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Newton Vs Lagrange Vs Hamilton |
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| Mar25-05, 04:54 PM | #1 |
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Newton Vs Lagrange Vs Hamilton
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>What is the relationship between:\n\n1. Newtonian Mechanics\n2. Lagrangian Mechanics\n3. Hamiltonian Mechanics\n\nCan they be declared to be *rigorously* mathematically equivalent?\n\nDoes it suffice to say that they are reformulations of each other,\neach well-adapted to a particular class of problems?\n\nIf so, can we describe reasonably clearly which type of problems are\nmore easily represented and/or solved using each version of mechanics?\n\nFinally, is there a book on classical mechanics which treats all three\nof these approaches to mechanics, along with their\ninter-relationships, with mathematical rigour? (I\'m afraid I don\'t\nmuch like Goldstein\'s widely cited book).\n\nVonny N.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>What is the relationship between:
1. Newtonian Mechanics 2. Lagrangian Mechanics 3. Hamiltonian Mechanics Can they be declared to be *rigorously* mathematically equivalent? Does it suffice to say that they are reformulations of each other, each well-adapted to a particular class of problems? If so, can we describe reasonably clearly which type of problems are more easily represented [itex]and/or[/itex] solved using each version of mechanics? Finally, is there a book on classical mechanics which treats all three of these approaches to mechanics, along with their inter-relationships, with mathematical rigour? (I'm afraid I don't much like Goldstein's widely cited book). Vonny N. |
| Mar26-05, 10:07 PM | #2 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n"Vonny N" <vonnyn@hotmail.com> wrote in message\nnews:df799de5.0503251424.2f339940@posting.google.com...\n> What is the relationship between:\n>\n> 1. Newtonian Mechanics\n> 2. Lagrangian Mechanics\n> 3. Hamiltonian Mechanics\n>\n> Can they be declared to be *rigorously* mathematically equivalent?\n>\n> Does it suffice to say that they are reformulations of each other,\n> each well-adapted to a particular class of problems?\n>\n> If so, can we describe reasonably clearly which type of problems are\n> more easily represented and/or solved using each version of mechanics?\n>\n> Finally, is there a book on classical mechanics which treats all three\n> of these approaches to mechanics, along with their\n> inter-relationships, with mathematical rigour? (I\'m afraid I don\'t\n> much like Goldstein\'s widely cited book).\n>\n> Vonny N.\n>\n\nA very brief reply from me cos my girlfriend is due around in 5 mins, I\nhaven\'t cleaned the house like I promised, and I haven\'t cooked the dinner\nlike I promised!\n\nPlease forgive the rather lax presentation as well\n\nIf you take a Lagrangian function L = difference in Kinetic T and Potential\nenergy V\ni.e. L = T - V. Apply a first order variation/Hamilton\'s primnciple and end\nup with the standard Lagrange equations d/dt(dL/dv) - dL/dx = 0. (little v =\nvelocity dx/dt)\n\nWith the standard, one dimensional T = 1/2mv^2 (v=dx/dt), V = 0 (freely\nmoving particle no field) and so L = 1/2.m.v^2 you will get mdv/dt = 0, with\na little flippancy we have d(mv)/dt = 0, i.e. rate of change of linear\nmomemtum = 0 when no force acts.\n\nIn short, if the Lagrangian is the difference of kinetic and potential\nenergy, you should get back to Newtonian mechanics. The real beauty is that\nthe Lagrangian can be expanded to to other functionals L. Do this and you\ncan are in the realm of field theory.\n\nHamilton takes Lagranges equations, which are second order in time, and\nconverts them to two first order equations. The beauty in these latter\nequations is that they are almost symmetric...\n\ncrikes my girlfriend has just turned up\n\nI\'m done for, back later\n\ncheers\n\nRichard Miller\n\n\n\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Vonny N" <vonnyn@hotmail.com> wrote in message
news:df799de5.0503251424.2f339940@posting.google.com... > What is the relationship between: > > 1. Newtonian Mechanics > 2. Lagrangian Mechanics > 3. Hamiltonian Mechanics > > Can they be declared to be *rigorously* mathematically equivalent? > > Does it suffice to say that they are reformulations of each other, > each well-adapted to a particular class of problems? > > If so, can we describe reasonably clearly which type of problems are > more easily represented [itex]and/or[/itex] solved using each version of mechanics? > > Finally, is there a book on classical mechanics which treats all three > of these approaches to mechanics, along with their > inter-relationships, with mathematical rigour? (I'm afraid I don't > much like Goldstein's widely cited book). > > Vonny N. > A very brief reply from [itex]me cos my[/itex] girlfriend is due around in 5 mins, I haven't cleaned the house like I promised, and I haven't cooked the dinner like I promised! Please forgive the rather lax presentation as well If you take a Lagrangian function L = difference in Kinetic T and Potential energy V i.e. [itex]L = T - V[/itex]. Apply a first order variation/Hamilton's primnciple and end up with the standard Lagrange equations [itex]d/dt(dL/dv) - dL/dx = .