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Text of the page (random words):
nian h 1 2 from euler lagrange equation to hamilton s equations 1 3 from stationary action principle to hamilton s equations 1 4 basic physical interpretation 2 example 3 deriving hamilton s equations 4 properties of the hamiltonian 5 hamiltonian as the total system energy toggle hamiltonian as the total system energy subsection 5 1 proof 5 2 application to systems of point masses 5 3 conservation of energy 6 hamiltonian of a charged particle in an electromagnetic field 7 from symplectic geometry to hamilton s equations toggle from symplectic geometry to hamilton s equations subsection 7 1 geometry of hamiltonian systems 7 2 riemannian manifolds 7 3 sub riemannian manifolds 7 4 poisson algebras 7 5 generalization to quantum mechanics through poisson bracket 8 see also 9 references 10 further reading 11 external links toggle the table of contents hamiltonian mechanics 41 languages afrikaans العربية беларуская български bosanski català čeština чӑвашла deutsch esperanto español eesti فارسی suomi français galego עברית हिन्दी bahasa indonesia italiano 日本語 한국어 മലയാളം nederlands norsk bokmål ਪੰਜਾਬੀ polski português română русский simple english slovenščina shqip српски srpski svenska ไทย türkçe українська oʻzbekcha ўзбекча tiếng việt 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikibooks wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia formulation of classical mechanics using momenta sir william rowan hamilton part of a series on classical mechanics f d p d t displaystyle textbf f frac d mathbf p dt second law of motion history timeline textbooks branches applied celestial continuum dynamics field theory kinematics kinetics statics statistical mechanics fundamentals acceleration angular momentum couple d alembert s principle energy kinetic potential force frame of reference inertial frame of reference impulse inertia moment of inertia mass mechanical power mechanical work moment momentum space speed time torque velocity virtual work formulations newton s laws of motion analytical mechanics lagrangian mechanics hamiltonian mechanics routhian mechanics hamilton jacobi equation appell s equation of motion koopman von neumann mechanics core topics damping displacement equations of motion euler s laws of motion fictitious force friction harmonic oscillator inertial non inertial reference frame motion linear newton s law of universal gravitation newton s laws of motion relative velocity rigid body dynamics euler s equations simple harmonic motion vibration rotation circular motion rotating reference frame centripetal force centrifugal force reactive coriolis force pendulum tangential speed rotational frequency angular acceleration displacement frequency velocity scientists kepler galileo huygens newton horrocks halley maupertuis daniel bernoulli johann bernoulli euler d alembert clairaut lagrange laplace poisson hamilton jacobi cauchy routh liouville appell gibbs koopman von neumann physics portal category v t e in physics hamiltonian mechanics is a reformulation of lagrangian mechanics that emerged in 1833 introduced by sir william rowan hamilton 1 hamiltonian mechanics replaces generalized velocities q i displaystyle dot q i used in lagrangian mechanics with generalized momenta both theories provide interpretations of classical mechanics and describe the same physical phenomena hamiltonian mechanics has a close relationship with geometry notably symplectic geometry and poisson structures and serves as a link between classical and quantum mechanics overview edit phase space coordinates p q and hamiltonian h edit let m l displaystyle m mathcal l be a mechanical system with configuration space m displaystyle m and smooth lagrangian l displaystyle mathcal l select a standard coordinate system q q displaystyle boldsymbol q boldsymbol dot q on the tangent bundle t m displaystyle tm the quantities p i q q t def l q i displaystyle textstyle p_ i boldsymbol q boldsymbol dot q t stackrel text def partial mathcal l partial dot q i are called momenta also generalized momenta conjugate momenta and canonical momenta for a time instant t displaystyle t the legendre transformation of l displaystyle mathcal l is defined as the map q q p q displaystyle boldsymbol q boldsymbol dot q to left boldsymbol p boldsymbol q right which is assumed to have a smooth inverse p q q q displaystyle boldsymbol p boldsymbol q to boldsymbol q boldsymbol dot q for a system with n displaystyle n degrees of freedom the lagrangian mechanics defines the energy function e l q q t def i 1 n q i l q i l displaystyle e_ mathcal l boldsymbol q boldsymbol dot q t stackrel text def sum _ i 1 n dot q i frac partial mathcal l partial dot q i mathcal l the legendre transform of l displaystyle mathcal l turns e l displaystyle e_ mathcal l into a function h p q t displaystyle mathcal h boldsymbol