formalisms:hamiltonian_formalism
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formalisms:hamiltonian_formalism [2018/04/08 17:13] georgefarr ↷ Links adapted because of a move operation |
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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| $$H \equiv \sum_j {\dot q}_j p_j - L$$ | $$H \equiv \sum_j {\dot q}_j p_j - L$$ | ||
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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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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
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| - | <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, | ||
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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> |
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formalisms/hamiltonian_formalism.1523200403.txt.gz · Last modified: 2018/04/08 15:13 (external edit)
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