formalisms:hamiltonian_formalism
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formalisms:hamiltonian_formalism [2018/04/08 14:13] jakobadmin [Concrete] |
formalisms:hamiltonian_formalism [2023/04/02 03:34] (current) edi [Concrete] |
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| ====== Hamiltonian Formalism ====== | ====== Hamiltonian Formalism ====== | ||
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
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| - | A short allegory will help to explain the different aims of [[frameworks:lagrangian_formalism|Lagrangian Mechanics]] and Hamilton’s Mechanics - and explain why we do bother. Imagine that we are keen on golf and want to improve our stroke. On Saturday, we are at the tee of hole number 18, we have selected our golf club, have an ample supply of identical golfballs, and proceed to hit 100 balls toward flag number 18. Exhausted, we walk over to the putting green and count up the number of balls we find there. The next day (Sunday) we again drive 100 balls, but just as we’re about to walk to hole 18 it starts to rain and we head, instead, for the clubhouse, where tea and scones awaits us. Fortunately, our companion used his smartphone to take photographs of each drive, and the phone has been programmed (using Lagrange’s Mechanics) to calculate the trajectory of a golfball, knowing the angle and speed at which it leaves the golf club, and so determine whether the given ball makes it to the putting green. | + | A short allegory will help to explain the different aims of [[formalisms:lagrangian_formalism|Lagrangian Mechanics]] and Hamilton’s Mechanics - and explain why we do bother. Imagine that we are keen on golf and want to improve our stroke. On Saturday, we are at the tee of hole number 18, we have selected our golf club, have an ample supply of identical golfballs, and proceed to hit 100 balls toward flag number 18. Exhausted, we walk over to the putting green and count up the number of balls we find there. The next day (Sunday) we again drive 100 balls, but just as we’re about to walk to hole 18 it starts to rain and we head, instead, for the clubhouse, where tea and scones awaits us. Fortunately, our companion used his smartphone to take photographs of each drive, and the phone has been programmed (using Lagrange’s Mechanics) to calculate the trajectory of a golfball, knowing the angle and speed at which it leaves the golf club, and so determine whether the given ball makes it to the putting green. |
| One might think that there’s not much to choose between the methods employed on Saturday and then on Sunday (apart from the fact that in one case we had need of a clever computing device) but there’s a world of difference: on Saturday, we count the number of balls on the green after their arrival; on Sunday, we calculate the whole trajectory of a given ball and so we know whether the ball arrives, and when. We can say that Saturday’s and Sunday’s results occur in different ‘spaces’. In the ‘Sunday space’, we can reconstruct the entire history of each and every golfball; in the ‘Saturday space’, we are happy to forego this detailed knowledge because we really just want to know what pro- portion of our drives do in fact make it to the putting green. We could also investigate other questions of a general nature, such as whether any golfballs at all will make it through a certain gap in the trees, and what overall difference the choice of golfclub makes, and so on. (If we need to know more about one specific ball or another, this more detailed knowledge can be reconstructed afterward, if we supply the appropriate extra data.) | One might think that there’s not much to choose between the methods employed on Saturday and then on Sunday (apart from the fact that in one case we had need of a clever computing device) but there’s a world of difference: on Saturday, we count the number of balls on the green after their arrival; on Sunday, we calculate the whole trajectory of a given ball and so we know whether the ball arrives, and when. We can say that Saturday’s and Sunday’s results occur in different ‘spaces’. In the ‘Sunday space’, we can reconstruct the entire history of each and every golfball; in the ‘Saturday space’, we are happy to forego this detailed knowledge because we really just want to know what pro- portion of our drives do in fact make it to the putting green. We could also investigate other questions of a general nature, such as whether any golfballs at all will make it through a certain gap in the trees, and what overall difference the choice of golfclub makes, and so on. (If we need to know more about one specific ball or another, this more detailed knowledge can be reconstructed afterward, if we supply the appropriate extra data.) | ||
| - | This allegory nicely demonstrates the sorts of differences we find between [[basic_tools:phase_space|phase space]] (the ‘Saturday space’) and [[basic_tools:configuration_space|configuration space]] (the ‘Sunday space’). In phase space we obtain qualitative information, about more golfballs, all in one go - we obtain ‘less from more’. Before we give some examples, let’s first explain how a plot in phase space is constructed. | + | This allegory nicely demonstrates the sorts of differences we find between [[basic_tools:phase_space|phase space]] (the ‘Saturday space’) and [[basic_tools:configuration_space|configuration space]] (the ‘Sunday space’). In phase space we obtain qualitative information, about more golfballs, all in one go - we obtain ‘less from more’. |
