Lagrangian mechanics Definition and 188 Threads

In Chapter 8: Central-Force Motion, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 323, Problem 8-5, we are asked to show that the two particles will collide after a time ##\tau/4√2##. I don't have any problems with the derivations and with...

Homework Statement Consider a particle moving over the curve ##z=a-bx^2## under the force of gravity. If the particle starts from rest at point ##(0,0)## (I'm guessing it means point ##(0,a)##), tell if the particle ever separates from the curve; if yes, find the point at which it does...

How is it that when using "natural" units we drop the units themselves. I understand that you can arbitrarily change the magnitude of a parameter by choosing a new unit. For example Oliver R. Smoot is exactly 1 smoot tall. However, in natural units with [c]=[h/(2π)]=1 the "smoot" part is...

<<Moderator's note: Moved from a technical forum, no template.>> 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...

Lagrangian Mechanics uses generalized coordinates and generalized velocities in configuration space. Hamiltonian Mechanics uses coordinates and corresponding momenta in phase space. Could anyone please explain the difference between configuration space and phase space. Thank you in advance for...

A very simple question. How do we represent a vector with Newton's notation (writing the arrow symbol with the overdot notation)? Can we write them both above each other. First, the overdot notation and then the arrow symbol? Thank you a lot for your help...

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...

Homework Statement Find the acceleration of a uniform solid sphere (of mass ##m## and radius ##R##) rolling without slipping down an incline at angle ##\alpha## using the Lagrangian method. Homework Equations Euler-Lagrange equation which says, $$\frac{\partial\mathcal{L}}{\partial...

(note: I'm going to represent the lagrangian as simply L because I don't know how to do script L in latex.) Homework Statement Two particles of equal masses m are confined to move along the x-axis and are connected by a spring with potential energy ##U = \frac{1}/{2}kx^2## (here x is the...

Homework Statement The carbon dioxide molecule can be considered a linear molecule with a central carbon atom, bound to two oxygen atoms with a pair of identical springs in opposing directions. Study the longitudinal motion of the molecule. If three coordinates are used, one of the normal...

Homework Statement Take the x-axis to be pointing perpendicularly upwards. Mass ##m_1## slides freely along the x-axis. Mass ##m_2## slides freely along the y-axis. The masses are connected by a spring, with spring constant ##k## and relaxed length ##l_0##. The whole system rotates with...

Homework Statement The Attempt at a SolutionSo I first tried by saying consider a time t in which mass m is directly above the origin O. I.e., mass m at the Cartesian coordinate (0, 4l/3). I wrote a = a(t) as the extension function of the spring, which has 0 natural length. So, I applied the...

Homework Statement Hello! I have some problems with constructing Lagrangian for the given system: (Attached files) Homework Equations The answer should be given in the following form: L=T-U=... The Attempt at a Solution I tried to construct the Lagrangian, but I'm not sure if I did it...

Hello, I am trying to "integrate into my understanding" the difference between Hamiltonian and Lagrangian mechanics. In a nutshell: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to the minimization principle? YES; I KNOW about Hamilton's Second...

I’m a bit confused about what exactly lagranigian mechanics is. I know that L = Ke - Pe I also know the equation d/dt(∂L/∂x’) - ∂L/∂x = 0 1.) Apparentaly solving this equation gives the equations of motion. What exactly does that mean though? I solved a very simple problem and got the...

Hello. I solve this problem: 1. Homework Statement The particles of mass m moves without friction on the inner wall of the axially symmetric vessel with the equation of the rotational paraboloid: where b>0. a) The particle moves along the circular trajectory at a height of z = z(0)...

What books include the theory of lagrangian mechanics? And where can i also find some problems?Could lagrangian mechanics help me in solving problems with oscillations?

Hi, i know that The homogeneity of space and time implies that the Lagrangian cannot contain explicitly either the radius vector r of the particle or the time t, i.e. L must be a function of v only but the lagrangian definition is ##L=\int L(\dot q,q,t)##, so velocity appears in the definition...

Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be...

In the paper http://physics.unipune.ernet.in/~phyed/26.2/File5.pdf, the author solves the LC-circuit using Euler-Lagrange equation. She assumes that the Lagrangian function for the circuit is $$L=T-V$$ where $$T=L\dot q^2/2$$ is the kinetic energy part $$V=q^2 / 2C$$ is the potential energy.She...

I can't for the life of me figure out what virtual work or D'alemberts principle mean and what the intuition behind them is. As far as I'm concerned D'alemberts principle is just a restatement of Newton's second law but considering the work instead of just the forces. What am I missing? I'm...

Hello everyone, Reading Landau and Lifshitz Course of Theoretical Physics Volume 1: Mechanics (page 3) I got suck in the following step (and I cite in italics): The change in S when q is replaced by q+δq is \int_{t_1}^{t_2} L(q+δq, \dot q +δ\dot q, t)dt - \int_{t_1}^{t_2} L(q, \dot q, t)dt...

Homework Statement A homogeneous hollow cylinder (mass M, radius R) is in the gravitational field and a horizontal axis through the center P rotatably mounted (central axis of the cylinder is fixed and can be rotated). A small, homogeneous solid cylinder (mass m, radius r) is rolling inside...

Consider a sphere constrained to roll on a rough FLAT HORIZONTAL surface. A book on classical mechanics says it requires 5 generalized co-ordinates to specify sphere's configuration: 2 for its centre of mass and 3 for its orientation. I did not understand why 3 for orientation. I guess only 2...

Homework Statement From the homework: In General Relativity it is found that the radial equation of an object orbiting a non-rotating black hole has the form $$\dot r^2 + (1 - 2 \frac {V_o} {r} ) (\frac {l^2} {r^2} + 1) = E^2$$ where ##r## is the radial coordinate, ##l## is the angular...

In lagrangian variation we are trying to minimize the action S = ∫t2t1 L dt. Consider a simple case of free particle. Imagine In a world that everyone one only knows how to solve ODE, Using euler lagrange equation, one has d2x/dt2 = 0 , give that we know the initial position of particle in the...

Homework Statement String is wrapped around two identical disks of mass m and radius R. One disk is fixed to the ceiling but is free to rotate. The other is free to fall, unwinding the string as it falls. Find the acceleration of the falling disk by finding the lagrangian and lagrange's...

In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...

Homework Statement Find the Lagrangian for the double pendulum system given below, where the length of the massless, frictionless and non-extendable wire attaching m_1 is l. m_2 is attached to m_1 through a massless spring of constant k and length r. The spring may only stretch in the m_1-m_2...

Homework Statement A uniform cylindrical drum of mass M and radius a is free to rotate about its axis, which i is horizontal. An elastic cable of negligible mass and length l is wrapped around the drum and carries on its free end a mass m. The cable has elastic potential energy \tfrac12...

Hello, I'm a second year physics student. We are going to use "hand and finch analytical mechanics", however the reviews I saw about this book are bad. I've already taken calculus for mathematicians, linear algebra, classical mechanics, special relativity, and electromagnetism. The topics it...

I am trying to establish a Rationalist approach to Physics as a side project, and have taken Hamilton's Principle as one of the few postulates in my work. I've developed the concept enough to arrive at the usual stuff, like the Euler-Lagrange equations, Newton's First Law and Nöther's Theorem...

Homework Statement what will be Lagrange,s equation of motion for a particle confined to move on surface of sphere whose radius is expanding such that Homework Equations Euler-lagranges equation of motion d/dt(∂L/∂{dq/dt})-∂L/∂q=0 The Attempt at a Solution Z=(R+R0e^at)cosθ...

Just refer to my profile picture to see what the issue is! :biggrin: Here's the problem: a ball of mass m is connected to a vertical pole with an inextensible, massless string of length r. The angle between the string and the pole is θ. The pole rotates around the z axis with a constant angular...

