Hamiltonian mechanics Definition and 39 Threads

Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics. Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. Like Lagrangian mechanics, Hamiltonian mechanics is equivalent to Newton's laws of motion in the framework of classical mechanics.

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  1. G

    I Continuity of Hamiltonian at separatrix in action-angle variables

    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...
  2. G

    Variational symmetries for the Emden-Fowler equation

    So firstly I calculated the partial derivatives of f to be: $$ \frac{\partial f}{\partial x} = \frac{2x}{2} (y'^2 - \frac{1}{3} y^6) + \frac{x^2}{2} (2y'y'' - 2y^5y') = xy'^2 - \frac{1}{3} y^6 + x^2y'(y'' - y^5) $$ $$ \frac{\partial f}{\partial y} = \frac{x^2}{2}*\frac{1}{3}*6y^5 = x^2y^5 $$ $$...
  3. G

    I Why must partialS/partialα_n = -β_n where S is the complete integral?

    As stated in the TLDR, I don't understand why the derivative of the Jacobi complete integral with respect to the constant α must be another constant, and furthermore why that constant is negative. The textbook I'm following, van Brunt's The Calculus of Variations proves it by taking: $$...
  4. cianfa72

    A Hamiltonian formulation of classical mechanics as symplectic manifold

    Hi, in the Hamiltonian formulation of classical mechanics, the phase space is a symplectic manifold. Namely there is a closed non-degenerate 2-form ##\omega## that assign a symplectic structure to the ##2m## even dimensional manifold (the phase space). As explained here Darboux's theorem since...
  5. ardaoymakas

    Hamiltonian Function thru new Variables Q,P -- Show that Q is cyclic

    I took the derviative of the Hamiltonian function with respect to Q and assumed that it was equal to 0 in order to find the Konstant A. I did find the Konstant A as -1/2m^2g but I still cant write the Hamiltonian equation without having the Q as a variable. Can someone please help? Translation...
  6. astroholly

    Deriving the Hamiltonian of a system given the Lagrangian

    I have found the Hamiltonian to be ##H = L - 6 (q_1)^2## using the method below: 1. Find momenta using δL/δ\dot{q_i} 2. Apply Hamiltonian equation: H = sum over i (p_i \dot{q_i}) - L 3(q_1)^2. Simplifying result by combining terms 4. Comparing the given Lagrangian to the resulting Hamiltonian I...
  7. Omega0

    B Lagrangrian and Hamiltonian mechanics: A historical picture

    Hi, I believe that I have an acceptable level of understanding where SRT, GRT, QM and QFT come from. This is not true for me regarding the "good old stuff". Newton, okay, this is relatively (:wink:) clear to me but do you know something about the historical motivation for Lagrangian and...
  8. D

    I Exploring the Flexibility of Coordinates in Euler-Lagrange Equations

    Hello all, so I’ve been reading Jennifer Coopersmith’s The Lazy Universe: An Introduction to the Principle of Least Action, and on page 72 it says: If I understand it right, she’s saying that in our Euler-Lagrange equation ## \frac {\partial L} {\partial q} - \frac {d} {dt} \frac {\partial L}...
  9. LuccaP4

    Hamiltonian mechanics: phase diagram

    The issue here is that I don't know how to operate the final equations in order to get the phase diagram. I suppose some things are held constant so I can get a known curve such as an ellipse. I attach the solved part, I don't know how to go on.
  10. LCSphysicist

    What is the minimum mathematic requirement for learning Lagrangian and Hamiltonian mechanics?

    Homework Statement:: ... Relevant Equations:: . What is the minimum mathematic requirement to the Lagrangian and hamiltonian mechanics? Maybe calc 3 and linear algebra?
  11. O

    What was the intuition behind the development of Hamiltonian mechanics?

    Is Hamiltonian mechanics a mathematical generalization of Newtonian mechanics or is it explaining some fundamental relationship that has a meaning that extends into our nature ? I guess my question is what would led William Rowan Hamilton to come up with his type of mechanics or anything...
  12. J

    Other Math required for advanced classical mechanics

    What types of math should a student be comfortable with going into a classical mechanics class at the level of Landau and Lifshitz? And are there any additional types of math that aren’t required, per se, but would be beneficial to know (for said course)?
  13. dRic2

    Hamiltonian Mechanics: why paths in state space never cross each other

    I'm reading a book about analytical mechanics and in particular, in a chapter on hamiltonian Mechanics it says: "In the state space (...) the complete solutionbof the canonical equations is pictured as an infinite manifold of curves which fill (2n+1)-dimensional space. These curves never cross...
  14. B

    Textbook for calculus of variations? Hamiltonian mechanics?

