# What is Conserved quantities: Definition and 39 Discussions

In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables the value of which remains constant along each trajectory of the system.Not all systems have conserved quantities, and conserved quantities are not unique, since one can always apply a function to a conserved quantity, such as adding a number.
Since many laws of physics express some kind of conservation, conserved quantities commonly exist in mathematical models of physical systems. For example, any classical mechanics model will have mechanical energy as a conserved quantity as long as the forces involved are conservative.

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15. ### I Origin of mass and Noethers theorem

Let me set up the question briefly. Emmy Noether's theorem relates symmetry to conserved quantities, e.g. invariance under translations in time => conservation of energy. A fundamental truth revealed. Massive gauge bosons, leptons and quarks all appear to acquire mass through the spontaneous...
16. ### Conserved quantities for massless particles (Schwarzschild)

If a metric admits a Killing vector field ##V ## it is possible to define conserved quantities: ## V^{\mu} u_{\mu}=const## where ## u^{\mu}## is the 4 velocity of a particle. For example, Schwarzschild metric admits a timelike Killing vector field. This means that the quantity ##g_{\mu 0}...
17. ### Noether's theorem and constructing conserved quantities

Homework Statement A particle of mass m and charge e moving in a constant magnetic field B which points in the z-direction has Lagrangian ##L = (1/2) m( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 ) + (eB/2c)(x\dot{y} − y\dot{x}). ## Show that the system is invariant under spatial displacement (in any...
18. ### Identifying conserved quantities using Noether's theorem

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...
19. ### Conserved quantities in the cart and pendulum problem

A problem on an assignment I'm doing deals with a cart of mass m1 which can slide frictionlessly along the x-axis. Suspended from the cart by a string of length l is a mass m2, which is constrained to move in the x-y plane. The angle between the pendulum and vertical is notated as phi. The...
20. ### How to get the conserved quantities of a integrable quantum system?

If I have an arbitrary quantum many-body model, what is the method to calculate the the conserved quantities if the model is integrable. If it is hard to explain, can you recommend some relevant books for me? Thanks a lot!
21. ### Finding Conserved Quantities for Analytical Mechanics Exam Questions

Hi all, I am preparing for my "second chance exam" in analytical mechanics. It is a graduate course i.e. based on geometry. (Our course notes are roughly based on Arnold's book). I was able to find some old exam questions and one of those has me stumped, completely. The question gives 3...
22. ### Trajectories with Conserved Quantities

Homework Statement A particle moves along a trajectory with constant magnitude of the velocity |\stackrel{→}{v}|=\stackrel{→}{v0} and constant angular momentum L⃗ = L⃗0. Determine the possible trajectories. Homework Equations d(L⃗)/(dt)=\stackrel{→}{N} where \stackrel{→}{N}=torque...
23. ### Conserved quantities as symmetry generators

Suppose we have a Lagrangian \mathcal{L(\phi, \partial_\mu \phi)} over a field \phi, and some variation on the field \delta \phi. If this variation induces a variation \delta \mathcal{L} = \partial_\mu F^\mu for some function F^\mu, then Noether's Theorem tells us that if we construct the...
24. ### How to spot conserved quantities from Lagrangians

Hi guys, The title pretty much says it. Say you have a very simple 3D Lagrangian: L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - V So How do you tell what is conserved from a generic potential? I know for example that if V = V(x,y,z) then the total linear momentum is not...
25. ### Finding Conserved Quantities of a Given Lagrangian

Homework Statement Find two independent conserved quantities for a system with Lagrangian L = A\dot{q}^{2}_{1} + B\dot{q_{1}}\dot{q_{2}} + C\dot{q}^{2}_{2} - D(2q_{1}-q_{2})^{4}\dot{q_{2}} where A, B, C, and D are constants. Homework Equations None.The Attempt at a Solution I've only found...
26. ### Conserved quantities in mechanics

So having been through translation and rotation I can conclude that my book has found 3 conserved quantities in classical dynamics: Energy Angular momentum Momentum That is 7 separate quantities which are conserved E,Lx,Ly,Lz,px,py,pz But this question is bothering me: How do we know that...
27. ### Can someone explain conserved quantities in this activity

Hello, Can Someone help me with this activity - http://www.particleadventure.org/other/education/five_s.html - its activity 5 The rules of the Game The answer to the activity is - http://www.particleadventure.org/other/education/five_s.html I don't understand how they got the answer for...
28. ### Conserved quantities in the Doran Metric?

I've been doing some amateur simulations of particle trajectories in a few well-known metrics (using GNU Octave and Maxima), and in the case of the Schwarzschild and Gullstrand-Painleve metrics I have the ability to check my results using freely available equations for conservation of energy and...
29. ### Symmetry Groups Algebras Commutators Conserved Quantities

Symmetry, Groups, Algebras, Commutators, Conserved Quantities OK, maybe this is asking too much, hopefully not. I'm trying to understand the connection between all of these constructions. I wonder if a summary about these interrelationship can be given. If I understand what I'm reading, there...
30. ### Conserved quantities in the Feynman diagrams.

I'm currently reading Griffiths book (I'm at chapter 4) on Particle physics, and I had a question about Feynman diagrams. In every "node" of a Feynman diagram, what quantities are conserved? Further, what quantities are conserved over the entire diagram?
31. ### Lagrangian of a particle + conserved quantities

Homework Statement Consider the spherical pendulum. In other words a particle with mass m constrained to move over the surface of a sphere of radius R, under the gravitational acceleration \vec g. 1)Write the Lagrangian in spherical coordinates (r, \phi, \theta) and write the cyclical...
32. ### Lagrangians and conserved quantities

Hi, I have a relatively straight forward question. If we have a Lagrangian that only depends on time and the position coordinate (and its derivative), how can I decide whether angular momentum is conserved? That is, if the Lagrangian specifically does not have theta or phi dependence, does...
33. ### Galilean invariance and conserved quantities

Hi I have a simple question what is the conserved quantity corresponding to the symmetry of galilean invariance? and Lorentz invariance? cheers M
34. ### Conserved Quantities from Boost/Rotation of Maxwell Lagrangian

I've calculated the conserved quantity for a boost or rotation of the Maxwell Lagrangian using the field form of Noether's theorem. If I calculated right, the components of a conserved four vector "current" considering boosts along in the x-axis appear to be: C^\mu = \eta^{\mu\nu} (F_{\nu 0}...
35. ### Differential geometry in quantum mechanics - conserved quantities

Hi everyone. This is kind of a geometry/quantum mechanics question (hope this is the right place to post this). So, in quantum mechanics we consider operators that reside in an infinite dimensional Hilbert space (to speak rather informally). We even have the cool commutator relation, which is...
36. ### Conserved quantities for geodesics

Homework Statement In comoving coordinates, a one dimensional expanding flat universe has a metric ds^2 = -c^2dt^2 + at(t)^2dr^2. Derive an expression for a conserved quantity for geodesics in terms of a, \tau and r, where \tau is the time measured in the rest frame of the freely falling...