[/itex] (little v = velocity [itex]dx/dt)[/itex] With the standard, one dimensional [itex]T = 1/2mv^2 (v=dx/dt), V =[/itex] (freely moving particle no field) and [itex]so L = 1/2[/itex].m.[itex]v^2[/itex] you will get [itex]mdv/dt = 0,[/itex] with a little flippancy we have [itex]d(mv)/dt = 0, i[/itex].e. rate of change of linear momemtum = when no force acts. In short, if the Lagrangian is the difference of kinetic and potential energy, you should get back to Newtonian mechanics. The real beauty is that the Lagrangian can be expanded to to other functionals L. Do this and you can are in the realm of field theory. Hamilton takes Lagranges equations, which are second order in time, and converts them to two first order equations. The beauty in these latter equations is that they are almost symmetric... crikes my girlfriend has just turned up I'm done for, back later cheers Richard Miller |
| Mar27-05, 02:56 AM | #3 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>vonnyn@hotmail.com (Vonny N) wrote in message news:<df799de5.0503251424.2f339940@posting.google.com>...\n> What is the relationship between:\n>\n> 1. Newtonian Mechanics\n> 2. Lagrangian Mechanics\n> 3. Hamiltonian Mechanics\n\n1. --> 2. In case of conservative forces, one can study a Lagrangian\nof the form L = T - V with V the potential energy. This is equivalent\nto the original Newtonian setup. There are, however, Lagrangians\nwhich are not of the form L = T - V and therefore are not\nrepresentable in Newtonian form. (This should be taken with a grain\nof salt).\n\n2. --> 3. To construct a Hamiltonian from a given Lagrangian, one\nstudies a map called the Legendre transform (this expresses the p\'s as\nderivatives of the Lagrangian). In order to do this construction, one\nhas to invert the Legendre transformation, so the transition to the\nHamiltonian formalism only works when this map is actually invertible.\n\nIf you know that p = dL/dv, it\'s easy to see that invertibility is\nequivalent to the matrix d^2 L/dv^2 being invertible. Now, this is\nalways the case for a Lagrangian of mechanical type (one of the form L\n= T - V, with T the kinetic energy).\n\nThe above should again be taken with a grain of salt. Even if the\nLagrangian is degenerate, there are more sophisticated ways of going\nto the Hamiltonian formalism. In this case, some constraints will\narise and these will determine a subset of the phase space. Take the\nexample where the Lagrangian does not depend on a certain velocity\ncoordinate v_0. Then the associated momentum p_0 = dL/dv_0 will be\nidentically zero.\n\nFurthermore, the above treatment only goes through in the case of\nfirst-order theories, i.e. depending on x, v, but not the derivatives\nof v. In that case, one can still make sense of a lot of these\nthings, but it is all a lot more involved. It certainly is no mere\nextension of the first order case.\n\n> Can they be declared to be *rigorously* mathematically equivalent?\n>\n> Does it suffice to say that they are reformulations of each other,\n> each well-adapted to a particular class of problems?\n\nThe prevailing mathematical point of view is that Lagrangian mechanics\ntakes place on the tangent bundle of the configuration space, whereas\nthe Hamiltonian formalism is dealt with on the cotangent bundle.\nThere are various intrinsic objects defined on these bundles, and the\nequations of motion can be expressed by use of them.\n\nAs to which formulation to use, I think it mostly depends (in the\nregular case) on the formulation of the problem. Sometimes one is\npreferred over the other: in the Lagrangian picture one has a\n"variational principle"; i.e. the equations of motion are derived by\nextremizing a certain functional. This can sometimes be exploited,\ne.g. in the construction of numerical schemes which preserve (some of)\nthe geometrical content of the problem.\n\nOn the other hand, Hamiltonian mechanics is more directly amenable to\ntechniques from symplectic geometry, e.g. symplectic reduction,\ndefinitions involving integrability, etc.\n\n> If so, can we describe reasonably clearly which type of problems are\n> more easily represented and/or solved using each version of mechanics?\n>\n> Finally, is there a book on classical mechanics which treats all three\n> of these approaches to mechanics, along with their\n> inter-relationships, with mathematical rigour? (I\'m afraid I don\'t\n> much like Goldstein\'s widely cited book).\n>\n\nSee the book "foundations of mechanics" by Abraham and Marsden or\n"introduction to mechanics and symmetry" by Marsden and Ratiu. These\nmake use of a lot of differential geometry so they might not be very\neasy going on a first reading (especially FoM), but they tell you just\nabout everything you\'ll ever want to know on this topic. There\'s also\nthe book by V.I. Arnold "Mathematical Methods of Classical Mechanics",\nwhich is very good.\n\nN.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>vonnyn@hotmail.com (Vonny N) wrote in message news:<df799de5.0503251424.2f339940@p...google.com>...
