Linked Questions

53
votes
5answers
6k views

Is the converse of Noether's first theorem true: Every conservation law has a symmetry?

Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Is the converse true: Any conservation law of a physical ...
26
votes
4answers
4k views

Equivalence between Hamiltonian and Lagrangian Mechanics

I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me. The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the Hamiltonian ...
16
votes
2answers
3k views

Is there a kind of Noether's theorem for the Hamiltonian formalism?

The original Noether's theorem assumes a Lagrangian formulation. Is there a kind of Noether's theorem for the Hamiltonian formalism?
15
votes
2answers
4k views

Invariance of Lagrangian in Noether's theorem

Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$. However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (...
5
votes
3answers
186 views

What does a symmetry that changes the Lagrangian by a total derivative do to the Hamiltonian $H$?

A tiny symmetry transformation may change the Lagrangian $L$ by a total time derivative of some function $f$. This is a basic fact used in the proof of Noether's theorem. How can we see the effect of ...
4
votes
1answer
402 views

Prove that Noether's Theorem produces generators of the symmetry

Suppose we have a classical Lagrangian $L(q,\dot{q})$. Here $q = q(s,t)$ is a generalized coordinate as a function of time and some parameter $s$ corresponding to a transformation. If this is a ...
2
votes
1answer
232 views

Proving that Noether charge generates symmetries in the Lagrangian formalism

I know that similar questions have been asked on this site before, but I haven't been able to find the answer to my specific question. I want to show that the Noether charge defined in Lagrangian ...
4
votes
1answer
93 views

Why are some symmetries invisible to the configuration space Lagrangian $L(q, \dot q,t)$?

Usually, when people talk about Lagrangians they are talking about a function of configuration space variables $q_i$ and their time derivatives $\dot q_i$. This is a function $L = L(q_i, \dot q_i,t)$. ...
1
vote
0answers
66 views

Legendre transformation and correspondance between Noether charges and quasi-symmetries

I have been trying to understand the Legendre transformation (in mechanics, in the hyperregular case: when the Legendre transformation is one-to-one) and the correspondence between symmetry $\to$ ...
2
votes
0answers
58 views

Are first Integrals the same in Lagrangian and Hamiltonian formalism?

A first integral in Lagrangian formalism is defined as a function which is constant along the solutions $(q,\dot{q})$ where $q$ are the generalized coordinates; while a first integral in Hamiltonian ...