    I need to learn about Hamiltonian mechanics involving functional and functional derivative... Also, I need to learn about generalized real and imaginary Hamiltonian... I only learned the basics of Hamiltonian mechanics during undergrad and now those papers I read show very generalized version...
  15. QuasarBoy543298

    Hamiltonian mechanics - the independence of p and q

    in the Lagrangian mechanics, we assumed that the Lagrangian is a function of space coordinates, time and the derivative of those space coordinates by time (velocity) L(q,dq/dt,t). to derive the Hamiltonian we used the Legendre transformation on L with respect to dq/dt and got H = p*(dq/dt) -...
  16. J

    Discrepancy in Lagrangian to Hamiltonian transformation?

    I know, $$ L=T-V \;\;\; \; \;\;\; [1]\;\;\; \; \;\;\; ( Lagrangian) $$ $$ H=T+V \;\;\; \; \;\;\;[2] \;\;\; \; \;\;\; (Hamiltonian)$$ and logically, this leads to the equation, $$ H - L= 2V \;\;\; \; \;\;\...
  17. chmodfree

    I Generalized Momentum is a linear functional of Velocity?

    Generalized momentum is covariant while velocity is contravariant in coordinate transformation on configuration space, thus they are defined in the tangent bundle and cotangent bundle respectively. Question: Is that means the momentum is a linear functional of velocity? If so, the way to...
  18. Gopal Mailpalli

    Classical Good book for Lagrangian and Hamiltonian Mechanics

    This book should introduce me to Lagrangian and Hamiltonian Mechanics and slowly teach me how to do problems. I know about Goldstein's Classical Mechanics, but don't know how do I approach the book.
  19. M

    What are Lagrangian and Hamiltonian mechanics?

    Only thing I know about them is that they are alternate mechanical systems to bypass the Newtonian concept of a "force". How do they achieve this? Why haven't they replaced Newtonian mechanics, if they somehow "invalidate" it or make it less accurate, by the Occam's razor principle? Thanks in...
  20. Ahmad Kishki

    Classical Lagrangian and Hamiltonian mechanics

    Recommend an easy going introduction to lagrangian and hamiltonian mechanics (for self study)
  21. N

    Hamiltonian mechanics: ∂H/∂t = ?

    According to my book, and wiki http://en.wikipedia.org/wiki/Hamiltonian_mechanics#As_a_reformulation_of_Lagrangian_mechanics, ##\frac{\partial{H}}{\partial{t}} = - \frac{\partial{L}}{\partial{t}}##, where ##L## is the Lagrangian. But how can this be? This assumes the generalized...
  22. D

    Generating Functions in Hamiltonian Mechanics

    Hi I have been looking at canonical transformation using generating functions. I am using the Goldstein book and it gives the following example - F1=qQ ⇔ Q=p and P=-q F2=qP ⇔ Q=q and P=p F3=pQ ⇔ Q=-q and P=-p F4=pP ⇔ Q=p and P=-q I'm confused ! Obviously functions 1 and 4 give the...
  23. H

    Derivation of ideal gas law by Hamiltonian mechanics

    Hi! I am trying to understand the statistical mechanics derivation of the ideal gas law shown at: http://en.wikipedia.org/wiki/Ideal_gas_law inder "Derivations". First of all, the statement "Then the time average momentum of the particle is: \langle \mathbf{q} \cdot \mathbf{F} \rangle=...
  24. D

    When is Hamiltonian mechanics useful

    Generally, what sort of problems are handled better by Hamiltonian mechanics than by Lagrangian mechanics? Can anyone give a specific example?
  25. B

    Unraveling the Mystery of Hamiltonian Mechanics

    Hi, A fundamental aspect in the Hamiltonian framework of mechanics is that the q's and p's are independent. I feel like I understand the steps in the Legendre transform from Lagrangian to Hamiltonian mechanics, but I don't see how you can go from a system where only the q's are independent...
  26. B

    What Does a Zero Hamiltonian Indicate About System Dynamics?

