# What is Hamilton's principle: Definition and 37 Discussions

In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.

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1. ### 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...
2. ### A Virtual work of constraint forces in Hamilton’s principle

Goldstein 2ed pg 36 So in the case of holonomic constraints we can move back and forth between Hamiltons principle and Lagrange equations given as ##\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)-\frac{\partial L}{\partial q_{j}}=0## But the Lagrange equations were...
3. ### A Are there real-world applications of Hamilton's Principle in mechanics?

Goooood Morning all! I am going through a few problems in advanced dynamics with Hamilton's Principle. One of them is shown above (this is NOT a question about the solution) The spring constants, damping, mass, force, are all given. So, too, is the constraint: large disk rolls without...
4. ### Why is Hamilton's Principle assumed to be valid for non-holonomic systems?

I am using Nivaldo Lemos' "Analytical Mechanics" textbook and on section 2.4 (Hamilton's Principle in the Non-Holonomic Case) he uses Hamilton's Principle and Lagrange Multipliers to arrive at the Lagrange Equations for the non-holonomic case. I don't understand why it is assumed that the...
5. ### A Can position and velocity vary independently in Hamilton's Principle?

To carry out the machinery of Hamilton's Principle though the calculus of variations, we desire to vary the position and velocity, independently. We proceed by varying at action, and set the variation to zero (I will assume ONE generalized variable: q1) In the above, I can see how we vary...
6. ### Invariance of a volume element in phase space, What does it mean?

The invariance of this volume element is shown by writing the infinitesimal volume elements $$d\eta$$ and $$d\rho$$ $$d\eta=dq_1.....dq_ndp_1......dp_n$$ $$d\rho=dQ_1.......dQ_ndP_1....dP_n$$ and we know that both of them are related to each other by the absolute value of the determinant of...
7. ### I Hamilton's Principle (HP), Lagrangian

I understand the process of the calculus of variations. I accept that a proper Lagrangian for Dynamics is "Kinetic minus potential" energy. I understand it is a principle, the same way F=ma is a law (something one cannot prove, but which works) Still... what do you say to students who say "I...
8. ### I Where does Hamilton's Principle come from?

Hamiltons Principle and the physcial entity action are the terms in which modern physics is formulated. How do you know that you can always find a Lagrangian for a System which is then used for Hamitlons Principle and the formulation of Action? Why is the action stationary?
9. ### Dynamics of Four Bar Link using Hamilton's Principle with Loop Closure

Summary:: Can someone point me to an example solution? Hello The attached figure is a four bar link. Each of the four bars has geometry, mass, moment of inertia, etc. A torque motor drives the first link. I am looking for an example (a simple solution so I can ground my self before...
10. ### A When is Hamilton's principle valid?

When is Hamilton's principle##\delta \int L d t=0## valid ? Is it only valid for monogenic and holonomic systems? What about monogenic and non holonomic systems? (I'm asking this because I got confused because I've found that Goldstein has got something wrong related to this in his 3rd edition)
11. ### 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...
12. ### A Exploring Why Hamilton's Principle Uses L=KE-PE Lagrangian

As I understand, Hamilton's Principle with the L = KE - PE Lagrangian, leads to Newton's equation. Why MINUS and not PLUS? I have seen many attempts at such a proof and I now realize that one does not prove PRINCIPLES. Like Laws, they are observations. It produces an equation that describes...
13. ### I D'Alembert's principle vs Hamilton's principle

In the first two chapters of Goldstein mechanics, the Lagrange equations are derived from both D'Alembert's principle and Hamilton's principle. I want to know what're the applicability of these two approaches to Lagrangian mechanics? Is one more powerful than the other or are they completely...
14. ### Hamiltonian of this system?

Let a mass m charged with q, attached to a spring with constant factor k = mω ^2 in an electric field E(t) = E0(t/τ) x since t=0. (Equilibrium position is x0 and the deformation obeys ξ = x - x0) What would the hamiltonian and motion equations be in t ≥ 0, in terms of m and ω?? Despise magnetic...
15. ### A Extending Hamilton's principle to systems with constraints

I'm working my way through Goldstein's Classical Mechanics and have followed the arguments until section 2.4 (Extending Hamilton's Principle to Systems with Constraints). In the second paragraph, Goldstein states that "When we derive Lagrange's equations from either Hamilton's or D'Alembert's...
16. ### I Confusion in variation derivative

This link shows us how to derive Hamilton's generalised principle starting from D'Alembert's principle. While I had no trouble understanding the derivation I am stuck on this particular step. I can't justify why ## \frac{d}{dt} \delta r_i = \delta [\frac{d}{dt}r_i] ##. This is because if I...
17. ### Hamilton's principle and minimum potential energy

Dear all, I was wondering what exactly the correspondence/relation is between Hamilton's principle (extremizing the action gives the allowed configurations) and the fact that a system wants to configurate such as to minimize its potential energy. Is there any? Somehow I can't find a decent...
18. ### Classical Spivak's Physics for Mathematicians: Mechanics

Hello, I will be enrolling in an undergraduate Classical Mechanics course and I was wondering if the book by Spivak "Physics for Mathematicians: Mechanics" would help me understand the concepts more in depth than usual. Until the time that I will be taking the course, I will already have...
19. ### I Classical/QM justification of principle of least action

Hi. Is the principle of least (better: stationary) action only an axiom in classical mechanics, or can it be derived from a more profound (classical) principle? As far as I know, it can be derived from the path integral formulation of QM. Is this a more profound justification for the principle...
20. ### I Is Hamilton's principle an empirical statement?

