Why is it important that Hamilton’s equations have the four symplectic properties and what do they mean?

The symplectic properties are: time invariance conservation of energy the element of phase space volume is invariant to coordinate transformations the volume the phase space element is invariant with respect to time I’m most inerested in what 3 and 4 mean and why they are important. Answer Coordinate invariance guarantees that the phase space M … Read more

Phase space Lagrangian?

Reading out of this lecture series we define a phase space Lagrangian L to be a function of 4n+1 variables namely q,˙q,p,˙p,t. My question is, what space is this function defined on? (I know that the ˙p is there for names sake only). My stab at an answer is it is a product space between … Read more

What does it mean to say a ‘double’ Legendre transform?

Ref. 1 mentons that you can achieve the momentum space Lagrangian by doing a so called double Legendre transform. It goes on to write:K(p,˙p,t) = L(q,˙q,t)−p˙q−q˙p, where K(p,˙p,t) is the momentum space Lagrangian. I don’t see how this is a Legendre transform as there seems to be a sign error in the way the Legendre transform has … Read more

qA⋅vq\mathbf{A}\cdot\mathbf{v} term in potential energy

In the famous Goldstein mechanics book, there is an example about a single (non-relativistic) particle of mass m and charge q moving in an E&M field. It says the force on the charge can be derived from the following velocity-dependent potential energy U=qϕ−qA⋅v. (eq 1.62 of 3rd ed.) I can see where the expression came … Read more

Poisson brackets of three dimensional angular momentum and its Lie algebra

I’ve recently noticed that the Poisson brackets of the three dimensional angular momentum {Li,Lj} in classical mechanics follow the same commutator relations as the standard basis of the Lie algebra so(3). This means that these to Lie algebras are isomorphic. Also so(3) is the Lie algebra of the Lie group SO(3), which is the group … Read more

How to prove a vector field is hamiltonian?

I’ve been studying Hamiltonian mechanics lately with kind of a more “differential geometry based” approach , but I’m stuck at a point where it is required to understand how you can prove a vector field is Hamiltonian. For example if we are given the following simple system of equation, representing a one dimentional system on … Read more

Proof of Liouville’s theorem: Relation between phase space volume and probability distribution function

I understand the proof of Liouville’s theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of any initial volume element is 0. From here, how do we say that probability distribution function is constant as we flow in … Read more

Proof of constructing action-angle coordinates on Hamiltonian system

By Liouville-Arnold Theorem, we know we can construct action-angle coordinates such that the Hamiltonian system, when described in these coordinates, will have a form that is integrable by quadratures. I am looking at a proof of the construction of these coordinates, and I am not certain of a certain part. On page 180 of Mathematical … Read more

The complex form of Hamilton canonical equations

I found an excerpt on page 171 of “The variational principles of mechanics” written by Cornelius Lanczos stated that If, however, the conjugate variables qk, pk are replaced by the complex variables qk+ipk√2=uk and qk−ipk√2=u∗k, then the double set of canonical equations ˙qi=∂H∂pi,˙pi=−∂H∂qi can be replace by the following single set of complex equations dukidt=−dHdu∗k. … Read more

Requirements for theory where Hamilton is the generator for time evolution

Introduction It is well known, that the Hamilton operator is the generator for the time evolution. In Heisenberg picture −i ∂tϕ(x,t)=[H,ϕ(x,t)]. For a Quantum Field Theory, we usually start with the Lagrangian L(ϕ(x,t),∂μϕ(x,t)), and then construct the Hamilton from there H(t):=∫d3x (∂L∂(∂0ϕ)⏟π(x,t) ∂0ϕ(x,t)−L). The other axioms besides how the Lagrangian looks like are the Canonical Commutator Relations for … Read more