    Homework Statement Basically, I'm given a Hamiltonian H = H(p,q) and asked to find a new Hamiltonian K = K(Q,P,t) using the generating functions method H = 1/2 (p^2 + q^2) Generating function f(q,P,t) = qp sec ( t ) - 1/2 (q^2 + P^2) tan ( t ) So, I have no problem finding the new...
  27. L

    Hamiltonian mechanics electromagnetic field

    Homework Statement Let (V(x,t) , A(x,t)) be a 4-vector potential that constructs the electromagnetic field (in gaussian Units) by E(x,t) = -∇V(x,t) - (1/c)δtA(x,t) , B = ∇xA , (x,t) elements of R3xRt Consider the lagrangian L=.5mv2 - eV(x,t) + (ev/c)(dot)A(x,t) a) compute and interpret the...
  28. snoopies622

    Hamiltonian mechanics and the electric field

    The dimensions of action divided by the dimensions of electric field strength are distance x time x charge. Does this mean that distance x time x charge - whatever one might call that - is the "conjugate momentum" of an electric field? If so - is there any physical significance to this...
  29. I

    Hamiltonian Mechanics: Constants of Motion & Calculation

    1. A particle of mass m is in the environment of a force field with components: Fz=-Kz, Fy=Fx=0 for some constant K. Write down the Hamiltonian of the particle in Cartesian coordinates .What are the constant of motion? 2. H=kinetic energy +potential energy [b]3. Is the Hamiltonian...
  30. N

    Is Newtonian Mechanics more general than Hamiltonian Mechanics?

    Apparently things like the Lorentz' force can't be handled as a hamiltonian system. I heard other people describe the hamiltonian mechanics as an equivalent characterization of classical mechanics, but this is wrong, then?
  31. N

    Do forces always work in the hamiltonian mechanics?

    Do forces always "work" in the hamiltonian mechanics?... Hello, I'll phrase my question two times: once for the people in a hurry, and a second time in a broader way: 1) short If I write down the equation of a force, can it be that Hamiltonian mechanics doesn't 'apply' to it? I was...
  32. J

    Coupled Generalized Momentum & Hamiltonian Mechanics

    I have a brilliantly engineered system of a bead-on-a-circular-loop (mass=m) rigidly attached to a massive block (mass=M) on one side and a spring on the other. The spring motion is constrained to be in x-direction only, while the bead is free to move on the wire anyway it wants to (no \phi...
  33. G

    Correspondence between Hamiltonian mechanics and QM

    To which entitity (operators, wavefunctions etc) in quantum mechanics do the dynamical variables and the hamiltonian vector fields that they generate (through Symplectic structure of classical mechanics) correspond to?
  34. F

    Difference between Newtonian, Langrange, and Hamiltonian Mechanics

    Hey I was just wondering what the differences between the three forms of mechanics were. I've only studied basic Newtonian mechanics so I'm not really sure about the other two. Could anyone elaborate?
  35. V

    Is Hamiltonian Mechanics Essential for Studying Quantum Mechanics?

    hello all, i'm an EE student,and I've recently started studying quantum mechanics. most textbooks start with schrodinger's equation directly but a few others (like say Liboff) start with the concept of hamiltonian from hamiltonian mechanics. is a knowledge of the same i.e...
  36. quasar987

    Hamiltonian mechanics: canonical transformations

    Say I have a canonical transformation Q(q,p), P(q,p). In the {q,p} canonical coordinates, the Hamiltonian is H(q,p,t)=p\dot{q}-L(q,\dot{q},t) And the function K(Q,P,t)=H(q(Q,P),p(Q,P),t) plays the role of hamiltonian for the canonical coordinates Q and P in the sense that...
  37. L

    Hamiltonian Mechanics: Central Potential V(r) w/ Spherical Coordinates

    Homework Statement Using spherical coordinates (r, \theta, \phi), obtain the Hamiltonian and the Hamilton equations of motion for a particle in a central potential V(r). Study how the Hamilton equations of motion simplify when one imposes the initial conditions p_{\phi}(0) = 0 and \phi (0)...
  38. kakarukeys

    Conserved charge generates symmetry transformation in Hamiltonian Mechanics

    Q is a conserved charge if \{Q, H\} = 0 Show that q+\epsilon\delta q satisfies the equation of motion. \delta q = \{q, Q\} I couldn't find the proof. Anybody knows? My workings: \delta q = \{q, Q\} \delta\dot{q} = \{\{q,Q\},H\} = - \{\{Q,H\},q\} - \{\{H,q\},Q\} \delta\dot{q} = \{\{q,Q\},H\} =...
  39. L

    What are Hamiltonian Mechanics?

    Can anyone give me a basic definition of Hamiltonian Mechanics without all the fancy mathematics, and perhaps could supply a few examples as to this? I am trying to make sense of this, but everywhere I go, I run into very large mathematical equations and no defintions I can understand...
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