| <cite>The Lazy Universe by Coopersmith</cite></blockquote> | <cite>The Lazy Universe by Coopersmith</cite></blockquote> | ||
| - | The Hamiltonian formalism describes mechanics by trajectories in [[basic_tools:phase_space|phase space]], while [[frameworks:lagrangian_formalism|Lagrangian mechanics]] uses trajectories in [[basic_tools:configuration_space|configuration space]]. | + | The Hamiltonian formalism describes mechanics by trajectories in [[basic_tools:phase_space|phase space]], while [[formalisms:lagrangian_formalism|Lagrangian mechanics]] uses trajectories in [[basic_tools:configuration_space|configuration space]]. |
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| <tabbox Concrete> | <tabbox Concrete> | ||
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| + | **Relationship to the Lagrangian formalism** | ||
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| + | The Hamiltonian is defined as the [[advanced_tools:legendre_transformation|Legendre transformation]] of the Lagrangian | ||
| $$H \equiv \sum_j {\dot q}_j p_j - L$$ | $$H \equiv \sum_j {\dot q}_j p_j - L$$ | ||
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| + | ---- | ||
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| + | **Graphical Summary of Newtonian, Lagrangian, and Hamiltonian Formalism** | ||
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| + | [{{ :frameworks:newton_lagrange_hamilton.jpg?nolink }}] | ||
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| + | For some concrete examples worked out in all three frameworks see [[https://esackinger.wordpress.com/blog/classical-mechanics|Fun with Symmetry]]. | ||
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| <blockquote> | <blockquote> | ||
| - | In the [[frameworks:lagrangian_formalism|Lagrangian approach]] we focus on the position and velocity of a particle, and | + | In the [[formalisms:lagrangian_formalism|Lagrangian approach]] we focus on the position and velocity of a particle, and |
| compute what the particle does starting from the Lagrangian $L(q, q˙)$, which is a function | compute what the particle does starting from the Lagrangian $L(q, q˙)$, which is a function | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | <WRAP group> | + | |
| - | <WRAP half column> | + | |
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| - | </WRAP> | ||
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| - | <WRAP half column> | ||
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| - | <blockquote>[E]verybody loves Hamilton’s equations: there are just two, and they summarize the entire essence of classical mechanics. | ||
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| - | <cite>[[https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/|John Baez]]</cite></blockquote> | ||
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| - | </WRAP> | ||
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| - | </WRAP> | ||
| <blockquote>In fact, | <blockquote>In fact, | ||
| with suitable choices of H, Hamilton's equations still hold true for any system | with suitable choices of H, Hamilton's equations still hold true for any system | ||
| of classical equations whatever, not just for Newton's equations. In particular, this will be the case for the Maxwell(-Lorentz) theory that we shall be | of classical equations whatever, not just for Newton's equations. In particular, this will be the case for the Maxwell(-Lorentz) theory that we shall be | ||
| - | considering shortly. Hamilton's equations also hold true for [[theories:special_relativity|special relativity]]. | + | considering shortly. Hamilton's equations also hold true for [[models:special_relativity|special relativity]]. |
| - | Even[[theories:general_relativity| general relativity]] can, if due care is exercised, be subsumed into the | + | Even[[models:general_relativity| general relativity]] can, if due care is exercised, be subsumed into the |
| Hamiltonian framework. Moreover, as we shall see later with [[equations:schroedinger_equation|Schrodinger's | Hamiltonian framework. Moreover, as we shall see later with [[equations:schroedinger_equation|Schrodinger's | ||
| equation]] (p. 288), this Hamiltonian framework provides the taking-off point | equation]] (p. 288), this Hamiltonian framework provides the taking-off point | ||
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| <blockquote>Hamiltonian Mechanics is geometry in [[basic_tools:phase_space|phase space]]. [...] | <blockquote>Hamiltonian Mechanics is geometry in [[basic_tools:phase_space|phase space]]. [...] | ||
| - | [[frameworks:lagrangian_formalism|Lagrangian mechanics]] is contained in hamiltonian mechanics as a special case (the phase space in this case is the cotangent bundle of the configuration space, and the hamiltonian function is the Legendre transform of the lagrangian function). | + | [[formalisms:lagrangian_formalism|Lagrangian mechanics]] is contained in hamiltonian mechanics as a special case (the phase space in this case is the cotangent bundle of the configuration space, and the hamiltonian function is the Legendre transform of the lagrangian function). |