Given a system of charged particles interacting with an EM field. Is the canonical momentum always conserved? If so, what is the associated symmetry? Thanks in advance.

Homework Statement Attached. Homework Equations I am assuming the coordinate transformation is \vec{x}' = \vec{x} + \alpha\vec{\gamma} ? Then you have \vec{v}' = \vec{v} + \alpha\frac{d\vec{\gamma}}{dt} And r is the magnitude of the x vector. The Attempt at a Solution Part A. So to get the...

Is the motivation for the action principle purely from empirical evidence, or theoretical arguments, or a mixture of the two? As I understand it, there was some empirical evidence from Fermat's observations in optics, i.e. that light follows the path of least time, notions of virtual work and...

I'm trying to understand how we set up the lagrangian for a charged particle in an electromagnetic field. I know that the lagrangian is given by $$L = \frac{m}{2}\mathbf{\dot{r}}\cdot \mathbf{\dot{r}} -q\phi +q\mathbf{\dot{r}}\cdot \mathbf{A} $$ I can use this to derive the Lorentz force law...

Homework Statement We have a particle of mass m moving in a plane described by the following Lagrangian: \frac{1}{2}m((\dot{x}^2)+(\dot{y}^2)+2(\alpha)(\dot{x})(\dot{y}))-\frac{1}{2}k(x^2+y^2+(\beta)xy) for k>0 is a spring constant and \alpha and \beta are time-independent. Find the normal...

Homework Statement We have a mas m attached to a vertical spring of length (l+x) where l is the natural length. Homework Equations Find the Lagrangian and the hamiltonian of the system if it moves like a pendulum The Attempt at a Solution we know that the lagrangian of a system is defined as...

Homework Statement Mass 1 can slide on a vertical rod under the influence of a constant gravitational force and and is connected to the rod via a spring with the spring konstant k and rest length 0. A mass 2 is connected to mass 1 via a rod of length L (forms a 90 degree angel with the first...

Lately when doing a simulation for a quadrocopters most reports I've come across regarding modeling use Eulers equation of motion. That makes sense, as the quadrocopter is a body rotating in 3 dimensions. Then I tried to model the system using Lagrange equations instead but I don't get the...

Homework Statement Please see attached image :) Homework Equations Euler-Lagrange Equation \frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}}} = 0 L = T - V The Attempt at a Solution a. The potential energy V is the potential energy from the spring and the...

Homework Statement Hi everybody! As always, I struggle with my special relativity class and here is a new problem I'd like to have some indications about: A masspoint m moves in the x-y-plane under the influence of gravity on a circular path of radius r (see attached pic). Which constraining...

I know how to implement Lagrangian mechanics at a mathematical level and also know that it follows the approach of calculus of variations (i.e. optimisation of functionals, finding their stationary values etc.), however, I'm unsure whether I've grasped the physical intuition behind the...

Homework Statement A simple pendulum of length ξ and mass m is suspended from a point on the circumference of a thin massless disc of radius α that rotates with a constant angular velocity ω about its central axis as shown in Figure. Find the equation of motion of the mass m. Homework...

Homework Statement [/B] Consider a mass m moving in a frictionless plane that slopes at an angle \alpha with the horizontal. Write down the Lagrangian \mathcal{L} in terms of coordinates x measured horizontally across the slope, and y, measured down the slope. (Treat the system as...

Homework Statement A rigid “T” consists of a long rod glued perpendicular to another rod of length l that is pivoted at the origin. The T rotates around in a horizontal plane with constant frequency ω. A mass m is free to slide along the long rod and is connected to the intersection of the...

Homework Statement So, a particle is moving in a plane under the action of a force F that is oriented at all times to the direction of the center of the force.may r be the distance from the particle to the center of the force generator. Find the potential generator expression that occurs and...

Homework Statement This is not really a homework questions, just part of my notes confusing me a bit. This is the derivation of total energy for a pendulum of mass m2 with movable pivot of mass m1. I don't understand how frequency can be read off. What am I missing? Homework Equations See...