The principle of the least action, that the particle will take the path of least Lagrangian, here given as T-U, is Hamilton's principle in classical mechanics. I am wondering if this is just an empirical, experimental observation that is not mathematically driven from elsewhere, just like...
21. ### Derivation of Lagrange's eqs

Homework Statement So I'm deriving Lagrange's equations using Hamilton's principle which states that the motion of a dynamical system follows the path, consistent with any constraints, that minimise the time integral over the lagrangian L = T-U, where T is the kinetic energy and U is the...
22. ### I Intuition for Hamilton's Principle

Is there an intuitive way to understand why nature selects the path that minimizes the action? I've seen it proven that the Euler-Lagrange equations are equivalent to Newton's laws (at least in Cartesian coordinates). So I can understand it mathematically. But on a more common-sense level...
23. ### Obtain Equation Using Hamilton's Principle

I want to obtain equation using Hamilton principle but I just couldn't figure it out; i have The kinetic energy : E_{k}=\dfrac{1}{2}m_{z} \displaystyle\int\limits_{0}^{L}\ \left[ \left( \dfrac{\partial w(x,t)}{\partial t}\right)^{2}+\left( \dfrac{\partial v(x,t)}{\partial...
24. ### How did we arrive to Hamilton's principle ?

Hi. The Lagrange equations can be derived simply by using Newton's laws and defining potential, work and kinetic energy. So that's just a mathematical reformulation of known results. Is it the same with Hamilton's principle? Is the concept of "action" and it being stationary just another...
25. ### Hamilton's Principle and the Principle of Virtual Work

I understand that accepting Hamilton's principle will yield identical results as accepting Newton's laws. However, simply accepting that the integral of the difference between kinetic and potential energies is an extrema seems not intuitively obvious. The textbook that I used for my classical...
26. ### Initial-Value Version of Hamilton's Principle?

As far as I understand, Hamilton's principle (a.k.a. "least action" or "stationary action") requires that you know both the initial and final location of a particle. Then, based on the requirement that the action must be stationary along any "possible" path, it will tell you what path(s) the...
27. ### Questions on Hamilton's Principle and Quantum Mechanics

Hi, reading Sakurai pages 102-103 (see http://www.scribd.com/doc/3035203/J-J-Sakurai-Modern-Quantum-Mechanics ) I found one thing hard to understand: If the Phase times Planck's constant equals Hamilton's Principal function in the classical limit (i.e. the action for the physically...
28. ### Understand Hamilton's Principle Intuitively

While I understand how the Euler-Lagrange equations are derived by minimizing the integral of the Lagrangian, I don't intuitively understand why Hamilton's principle is true. Specifically, what physical quantity does the Lagrangian represent and what does minimizing it mean? I'd just like to get...
29. ### Where does Hamilton's principle come from?

Does it have any deeper theoretical foundation or is it just true empirically?
30. ### Does Hamilton's principle follow from the Feynman formulation of QM?

Do you think Hamilton's principle from classical mechanics can be deduced from Feynman's path integral in quantum mechanics? (We get across this question in another discussion: https://www.physicsforums.com/showthread.php?t=609087&page=5) Of course, there is a loose connection, since...
31. ### History of Hamilton's principle

How did Hamilton work out that action is the stationary quantity for a mechanical system? I've seen proofs that action is stationary, but it's unclear to me how Hamilton worked out that action as opposed to some other quantity should be stationary.
32. ### Hamilton's principle: why dA rather than dA/dt?

Hamilton's principle is described as \deltaI=\delta\intL dt = 0 so as the action is stationary. This does not seem to be the same as dI/dt = 0, which is how I understand the condition for a function being stationary. Am I misinterpreting the equation?
33. ### Vibration Problem: Equations of Motion using Hamilton's Principle

[b]1. Frame made up of 3 rigid bars linked to each other and the ground by rotational springs, k, a viscous damper, c which connects opposite corners of the frame. Subject to base lateral motion (s(t)), and the two vertical bars rotate by an amount theta(t). Assuming small displacements...
34. ### Hamilton's Principle Equations Work Shown

Hamilton's Principle Equations! Work Shown Please Help! A particle of mass m moves under the influence of gravity alng the helix z=k(theta), and r=R, where R and k are constants and z is vertical. a.) Using cartesian co-ordinates, write down the expressions for the kinetic energy of the...
35. ### Principle of Least Action OR Hamilton's Principle

Are the principle of least action(http://astro.berkeley.edu/~converse/Lagrange/Kepler%27sFirstLaw.htm) and the hamilton principle 'exactly' the same? As far as I know, yes. How do I go from one to the other
36. ### Extendidng Hamilton's Principle to Non-Holonomic sytems

Can someone explain me how to extend Hamilton's principle to non-holonomic system's thru the lagrange undetermined multipliers? PS:Assume the system is Semi-Holonomic that is f_{\alpha}(q_{i},q_{2} \cdots q_{n} ,\dot{q_{1}}, \dot{q_{2}}, \cdots \dot{q_{n}})=0 such a equation exists for...
37. ### Hamilton's principle and Lagrangian mechanics

I have seen that Lagrange's equations are some times derived from Hamilton's principle. This makes me wonder what the historical development of these ideas was. Hamilton lived in the nineteenth century while Lagrange lived in the eighteenth century. The principle that minimizes the integral of...