| The hamiltonian point of view allows us to solve completely a series of mechanical problems which do not yield solutions by other means (for example, the problem of attraction by two stationary centers and the problem of geodesics of the triaxial ellipsoid. The hamiltonian point of view has even greater value for the approximate methods of perturbation theory (celestial mechanics), for understanding the general character of motion in complicated mechanical systems (ergodic theory, statistical mechanics) and in connection with other areas of mathematical physics (optics, quantum mechanics, etc.) | The hamiltonian point of view allows us to solve completely a series of mechanical problems which do not yield solutions by other means (for example, the problem of attraction by two stationary centers and the problem of geodesics of the triaxial ellipsoid. The hamiltonian point of view has even greater value for the approximate methods of perturbation theory (celestial mechanics), for understanding the general character of motion in complicated mechanical systems (ergodic theory, statistical mechanics) and in connection with other areas of mathematical physics (optics, quantum mechanics, etc.) | ||
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| <cite>Mathematical Methods of Classical Mechanics [[https://books.google.de/books?id=5OQlBQAAQBAJ&lpg=PA160&ots=u7Qs-TMaNb&dq=%22Hamiltonian%20Mechanics%20is%20geometry%20in%20phase%20space.%22&hl=de&pg=PA160#v=onepage&q&f=false|Vladimir Arnold]]</cite></blockquote> | <cite>Mathematical Methods of Classical Mechanics [[https://books.google.de/books?id=5OQlBQAAQBAJ&lpg=PA160&ots=u7Qs-TMaNb&dq=%22Hamiltonian%20Mechanics%20is%20geometry%20in%20phase%20space.%22&hl=de&pg=PA160#v=onepage&q&f=false|Vladimir Arnold]]</cite></blockquote> | ||
| + | <blockquote>As Weinberg points in his QFT book, in the Hamiltonian formalism it is easier to check the unitarity of the theory because unitarity is directly related to evolution, while in the Lagrangian formalism the symmetries that mix space with time are more explicit. Therefore the Hamiltonian formalism is usually more convenient in non-relativistic and galilean quantum theories. In order for a theory to be Poincare invariant, the Lagrangian needs to be a Poincare scalar, what it is easy to see. The equivalent condition in the Hamiltonian formalism is that there is a Poincare algebra with the Hamiltonian as the zero component of the 4-momentum. This condition needs to be checked, as it is not elemental to see.<cite>[[https://physics.stackexchange.com/a/33516/37286|Diego Mazón]]</cite></blockquote> | ||
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| + | * See also the discussion on the question [[https://physics.stackexchange.com/questions/89035/whats-the-point-of-hamiltonian-mechanics|What's the point of Hamiltonian mechanics?]] at StackExchange | ||
| <tabbox FAQ> | <tabbox FAQ> | ||
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| --> How is the Hamiltonian Formalism related to the Newtonian Formalism?# | --> How is the Hamiltonian Formalism related to the Newtonian Formalism?# | ||
| - | <blockquote>"Recall that we derived Hamilton’s equations for a particle moving in a force field $F = -dV/dx$ by writing down the equations of motion in the form $$ m \dot{x} = p , \quad \dot{p} = - \frac{\partial V}{\partial x} .$$ The observant reader will have noticed that these two equations are just one way to express Newton’s second law. More generally for a system of N point-like particles moving in three-dimensional physical space, Newton’s second law would be $$ m \dot{x_j} = p_j , \quad \dot{p}_j = - \frac{\partial V}{\partial x_j} .$$" <cite>The symplectic egg in classical and quantum mechanics by Maurice A. de Gosson</cite></blockquote> | + | <blockquote>Recall that we derived Hamilton’s equations for a particle moving in a force field $F = -dV/dx$ by writing down the equations of motion in the form $$ m \dot{x} = p , \quad \dot{p} = - \frac{\partial V}{\partial x} .$$ The observant reader will have noticed that these two equations are just one way to express Newton’s second law. More generally for a system of N point-like particles moving in three-dimensional physical space, Newton’s second law would be $$ m \dot{x_j} = p_j , \quad \dot{p}_j = - \frac{\partial V}{\partial x_j} .$$ <cite>The symplectic egg in classical and quantum mechanics by Maurice A. de Gosson</cite></blockquote> |
| <-- | <-- | ||
| - | --> Where do the Hamiltonian Equations come from?# | ||
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| - | "Hamilton’s equations and the [[equations:maxwell_relations|Maxwell relations]]—are mathematically just the same. They both say simply that partial derivatives commute." See https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/ | ||
| - | and https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/ | ||
| - | <-- | ||
| --> How is a Hamiltonian constructed from a Lagrangian with a Legendre transform# | --> How is a Hamiltonian constructed from a Lagrangian with a Legendre transform# | ||
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