...

Homework Statement I'm trying to do a little review of Lagrangian Mechanics through studying the two-body problem for a radial force. I have the Lagrangian of the system L=\frac{1}{2}m_1\dot{\vec{r_1}}^{2}+\frac{1}{2}m_2\dot{\vec{r_2}}^{2}-V(|{\vec{r_1}-\vec{r_2}}|) . Now I'm trying to find...

Homework Statement The problem is a Lagrangian problem that solves for a differential equation. I need to write a program to solve the Lagrangian numerically. My professor said you do not need mass for the program, but I'm not sure how. The problem is a vertical cone with a bead rolling around...

Hi all, I've recently been asked for an explanation as to why the Lagrangian is a function of the positions and velocities of the particles constituting a physical system. What follows is my attempt to answer this question. I would be grateful if you could offer your thoughts on whether this is...

The problem goes by this: A sphere of radius ##\rho## is constrained to roll without slipping on the lower half of the inner surface of a hollow cylinder of inside radius R. Determine the Lagrangian function, the equation of constraint, and Lagrange's equations of motion. Find the frequency of...

If I stated a problem that you have to find the solution [0,\infty[\;\to\mathbb{R},\quad t\mapsto x(t) to the problem x(0) = x_0 < R \dot{x}(0) = v_0 > 0 m\ddot{x}(t) = -\partial_x U\big(x(t)\big),\quad\quad m>0 where R, v_0, m are some constants, and the function U has been defined...

the question is that there is a particle in 3 spatial Euclidean dimensions in cylindrical coordinates. I want to find a symmetry for the lagrangian if the potential energy is function of r and k.theta+z V=V(r,k.theta+z) any help please ?

Hi guys. I hope this isn't a bad place to post my question, which is: I'm reading some lecture notes on Lagrangian mechanics, and we've just derived the Euler-Lagrange equations of motion for a particle in an electromagnetic field. It reads: m \ddot{\vec{r}} = -\frac{e}{c} \frac{\partial...

I have just started studying Lagrangian Mechanics, and I can find decent material on the internet that describes the theory behind it, several proofs on equivalence and even some good solved examples. However, I would really appreciate if someone could recommend a book that has some of the...

Homework Statement I've thought of a problem to help me with Lagrange multipliers but have got stuck. Consider a particle of mass m moving on a surface described by the curve y = x2, the particle is released from rest at t = 0 and a position x = l. I'm trying to work out the EOM's but have...

Hi there, I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are the same is presented in the book I am reading as a rule, commutativity, and...

Homework Statement This is the problem: http://i.imgur.com/OJyzfhz.png?1 Homework Equations $$\frac{dL}{dq_i}-\frac{d}{dt}\frac{dL}{d\dot{q_i}}=0$$ $$ L=\frac{1}{2}mv^2-U(potential energy)$$ The Attempt at a Solution This is my attempt at question A): http://i.imgur.com/IeJVGm3.jpgDoes...

I am learning Hamiltonian and Lagrangian mechanics and looking for a book that starts with Newtonian mechanics and then onto Lagrangian & Hamiltonian mechanics. It should have some historical context explaining the need to change the approaches for solving equation of motions. Also it should...

Hi we covered the Lagrange multiplier method in Lagrangian Mechanics and as far as I know, is the physical meaning behind this to be able to solve either some non-holonomic constraints or to get some information about the constraint forces. my problem is, i do not know the physical meaning of...

Write down the Lagrangian $\mathcal{L}(x_1,x_2,\dot{x}_1,\dot{x}_2)$ for two particles of equal masses, $m_1 = m_2 = m$, confined to the $x$ axis and connected by a spring with potential energy $U = \frac{1}{2}kx^2$. [Here $x$ is the extension of the spring, $x = (x_1 - x_2 - \ell)$ where $\ell$...

Homework Statement So we have started Lagrangian Mechanics in my class, and I really don't understand it at all. My teacher keeps doing the math on the board, but he hasn't really said what a Lagrangian is, and what an Action is. I really am lost from the start with these problems. Any help...

In the attached snip, the last few steps of the lagrangian equation is shown. I don't understand how the \frac{\delta V}{\delta\dot{q_j}}= 0. As an example let me take gravitational force. With change in velocity ( along the downwards direction obviously), there sure is a change in gravitational...

I wanted to solve the problem of a cone rotating on its side over a table, around an axis that pass through it's apex, like in the figure. What I want to find is the angular speed ω, the spin of the solid, such that the cone "stands" over it's apex. I don't know how to set the condition...