> What is the relationship between: > > 1. Newtonian Mechanics > 2. Lagrangian Mechanics > 3. Hamiltonian Mechanics 1. --> 2. In case of conservative forces, one can study a Lagrangian of the form [itex]L = T - V[/itex] with V the potential energy. This is equivalent to the original Newtonian setup. There are, however, Lagrangians which are not of the form [itex]L = T - V[/itex] and therefore are not representable in Newtonian form. (This should be taken with a grain of salt). 2. --> 3. To construct a Hamiltonian from a given Lagrangian, one studies a map called the Legendre transform (this expresses the p's as derivatives of the Lagrangian). In order to do this construction, one has to invert the Legendre transformation, so the transition to the Hamiltonian formalism only works when this map is actually invertible. If you know that [itex]p = dL/dv,[/itex] it's easy to see that invertibility is equivalent to the matrix [itex]d^2 L/dv^2[/itex] being invertible. Now, this is always the case for a Lagrangian of mechanical type (one of the form L [itex]= T - V,[/itex] with T the kinetic energy). The above should again be taken with a grain of salt. Even if the Lagrangian is degenerate, there are more sophisticated ways of going to the Hamiltonian formalism. In this case, some constraints will arise and these will determine a subset of the phase space. Take the example where the Lagrangian does not depend on a certain velocity coordinate [itex]v_0[/itex]. Then the associated momentum [itex]p_0 = dL/dv_0[/itex] will be identically zero. Furthermore, the above treatment only goes through in the case of first-order theories, i.e. depending on x, v, but not the derivatives of v. In that case, one can still make sense of a lot of these things, but it is all a lot more involved. It certainly is no mere extension of the first order case. > Can they be declared to be *rigorously* mathematically equivalent? > > Does it suffice to say that they are reformulations of each other, > each well-adapted to a particular class of problems? The prevailing mathematical point of view is that Lagrangian mechanics takes place on the tangent bundle of the configuration space, whereas the Hamiltonian formalism is dealt with on the cotangent bundle. There are various intrinsic objects defined on these bundles, and the equations of motion can be expressed by use of them. As to which formulation to use, I think it mostly depends (in the regular case) on the formulation of the problem. Sometimes one is preferred over the other: in the Lagrangian picture one has a "variational principle"; i.e. the equations of motion are derived by extremizing a certain functional. This can sometimes be exploited, e.g. in the construction of numerical schemes which preserve (some of) the geometrical content of the problem. On the other hand, Hamiltonian mechanics is more directly amenable to techniques from symplectic geometry, e.g. symplectic reduction, definitions involving integrability, etc. > If so, can we describe reasonably clearly which type of problems are > more easily represented [itex]and/or[/itex] solved using each version of mechanics? > > Finally, is there a book on classical mechanics which treats all three > of these approaches to mechanics, along with their > inter-relationships, with mathematical rigour? (I'm afraid I don't > much like Goldstein's widely cited book). > See the book "foundations of mechanics" by Abraham and Marsden or "introduction to mechanics and symmetry" by Marsden and Ratiu. These make use of a lot of differential geometry so they might not be very easy going on a first reading (especially FoM), but they tell you just about everything you'll ever want to know on this topic. There's also the book by V.I. Arnold "Mathematical Methods of Classical Mechanics", which is very good. N. |
| Mar27-05, 02:57 AM | #4 |
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Newton Vs Lagrange Vs Hamilton
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Vonny N wrote:\n> Finally, is there a book on classical mechanics which treats all\nthree\n> of these approaches to mechanics, along with their\n> inter-relationships, with mathematical rigour? (I\'m afraid I don\'t\n> much like Goldstein\'s widely cited book).\n>\n> Vonny N.\n\nThere are many excellent classical mechanics texts. But the only one\nthat I\'m aware of that is completely clear on the exact meaning of the\nequations, and avoids the formal ambiguity inherent in most\npresentations of the Euler-Lagrange equations, is _Structure and\nInterpretation of Classical Mechanics_, by Sussman and Wisdom. It is\nboth quite clear and quite rigorous. It uses a functional notation,\nsimilar to Spivak (Calculus on Manifolds, Differential Geometry).\n\nThe book is most useful if you also install a freely available Scheme\nvariant, which has a symbolic classical mechanics library, so you can\ndo all the exercises. But it\'s still quite valuable without the\nsoftware. Disclaimer - Sussman is my advisor. But see the Amazon\nreviews.\n\nWhat did you not like about Goldstein?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Vonny N wrote:
> Finally, is there a book on classical mechanics which treats all three > of these approaches to mechanics, along with their > inter-relationships, with mathematical rigour? (I'm afraid I don't > much like Goldstein's widely cited book). > > Vonny N. There are many excellent classical mechanics texts. But the only one that I'm aware of that is completely clear on the exact meaning of the equations, and avoids the formal ambiguity inherent in most presentations of the Euler-Lagrange equations, is _Structure and Interpretation of Classical [itex]Mechanics_,[/itex] by Sussman and Wisdom. It is both quite clear and quite rigorous. It uses a functional notation, similar to Spivak (Calculus on Manifolds, Differential Geometry). The book is most useful if you also install a freely available Scheme variant, which has a symbolic classical mechanics library, so you can do all the exercises. But it's still quite valuable without the software. Disclaimer - Sussman is my advisor. But see the Amazon reviews. What did you not like about Goldstein? |
| Mar28-05, 06:49 PM | #5 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Vonny N wrote:\n> What is the relationship between:\n> 1. Newtonian Mechanics\n> 2. Lagrangian Mechanics\n> 3. Hamiltonian Mechanics\n\nDefining "Newtonian Mechanics" to mean a system with a 2nd order\nequation of motion q^i\'\'(t) = A^i(q(t),q\'(t)) for i=1,...,N\n(q=(q^1,...,q^N)), then the equivalence is:\n\nNewtonian + Helmholz Conditions = Lagrangian\nLagrangian + Non-zero mass matrix -> Hamiltonian\nHamiltonian + Non-zero dispersion matrix -> Lagrangian\n\nwith the mass matrix m_{ij} and dispersion matrix W^{ij} defined by:\nm_{ij} = d^2L/dv^idv^j; W^{ij} = d^2H/dp_idp_j.\n\n(m and W are inverses when L and H are related by the Legendre\ntransform).\n\nIn Quantum Theory, if starting only with a 2nd order equation of motion\nplus the equal time commutators\n[q^i(t),q^j(t)] = 0 for all time t\nthen self-consistency will very nearly force the system to be\nLagrangian and Hamiltonian with the dispersion matrix being\nW^ij = [q^i(t),q^j\'(t)]/(i h-bar).\nStrictly speaking, this is true in the classical limit as h-bar -> 0,\nif the limiting value of W is non-singular. It is, however, probably\nalso true *before* going to the classical limit.\n\nIn the more general situation, what you have is roughly the following.\nIf W is singular, the singular modes can be factored out and yield\nclassical coordinates. The remaining modes gives you the quantum\ncoordinates, which must then comprise a quantum system that has a\nHamiltonian and non-singular Lagrangian as its classical limit.\n\nIf, further, the commutators [q,v] are c-numbers, then this forces the\nHamiltonian to be quadratic in the momenta. If, more generally, the\n[q,v] only commute with the q\'s (so that [q,[q,v]]\'s are 0), then the\nsame general conclusion may hold there, as well, but with the\ncorresponding mass matrix being non-constant functions of q. The\nequations of motion will then take on the form of a combination of the\ngeodesic law (in configuration space) with the mass matrix proportional\nto the metric and a Lorentz force law with respect to some Yang-Mills\npotential (but in configuration space).\n\nIf further requiring that the system be a representation of the\nappropriate space-time symmetry group (Galilei or Poincare\') then it\nmay also be true that the configuration space factors into a number of\ncopies of single particle states, each satisfying a version of geodesic\n+ Lorentz -- but this time with the laws holding in *ordinary*\nspacetime. The "factorability conjecture" hasn\'t been proven, or even\nproperly formulated, as far as I know.\n\nThis is material, when fully developed, will form an essential core of\nthe treatise on Quantum Theory that I am currently working on. Stay\ntuned for further developments ...\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Vonny N wrote:
> What is the relationship between: > 1. Newtonian Mechanics > 2. Lagrangian Mechanics > 3. Hamiltonian Mechanics Defining "Newtonian Mechanics" to mean a system with a 2nd order equation of motion [itex]q^i''(t) = A^i(q(t),q'(t))[/itex] for [itex]i=1,[/itex]...,N [itex](q=(q^1,...,q^N)),[/itex] then the equivalence is: Newtonian + Helmholz Conditions = Lagrangian Lagrangian + Non-zero mass matrix -> Hamiltonian Hamiltonian + Non-zero dispersion matrix -> Lagrangian with the mass matrix [itex]m_{ij}[/itex] and dispersion matrix [itex]W^{ij}[/itex] defined by: [itex]m_{ij} = d^{2L}/dv^idv^j; W^{ij} = d^{2H}/dp_idp_j[/itex]. (m and W are inverses when L and H are related by the Legendre transform). In Quantum Theory, if starting only with a 2nd order equation of motion plus the equal time commutators [itex][q^i(t),q^j(t)] =[/itex] for all time t then self-consistency will very nearly force the system to be Lagrangian and Hamiltonian with the dispersion matrix being [itex]W^{ij} = [q^i(t),q^j'(t)]/(i[/itex] h-bar). Strictly speaking, this is true in the classical limit as h-bar [itex]-> 0,[/itex] if the limiting value of W is non-singular. It is, however, probably also true *before* going to the classical limit. In the more general situation, what you have is roughly the following. If W is singular, the singular modes can be factored out and yield classical coordinates. The remaining modes gives you the quantum coordinates, which must then comprise a quantum system that has a Hamiltonian and non-singular Lagrangian as its classical limit. If, further, the commutators [q,v] are c-numbers, then this forces the Hamiltonian to be quadratic in the momenta. If, more generally, the [q,v] only commute with the q's (so that [q,[q,v]]'s are 0), then the same general conclusion may hold there, as well, but with the corresponding mass matrix being non-constant functions of q. The equations of motion will then take on the form of a combination of the geodesic law (in configuration space) with the mass matrix proportional to the metric and a Lorentz force law with respect to some Yang-Mills potential (but in configuration space). If further requiring that the system be a representation of the appropriate space-time symmetry group (Galilei or Poincare') then it may also be true that the configuration space factors into a number of copies of single particle states, each satisfying a version of geodesic + Lorentz -- but this time with the laws holding in *ordinary* spacetime. The "factorability conjecture" hasn't been proven, or even properly formulated, as far as I know. This is material, when fully developed, will form an essential core of the treatise on Quantum Theory that I am currently working on. Stay tuned for further developments ... |