p boldsymbol q t known as the hamiltonian the hamiltonian satisfies h l q q t e l q q t displaystyle mathcal h left frac partial mathcal l partial boldsymbol dot q boldsymbol q t right e_ mathcal l boldsymbol q boldsymbol dot q t which implies that h p q t i 1 n p i q i l q q t displaystyle mathcal h boldsymbol p boldsymbol q t sum _ i 1 n p_ i dot q i mathcal l boldsymbol q boldsymbol dot q t where the velocities q q 1 q n displaystyle boldsymbol dot q dot q 1 ldots dot q n are found from the n displaystyle n dimensional equation p l q displaystyle textstyle boldsymbol p partial mathcal l partial boldsymbol dot q which by assumption is uniquely solvable for q displaystyle boldsymbol dot q the 2 n displaystyle 2n dimensional pair p q displaystyle boldsymbol p boldsymbol q is called phase space coordinates also canonical coordinates from euler lagrange equation to hamilton s equations edit in phase space coordinates p q displaystyle boldsymbol p boldsymbol q the n displaystyle n dimensional euler lagrange equation l q d d t l q 0 displaystyle frac partial mathcal l partial boldsymbol q frac d dt frac partial mathcal l partial dot boldsymbol q 0 becomes hamilton s equations in 2 n displaystyle 2n dimensions d q d t h p d p d t h q displaystyle frac mathrm d boldsymbol q mathrm d t frac partial mathcal h partial boldsymbol p quad frac mathrm d boldsymbol p mathrm d t frac partial mathcal h partial boldsymbol q proof the hamiltonian h p q displaystyle mathcal h boldsymbol p boldsymbol q is the legendre transform of the lagrangian l q q displaystyle mathcal l boldsymbol q dot boldsymbol q thus one has l q q h p q p q displaystyle mathcal l boldsymbol q dot boldsymbol q mathcal h boldsymbol p boldsymbol q boldsymbol p dot boldsymbol q 1 where p l q displaystyle boldsymbol p partial mathcal l partial dot boldsymbol q by rearranging the equation p l q displaystyle boldsymbol p partial mathcal l partial dot boldsymbol q we can write q displaystyle dot boldsymbol q in terms of q displaystyle boldsymbol q and p displaystyle boldsymbol p as q q p displaystyle dot boldsymbol q boldsymbol q boldsymbol p so 1 becomes an equation in two variables q displaystyle boldsymbol q and p displaystyle boldsymbol p taking the partial derivative of both sides of 1 with respect to p displaystyle boldsymbol p i e keeping q displaystyle boldsymbol q fixed gives l q q p h p q p q p h p q displaystyle frac partial mathcal l partial dot boldsymbol q frac partial dot boldsymbol q partial boldsymbol p frac partial mathcal h partial boldsymbol p dot boldsymbol q boldsymbol p frac partial dot boldsymbol q partial boldsymbol p implies frac partial mathcal h partial boldsymbol p dot boldsymbol q taking the partial derivative of both sides of 1 with respect to q displaystyle boldsymbol q instead i e keeping p displaystyle boldsymbol p fixed gives l q l q q q h q p q q l q h q displaystyle frac partial mathcal l partial boldsymbol q frac partial mathcal l partial dot boldsymbol q frac partial dot boldsymbol q partial boldsymbol q frac partial mathcal h partial boldsymbol q boldsymbol p frac partial dot boldsymbol q partial boldsymbol q implies frac partial mathcal l partial boldsymbol q frac partial mathcal h partial boldsymbol q now the euler lagrange equations yield p d d t l q l q h q displaystyle dot boldsymbol p frac mathrm d mathrm d t frac partial mathcal l partial dot boldsymbol q frac partial mathcal l partial boldsymbol q frac partial mathcal h partial boldsymbol q from stationary action principle to hamilton s equations edit let p a b x a x b displaystyle mathcal p a b boldsymbol x _ a boldsymbol x _ b be the set of smooth paths q a b m displaystyle boldsymbol q a b to m for which q a x a displaystyle boldsymbol q a boldsymbol x _ a and q b x b displaystyle boldsymbol q b boldsymbol x _ b the action functional s p a b x a x b r displaystyle mathcal s mathcal p a b boldsymbol x _ a boldsymbol x _ b to mathbb r is defined via s q a b l t q t q t d t a b i 1 n p i q i h p q t d t displaystyle mathcal s boldsymbol q int _ a b mathcal l t boldsymbol q t dot boldsymbol q t dt int _ a b left sum _ i 1 n p_ i dot q i mathcal h boldsymbol p boldsymbol q t right dt where q q t displaystyle boldsymbol q boldsymbol q t and p l q displaystyle boldsymbol p partial mathcal l partial boldsymbol dot q see above a path q p a b x a x b displaystyle boldsymbol q in mathcal p a b boldsymbol x _ a boldsymbol x _ b is a stationary point of s displaystyle mathcal s and hence is an equation of motion if and only if the path p t q t displaystyle boldsymbol p t boldsymbol q t in phase space coordinates obeys the hamilton equations basic physical interpretation edit a simple interpretation of hamiltonian mechanics comes from its application on a one dimensional system consisting of one nonrelativistic particle of mass m the value h p q displaystyle h p q of the hamiltonian is the total