# What is Hamiltonian formalism: Definition and 22 Discussions

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. ### A Recover Hamilton equation from 2-form defined on phase space

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...
2. ### I Proof that canonical transformation implies symplectic condition

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...
3. ### A Regular vs stable orbits in spherically symmetric potentials

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...
4. ### Intro to Quantum Mechanics - Formalism normalisation

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
5. ### I Hamiltonian formalism and partition function

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...
6. ### I Lagrangian with generalized positions

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...
7. ### Change in Hamiltonian under Non-Canonical transformation

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...
8. ### A Hamilton's principle and virtual work by constraint forces

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...
9. ### A Non holonomic constraints in classical mechanics textbook

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
10. ### I Virtual displacement is not consistent with constraints

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...
11. ### I Energy in the Hamiltonian formalism from phase space evolution

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...
12. ### Given a set of equations, show if it is a Hamiltonian system

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...
13. ### Conservation laws in Newtonian and Hamiltonian (symplectic) mechanics

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...
14. ### Classical Classical Mechanics: Landau vs Goldstein Textbooks

Which textbook is better for an upper division course in classical mechanics - Goldstein’s book or L&L’s book?
15. ### Hamilton-Jacobi equation and particle-wave motion

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...
16. ### A One Hamiltonian formalism query - source is Goldstein's book

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...
17. ### 3-species Lotka-Volterra food chain in Hamiltonian formalism

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...
18. ### Classical mechanics, Hamiltonian formalism, change of variables

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\\...
19. ### Sources on explicitly time-dependent Hamiltonian formalism

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...
20. ### Can a quantum formalism exist without a Hamiltonian Formalism

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...
21. ### Why Lagrangian and Hamiltonian formalism

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
22. ### Lagrangian and hamiltonian formalism

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?