| Mar29-05, 02:20 AM | #6 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <1111898006.605616.297520@l41g2000cwc.googlegroups.com>,\nBob Hearn <bob.hearn@gmail.com> wrote:\n>\n> [...] _Structure and\n>Interpretation of Classical Mechanics_, by Sussman and Wisdom. It is\n>both quite clear and quite rigorous. It uses a functional notation,\n>similar to Spivak (Calculus on Manifolds, Differential Geometry).\n>\n>The book is most useful if you also install a freely available Scheme\n>variant, [...]\n\nI hadn\'t heard of that book before, but the title caught my eye because\nI\'ve read Abelson and Sussman\'s _Structure and Interpretation of Computer\nPrograms_. It\'s a true "classic" in computer science, taking a functional\napproach using Scheme. Based on that experience, I\'d expect SICM to be\nvery interesting. I\'ll have to try to carve out time to work through it\nthis summer.\n\n--\nJon Bell <jtbell@presby.edu> Presbyterian College\nDept. of Physics and Computer Science Clinton, South Carolina USA\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <1111898006.605616.297520@l41g2000cwc.googlegroups.com>,
Bob Hearn <bob.hearn@gmail.com> wrote: > > [...] _Structure and >Interpretation of Classical [itex]Mechanics_,[/itex] by Sussman and Wisdom. It is >both quite clear and quite rigorous. It uses a functional notation, >similar to Spivak (Calculus on Manifolds, Differential Geometry). > >The book is most useful if you also install a freely available Scheme >variant, [...] I hadn't heard of that book before, but the title caught my eye because I've read Abelson and Sussman's _Structure and Interpretation of Computer Programs_. It's a true "classic" in computer science, taking a functional approach using Scheme. Based on that experience, I'd expect SICM to be very interesting. I'll have to try to carve out time to work through it this summer. -- Jon Bell <jtbell@presby.edu> Presbyterian College Dept. of Physics and Computer Science Clinton, South Carolina USA |
| Mar29-05, 11:16 AM | #7 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Neil Price wrote:\n> vonnyn@hotmail.com (Vonny N) wrote in message news:<df799de5.0503251424.2f339940@posting.google.com>...\n>\n >>What is the relationship between:\n>>\n>>1. Newtonian Mechanics\n>>2. Lagrangian Mechanics\n>>3. Hamiltonian Mechanics\n\n1 is a special case of 3 which is a special case of 2.\n\n\n>>Can they be declared to be *rigorously* mathematically equivalent?\n\nNo.\n\n\n> On the other hand, Hamiltonian mechanics is more directly amenable to\n> techniques from symplectic geometry, e.g. symplectic reduction,\n> definitions involving integrability, etc.\n\nThere is also a symplectic view of Lagrangian mechanics, presented,\ne.g., in the book on Mechanics and Symmetry by Marsden and Ratiu.\n\n\n>>Finally, is there a book on classical mechanics which treats all three\n>>of these approaches to mechanics, along with their\n>>inter-relationships, with mathematical rigour?\n\nMarsden and Ratiu give an almost rigorous approach.\n\n\n> See the book "foundations of mechanics" by Abraham and Marsden or\n> "introduction to mechanics and symmetry" by Marsden and Ratiu.\n\nMarsden and Ratiu give an almost rigorous approach.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Neil Price wrote:
> vonnyn@hotmail.com (Vonny N) wrote in message news:<df799de5.0503251424.2f339940@p...google.com>... > >>What is the relationship between: >> >>1. Newtonian Mechanics >>2. Lagrangian Mechanics >>3. Hamiltonian Mechanics 1 is a special case of 3 which is a special case of 2. >>Can they be declared to be *rigorously* mathematically equivalent? No. > On the other hand, Hamiltonian mechanics is more directly amenable to > techniques from symplectic geometry, e.g. symplectic reduction, > definitions involving integrability, etc. There is also a symplectic view of Lagrangian mechanics, presented, e.g., in the book on Mechanics and Symmetry by Marsden and Ratiu. >>Finally, is there a book on classical mechanics which treats all three >>of these approaches to mechanics, along with their >>inter-relationships, with mathematical rigour? Marsden and Ratiu give an almost rigorous approach. > See the book "foundations of mechanics" by Abraham and Marsden or > "introduction to mechanics and symmetry" by Marsden and Ratiu. Marsden and Ratiu give an almost rigorous approach. Arnold Neumaier |
| Mar29-05, 06:41 PM | #8 |
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Recognitions:
 Science Advisor |
Check out, The Variational Principles of Mechanics, by C. Lanczos, a bible for us older types. In my view, it's as good as any physics book ever written, old (50 years) as it is. He makes tough stuff easy to understand. Well worth the trouble to find.
