Hamiltonian formalism Definition and 25 Threads

I've a doubt regarding the application of the principle of minimum action to real cases. Pick an inertial frame with a potential ##V## defined on it. The principle (aka Hamilton's principle) claims that the actual path taken from a body gives rise to a "stationary" action when calculated from a...

The book Classical Mechanics by Alexei Deriglazov defines as canonical a transformation Z=Z(z,t) that preserves the Hamiltonian form of the equation of motion for any H. After taking the divergence of the vector equation relating the components of the time derivative of Z in the two coordinate...

As said in the tl;dr: is the Hamiltonian necessarily differentiable (hence continuous) at the separatrix in the action-angle formalism? After all, the action variables are different depending on the type of motion. As far as I know the Hamiltonian H = H(J) can be found by inverting J for E, and...

Following Steinacker's book we can say that given the manifold ##N## of a configuration space and its tangent bundle ##TN## we define a differentiable function ##L(\gamma,\dot\gamma): TN\rightarrow \mathbb{R}## and call it the Lagrangian function. We know there's always an isomorphism between a...

Goldstein's Classical Mechanics makes the claim (pages 382 to 383) that given coordinates ##q,p##, Hamiltonian ##H##, and new coordinates ##Q(q,p),P(q,p)##, there exists a transformed Hamiltonian ##K## such that ##\dot Q_i = \frac{\partial K}{ \partial P_i}## and ##\dot P_i = -\frac{\partial...

I am struggling with Hamiltonian formulation of classical mechanics. I think I have grasped the idea of canonical transformations, including the idea of angle-action variables and invariant tori in phase space. Still, few points seem to elude my understanding... Let's talk about a particle...

I can't figure out how they get i/sqrt(2) for normalisation of c1. Why is it a complex number? If I normalise c1 I just get 1/sqrt(2) because i disappears in the absolute value squared. Thanks

In hamiltonian formalism we have the generalized coordinates ##q_i## and the conjugates moments ##p_i##. For a dipole in a give magnetic field ##B## the Hamiltonian is ##H=-\mu B cos \theta## where ##\theta## is the angle between ##\vec \mu## and ##\vec B##. Can i consider ##\theta## or ##cos...

Hi Pfs When instead of the variables x,x',t the lagrangiean depends on the trandformed variables q,q',t , time may be explicit in this lagrangian and q' (the velocity of q) may appear outside. I am looking for a toy model in which tine is not explicit in L but where the velocities appear somhere...

I know that if the transformation was canonical, the form of Hamilton's equation would remain invariant. If the generating function for the transformation was time independent, then the Hamiltonian would be invariant and we could directly replace q and p with the transformation equations to get...

Found a question on another website, I have the exact same question. Please help me Goldstein says : I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. How does the fact that "the same Hamilton's principle holds for both holonomic and...

I want to learn about the non holonomic case in lagrangian and Hamiltonian mechanics. I've seen that many people say that Goldstein 3rd ed is wrong there. Where should I go to learn it. My mathematics level is at the level Goldstein uses. Please help

Goldstein 3rd ed says "First consider holonomic constraints. When we derive Lagrange's equation from either Hamilton's or D'Alembert's principle, the holonomic constraint appear in the last step when the variations in the ##q_i## were considered independent of each other. However, the virtual...

The hamiltonian ´for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know that ##H=E##. For this hamiltonian, using the Hamilton's equations and initial conditions...

Hi! So this is my first homework ever of Hamiltonian dynamics and I am struggling with the understanding of the most basic concepts. My lecturer is following Saletan's and Deriglazov's and from what I have read and from my lectures, this is what I think I know. Please let me know if this is...

In Newtonian mechanics, conservation laws of momentum and angular momentum for an isolated system follow from Newton's laws plus the assumption that all forces are central. This picture tells nothing about symmetries. In contrast, in Hamiltonian mechanics, conservation laws are tightly...

Which textbook is better for an upper division course in classical mechanics - Goldstein’s book or L&L’s book?

I've seen somwhere a claim that Hamilton-Jacobi euqation is the only formulation of classical mechanics which can treat motion of particle as wave motion. There was something about hamilton prinicpal function, hamilton characteristic function and one of these change in time like wavefront or...

In 3rd edition of Goldstein's "Classical Mechanics" book, page 335, section 8.1, it is mentioned that : In Hamiltonian formulation, there can be no constraint equations among the co-ordinates. Why is this necessary ? Any simple example which will elaborate this fact ? But in Lagrangian...

Hello, I have been assigned to write a report on a topic of my choice for my Computational Physics class, and I chose to focus on the symplectic integration of Hamiltonian systems, in particular the Lotka-Volterra model. A 3-species model(\gamma eats \beta, \beta eats \alpha) is not, unlike the...

Homework Statement This problem has to do with a canonical transformation and Hamiltonian formalism. Below is my attempt at solving it, but I am not too sure about it. Please help! Let \theta be some parameter. And X_1=x_1\cos \theta-y_2\sin\theta\\ Y_1=y_1\cos \theta+x_2\sin\theta\\...

Not sure I am posting this in the right subforum, if this is not the case, please feel free to move it. Anyway, the title about sums it up - I need to find a good source which offers a thourough treatment of Hamiltonian formalism for the explicitly time-dependent case - could someone possibly...

I was thinking about this. In every problem I have worked, we suppose a hamiltonian exists which can describe the system. There are obviously Hamiltonians which are not possible classically, such as in the 1-D ising model of paramagnetism, where the Hamiltonian contains terms of s_i. s_j where...

Dear all, could please give my some links or references to material that justifies the mathematical and physical reasons for introducing these two formalisms in mechanics? Thanks. Goldbeetle

what is the difference between these two formalism and when are each used? also is it true the lagrangian formalism is used more in qft, if so i am curious to know why?