energy of the system in this case the sum of kinetic and potential energy traditionally denoted t and v respectively here p is the momentum mv and q is the space coordinate then h t v t p 2 2 m v v q displaystyle mathcal h t v qquad t frac p 2 2m qquad v v q t is a function of p alone while v is a function of q alone i e t and v are scleronomic in this example the time derivative of q is the velocity and so the first hamilton equation means that the particle s velocity equals the derivative of its kinetic energy with respect to its momentum the time derivative of the momentum p equals the newtonian force and so the second hamilton equation means that the force equals the negative gradient of potential energy example edit main article spherical pendulum a spherical pendulum consists of a mass m moving without friction on the surface of a sphere the only forces acting on the mass are the reaction from the sphere and gravity spherical coordinates are used to describe the position of the mass in terms of r θ φ where r is fixed r ℓ spherical pendulum angles and velocities the lagrangian for this system is 2 l 1 2 m ℓ 2 θ 2 sin 2 θ φ 2 m g ℓ cos θ displaystyle l frac 1 2 m ell 2 left dot theta 2 sin 2 theta dot varphi 2 right mg ell cos theta thus the hamiltonian is h p θ θ p φ φ l displaystyle h p_ theta dot theta p_ varphi dot varphi l where p θ l θ m ℓ 2 θ displaystyle p_ theta frac partial l partial dot theta m ell 2 dot theta and p φ l φ m ℓ 2 sin 2 θ φ displaystyle p_ varphi frac partial l partial dot varphi m ell 2 sin 2 theta dot varphi in terms of coordinates and momenta the hamiltonian reads h 1 2 m ℓ 2 θ 2 1 2 m ℓ 2 sin 2 θ φ 2 t m g ℓ cos θ v p θ 2 2 m ℓ 2 p φ 2 2 m ℓ 2 sin 2 θ m g ℓ cos θ displaystyle begin aligned h underbrace bigl tfrac 1 2 m ell 2 dot theta 2 tfrac 1 2 m ell 2 sin 2 theta dot varphi 2 bigr _ t underbrace bigl mg ell cos theta bigr _ v 2ex frac p_ theta 2 2m ell 2 frac p_ varphi 2 2m ell 2 sin 2 theta mg ell cos theta end aligned hamilton s equations give the time evolution of coordinates and conjugate momenta in four first order differential equations θ p θ m ℓ 2 φ p φ m ℓ 2 sin 2 θ p θ p φ 2 m ℓ 2 sin 3 θ cos θ m g ℓ sin θ p φ 0 displaystyle begin aligned dot theta p_ theta over m ell 2 6pt dot varphi p_ varphi over m ell 2 sin 2 theta 6pt dot p_ theta p_ varphi 2 over m ell 2 sin 3 theta cos theta mg ell sin theta 6pt dot p_ varphi 0 end aligned momentum p φ displaystyle p_ varphi which corresponds to the vertical component of angular momentum l z ℓ sin θ m ℓ sin θ φ displaystyle l_ z ell sin theta times m ell sin theta dot varphi is a constant of motion that is a consequence of the rotational symmetry of the system around the vertical axis being absent from the hamiltonian azimuth φ displaystyle varphi is a cyclic coordinate which implies conservation of its conjugate momentum deriving hamilton s equations edit hamilton s equations can be derived by a calculation with the lagrangian l displaystyle mathcal l generalized positions q i and generalized velocities q i where i 1 n displaystyle i 1 ldots n 3 here we work off shell meaning q i displaystyle q i q i displaystyle dot q i t displaystyle t are independent coordinates in phase space not constrained to follow any equations of motion in particular q i displaystyle dot q i is not a derivative of q i displaystyle q i the total differential of the lagrangian is d l i l q i d q i l q i d q i l t d t displaystyle mathrm d mathcal l sum _ i left frac partial mathcal l partial q i mathrm d q i frac partial mathcal l partial dot q i mathrm d dot q i right frac partial mathcal l partial t mathrm d t the generalized momentum coordinates were defined as p i l q i displaystyle p_ i partial mathcal l partial dot q i so we may rewrite the equation as d l i l q i d q i p i d q i l t d t i l q i d q i d p i q i q i d p i l t d t displaystyle begin aligned mathrm d mathcal l sum _ i left frac partial mathcal l partial q i mathrm d q i p_ i mathrm d dot q i right frac partial mathcal l partial t mathrm d t sum _ i left frac partial mathcal l partial q i mathrm d q i mathrm d p_ i dot q i dot q i mathrm d p_ i right frac partial mathcal l partial t mathrm d t end aligned after rearranging one obtains d i p i q i l i l q i d q i q i d p i l t d t displaystyle mathrm d left sum _ i p_ i dot q i mathcal l right sum _ i left frac partial mathcal l partial q i mathrm d q i dot q i mathrm d p_ i right frac partial mathcal l partial t mathrm d t the term in parentheses on the left hand side is just the hamiltonian h p i q i l textstyle mathcal h sum p_ i dot q i mathcal l defined previously therefore d h i l q i d q i q i d p i l t d t displaystyle mathrm d mathcal h sum _ i left frac partial mathcal l partial q i mathrm d q i dot q i mathrm d p_ i right frac partial mathcal l partial t mathrm d t one may also calculate the total differential of the hamiltonian h displaystyl...