Regards, Reilly Atkinson |
| Mar30-05, 02:03 PM | #9 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"richard miller" <richard@microscitech.freeserve.co.uk> wrote in message\nnews:d24a63\\$n3u\\$1@news8.svr.pol.co.uk...\n> "Vonny N" <vonnyn@hotmail.com> wrote in message\n> news:df799de5.0503251424.2f339940@posting.google.com...\n> > What is the relationship between:\n> >\n> > 1. Newtonian Mechanics\n> > 2. Lagrangian Mechanics\n> > 3. Hamiltonian Mechanics\n> >\n> > Can they be declared to be *rigorously* mathematically equivalent?\n> >\n> > Does it suffice to say that they are reformulations of each other,\n> > each well-adapted to a particular class of problems?\n> >\n> > If so, can we describe reasonably clearly which type of problems are\n> > more easily represented and/or solved using each version of mechanics?\n> >\n> > Finally, is there a book on classical mechanics which treats all three\n> > of these approaches to mechanics, along with their\n> > inter-relationships, with mathematical rigour? (I\'m afraid I don\'t\n> > much like Goldstein\'s widely cited book).\n> >\n> > Vonny N.\n> >\n>\n> A very brief reply from me cos my girlfriend is due around in 5 mins, I\n> haven\'t cleaned the house like I promised, and I haven\'t cooked the dinner\n> like I promised!\n>\n> Please forgive the rather lax presentation as well\n>\n> If you take a Lagrangian function L = difference in Kinetic T and\nPotential\n> energy V\n> i.e. L = T - V. Apply a first order variation/Hamilton\'s primnciple and\nend\n> up with the standard Lagrange equations d/dt(dL/dv) - dL/dx = 0. (little v\n=\n> velocity dx/dt)\n>\n> With the standard, one dimensional T = 1/2mv^2 (v=dx/dt), V = 0 (freely\n> moving particle no field) and so L = 1/2.m.v^2 you will get mdv/dt = 0,\nwith\n> a little flippancy we have d(mv)/dt = 0, i.e. rate of change of linear\n> momemtum = 0 when no force acts.\n>\n> In short, if the Lagrangian is the difference of kinetic and potential\n> energy, you should get back to Newtonian mechanics. The real beauty is\nthat\n> the Lagrangian can be expanded to to other functionals L. Do this and you\n> can are in the realm of field theory.\n>\n> Hamilton takes Lagranges equations, which are second order in time, and\n> converts them to two first order equations. The beauty in these latter\n> equations is that they are almost symmetric...\n>\n> crikes my girlfriend has just turned up\n>\n> I\'m done for, back later\n>\n> cheers\n>\n> Richard Miller\n>\n\nThree or so days later, I\'ve looked at the other replies, don\'t have much\ntoo add, except:\n\nThe Hamiltonian formulation gives you these gorgeous symmetric equations in\nthe generalised coordinates (q\'s), conjugate momenta (p\'s) i.e. your q\'s and\np\'s.\n\nwith H defined by p dq/dt - L, then for the p and q\'s you have\n\ndH/dp = dq/dt, dH/dq = -dp/dt\n\nBefore you even mention the words \'quantum mechanics\', you have conjugate\ncoordinates \'mometum and position\'. Uncertainty anyone?\n\nThe beauty is that if you just looked at Hamilton\'s equations, the p and q\nequations look rather symmetric. You might lose track of what is position\nand what is momenta. Then there\'s energy and time too.\n\nThat aside, Hamilton\'s formulation leads to the concept of phase space. It\nis quite neat in that usually we enquire afer the time evolution of a\nsystem, but something like a velocity versus position (distance) plot can be\njust as, if not more, interesting. For example, a pendulum swings (no big\ndeal); but the time evolution (angular position versus time) is of little\ninterest; it is the frequencies and amplitudes of the system that are of far\nmore interest. Via more \'Canonical transformations\' one gets to action angle\nvariables, frequencies and even stability of the system. This is where it\nall gets very interesting from the modern dynamics viewpoint. Stability of\nthe solar System, KAM Theorem (Kolmogrov-Arnold-Moser theorem, Mathworld)..\netc. This is Hamiltonian dynamics at its best - and lots of modern\n\'symplectic\' stuff to go.\n\nOK, I know you wanted a better answer than the above woffle. Here is a book\nlink (I hope it is not too elementary for you)\n\nA very good book from a modern perspective is Neil S. Rasband, \'Dynamics\',\nISBN 0471873985\n\nThen simply Google \'kolmogrov-Arnold-Moser\' etc. Stability of the Solar\nSystem.\n\nAnd finally, after much woffle,\n\nNewtonian = Lagrangian (and Hamilton\'s Principle) with a special Lagrangian\nT - V\n\nNewtonian = Hamiltonian with Hamiltonian = T + V\n\nLagragian = Hamilton, but without the mathematical beauty or theoretical\nadvancement or laughs\n\nI know this is simplistic, there are dissipative forces etc, but just for\nsimplicity...\n\nEnjoy\n\nRichard Miller\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"richard miller" <richard@microscitech.freeserve.co.uk> wrote in message