Homework Statement Write the lagrangian equations for: A simple pendulum whose suspension point oscillates horizontally in its plan according to the law x = a.cos(ωt)My problem is trying to know which are the generalized coordinates. i considered :x (θ) = a.cos(ωt) + l.sinθ y (θ) = l.cos...

In Newton's problem,and other central force problems in Classical Mechanics, you can get with decreasing the center of mass movement to the lagrangian: L=1/2m(r' ^2+r^2 \varphi'^2)-V(r) because \varphi is cyclic, you can write: \frac{d}{dt}(mr^2 \varphi')=0 or, defining the angular...

So in my internet readings on Lagrangian mechanics I started researching applications with non-potential and/or non-conservative forces and came across this page: http://farside.ph.utexas.edu/teaching/336k/Newton/node90.html This page is fascinating but I'm having a bit of difficulty...

In some texts about Lagrangian mechanics,its written that the generalized coordinates need not be length and angles(as is usual in coordinate systems)but they also can be quantities with other dimensions,say,energy,length^2 or even dimensionless. I want to know how will be the Lagrange's...

We all know the proof, from Newtonian mechanics, that the motion of the center of mass of a system of particles can be found by treating the center of mass as a particle with all the external forces acting on it. I want to prove the same think, but within the framework of Lagrangian mechanics...

Suppose we have a system with scleronomic constraints. Is the condition that ∂V/∂qj=0 for generalized coordinates qj a necessary condition for equilibrium? A sufficient condition? I managed to "prove" that the above condition is necessary and sufficient for any type of holonomic constaint...

i'm just ready to start QM and I looked at the text and I turned to Shro eq to see if I could understand it and they mentioned Hamiltonian operator. It looked like the book assumed knowledge of H and L mechanics. Do I need to know this stuff? I wasn't told by others that I needed this. I was...

First, to make sure i have this right, lagrangian mechanics, when describing a dynamic system, is the time derivative of the positional partial derivatives (position and velocity) of the lagrangian of the system, which is the difference between the kinetic and potential energy of the system...

Homework Statement (i) A particle of mass m moves in the x - y plane. Its coordinates are x(t) and y(t). What is the kinetic energy of this particle? (ii) The potential energy of this particle is V (y). The actual form of V will remain unspecied, except that it depends only on the y...

Homework Statement I want to be able to plot a trajectory wrt time of a ball that rolls without slip on a curved surface. Known variables: -radius/mass/moment of inertia of the ball. -formula for the curvature of the path (quadratic) -formula relating path length and corresponding height...

Homework Statement Essentially the problem that I am trying to solve is the same as in this topic except that it is for 3 springs and 3 masses https://www.physicsforums.com/showthread.php?t=299905 Homework Equations I have found similar equations as in the topic but I face a problem in...

Hello, Can anybody tell me why \nabla_{\textbf q} \langle \textbf q, P \textbf q \rangle = P \textbf q? Explanation of notations: q is an n-dimensional vector: \textbf q = (q_1, q_2, \cdots, q_n)^T P is an nxn-dimensional, real, Hermitian matrix \nabla_{\textbf q} :=...

I'm doing lagrangian mechanics and trying to understand my notes, are these three statements correct: 1. If kinetic energy is quadratic then energy equals generalised energy. 2. Saying kinetic energy of a system is quadratic is the same as saying none of the position vectors in a system...

The Wikipedia article regarding Lagrangian Mechanics mentions that we can essentially derive a new set of equations of motion, thought albeit non-linear ODEs, using Lagrangian Mechanics. My question is: how difficult is it usually to solve these non-linear ODEs? What are the usual numerical...

Is anyone good with Lagrangian mechanics applied to constrained systems? I had a question about the Lagrange multiplier method, maybe I should have posted it in this section. https://www.physicsforums.com/showthread.php?t=550139

Homework Statement A heavy particle is placed at the top of a vertical hoop. Calculate the reaction of the hoop on the particle by means of the Lagrange's undetermined multipliers and Lagrange's Equations. Find the height at which the particle falls of. Homework Equations \frac{d}{dt}...

Hello everyone I have difficulties in understanding some stuff in Lagrangian and Hamiltonian mechanics. This concerns the equations : \dot p = - \frac{\partial H}{\partial q} \frac{d}{dt} \frac{\partial L}{\partial \dot q} = \frac{\partial L}{\partial q} First I have to say that I'm a math...