news:d24a63$n3u$1@news8.svr.pol.co.uk... > "Vonny N" <vonnyn@hotmail.com> wrote in message > news:df799de5.0503251424.2f339940@posting.google.com... > > What is the relationship between: > > > > 1. Newtonian Mechanics > > 2. Lagrangian Mechanics > > 3. Hamiltonian Mechanics > > > > Can they be declared to be *rigorously* mathematically equivalent? > > > > Does it suffice to say that they are reformulations of each other, > > each well-adapted to a particular class of problems? > > > > If so, can we describe reasonably clearly which type of problems are > > more easily represented [itex]and/or[/itex] solved using each version of mechanics? > > > > Finally, is there a book on classical mechanics which treats all three > > of these approaches to mechanics, along with their > > inter-relationships, with mathematical rigour? (I'm afraid I don't > > much like Goldstein's widely cited book). > > > > Vonny N. > > > > A very brief reply from [itex]me cos my[/itex] girlfriend is due around in 5 mins, I > haven't cleaned the house like I promised, and I haven't cooked the dinner > like I promised! > > Please forgive the rather lax presentation as well > > If you take a Lagrangian function L = difference in Kinetic T and Potential > energy V > i.e. [itex]L = T - V[/itex]. Apply a first order variation/Hamilton's primnciple and end > up with the standard Lagrange equations [itex]d/dt(dL/dv) - dL/dx =[/itex] . (little v = > velocity [itex]dx/dt)[/itex] > > With the standard, one dimensional [itex]T = 1/2mv^2 (v=dx/dt), V =[/itex] (freely > moving particle no field) and [itex]so L = 1/2[/itex].m.[itex]v^2[/itex] you will get [itex]mdv/dt = 0,[/itex] with > a little flippancy we have [itex]d(mv)/dt = 0, i[/itex].e. rate of change of linear > momemtum = when no force acts. > > In short, if the Lagrangian is the difference of kinetic and potential > energy, you should get back to Newtonian mechanics. The real beauty is that > the Lagrangian can be expanded to to other functionals L. Do this and you > can are in the realm of field theory. > > Hamilton takes Lagranges equations, which are second order in time, and > converts them to two first order equations. The beauty in these latter > equations is that they are almost symmetric... > > crikes my girlfriend has just turned up > > I'm done for, back later > > cheers > > Richard Miller > Three or so days later, I've looked at the other replies, don't have much too add, except: The Hamiltonian formulation gives you these gorgeous symmetric equations in the generalised coordinates (q's), conjugate momenta (p's) i.e. your q's and p's. with H defined by [itex]p dq/dt - L,[/itex] then for the p and q's you have [tex]dH/dp = dq/dt, dH/dq = -dp/dt[/tex] Before you even mention the words 'quantum mechanics', you have conjugate coordinates 'mometum and position'. Uncertainty anyone? The beauty is that if you just looked at Hamilton's equations, the p and q equations look rather symmetric. You might lose track of what is position and what is momenta. Then there's energy and time too. That aside, Hamilton's formulation leads to the concept of phase space. It is quite neat in that usually we enquire afer the time evolution of a system, but something like a velocity versus position (distance) plot can be just as, if not more, interesting. For example, a pendulum swings (no big deal); but the time evolution (angular position versus time) is of little interest; it is the frequencies and amplitudes of the system that are of far more interest. Via more 'Canonical transformations' one gets to action angle variables, frequencies and even stability of the system. This is where it all gets very interesting from the modern dynamics viewpoint. Stability of the solar System, KAM Theorem (Kolmogrov-Arnold-Moser theorem, Mathworld).. etc. This is Hamiltonian dynamics at its best - and lots of modern 'symplectic' stuff to go. OK, I know you wanted a better answer than the above woffle. Here is a book link (I hope it is not too elementary for you) A very good book from a modern perspective is Neil S. Rasband, 'Dynamics', ISBN 0471873985 Then simply Google 'kolmogrov-Arnold-Moser' etc. Stability of the Solar System. And finally, after much woffle, Newtonian = Lagrangian (and Hamilton's Principle) with a special Lagrangian [itex]T - V[/itex] Newtonian = Hamiltonian with Hamiltonian [itex]= T + V[/itex] Lagragian = Hamilton, but without the mathematical beauty or theoretical advancement or laughs I know this is simplistic, there are dissipative forces etc, but just for simplicity... Enjoy Richard Miller |
| Mar30-05, 10:46 PM | #10 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Thanks very much for bringing this beautiful work to my attention.\n\nhttp://mitpress.mit.edu/SICM/book.html\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Thanks very much for bringing this beautiful work to my attention.
http://mitpress.mit.edu/SICM/book.html |
| Mar30-05, 10:46 PM | #11 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Neil Price wrote:\n> The prevailing mathematical point of view is that Lagrangian\nmechanics\n> takes place on the tangent bundle of the configuration space, whereas\n> the Hamiltonian formalism is dealt with on the cotangent bundle.\n\nThe velocity v = dq/dt resides on the tangent bundle TQ of the manifold\nQ where q lives. But the state (q,v) lives on the first jet J^1(Q),\nwhich is the manifold whose local coordinates are just (q,v)\nthemselves, with v in T_q(Q).\n\n> As to which formulation to use, I think it mostly depends (in the\n> regular case) on the formulation of the problem. Sometimes one is\n> preferred over the other: in the Lagrangian picture one has a\n> "variational principle"; i.e. the equations of motion are derived by\n> extremizing a certain functional.\n\nThe Lagrangian is the one that\'s fundamental. An action is formulated\nover a spacetime region, and (after applying the equations of motion)\nthe total variation will reduce to one on the spacetime boundary\ncomprising the integral (p.delta(q)) over the boundary (dR) of the\nregion R. For ordinary mechanics, with regions being intervals on the\ntime line, dR is the chain (T+) - (T-) comprising 2 points and the\ntotal variation is just the value\np(T+).delta(q(T+)) - p(T-).delta(q(T-)).\n\nIn a Lorentzian space (as opposed to a Newtonian space), one can also\ntake the spacetime region R so that (a) it\'s compact and (b) its\nboundary is EVERYWHERE spacelike, and of the form dR = (R+) - (R-),\nwith (R+) and (R-) connected by a homotopy, such that the common\nboundary d(R+) = d(R-) (= A, the "anchor") remains fixed. The homotopy\nparameter then plays the role of the time coordinate, the anchor A\nplaying the analogous role of asymptotic infinity. This way you can\navoid running all the integrals off to infinity and get consistent\ntheories, even in a quantum setting (noting that the essential block\nprovided by Haag\'s no-interaction theorem is directly connected with\nthe issue of using infinite volume integrals and global unitary\nevolution operators that encompass all of space).\n\nThese issues, too, may find their way into the treatise mentioned\nearlier that I\'m working on.\n\nFor a Hamiltonian, you need an additional prerequisite: a time-like\nvector flow, and the ability to do Lie derivatives on the fields in\nquestion. I think Lecture Notes in Physics 107 is the edition that\ncovers the general issues surrounding all the variational pictures.\nFor field theories, Legendre transforms are not unique, since you can\nchoose any of the infinity of time-like vector fields to do the\nvariation on.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Neil Price wrote:
> The prevailing mathematical point of view is that Lagrangian mechanics > takes place on the tangent bundle of the configuration space, whereas > the Hamiltonian formalism is dealt with on the cotangent bundle. The velocity [itex]v = dq/dt[/itex] resides on the tangent bundle TQ of the manifold Q where q lives. But the state (q,v) lives on the first jet [itex]J^1(Q),[/itex] which is the manifold whose local coordinates are just (q,v) themselves, with v in [itex]T_q(Q)[/itex]. > As to which formulation to use, I think it mostly depends (in the > regular case) on the formulation of the problem. Sometimes one is > preferred over the other: in the Lagrangian picture one has a > "variational principle"; i.e. the equations of motion are derived by > extremizing a certain functional. The Lagrangian is the one that's fundamental. An action is formulated over a spacetime region, and (after applying the equations of motion) the total variation will reduce to one on the spacetime boundary comprising the integral (p.[itex]\delta(q))[/itex] over the boundary (dR) of the region R. For ordinary mechanics, with regions being intervals on the time line, dR is the chain [itex](T+) - (T-)[/itex] comprising 2 points and the total variation is just the value [itex]p(T+)[/itex].[itex]\delta(q(T+)) - p(T-)[/itex].[itex]\delta(q(T-))[/itex]. In a Lorentzian space (as opposed to a Newtonian space), one can also take the spacetime region R so that (a) it's compact and (b) its boundary is EVERYWHERE spacelike, and of the form [itex]dR = (R+) - (R-),[/itex] with [itex](R+)[/itex] and [itex](R-)[/itex] connected by a homotopy, such that the common boundary [itex]d(R+) = d(R-) (= A,[/itex] the "anchor") remains fixed. The homotopy parameter then plays the role of the time coordinate, the anchor A playing the analogous role of asymptotic infinity. This way you can avoid running all the integrals off to infinity and get consistent theories, even in a quantum setting (noting that the essential block provided by Haag's no-interaction theorem is directly connected with the issue of using infinite volume integrals and global unitary evolution operators that encompass all of space). These issues, too, may find their way into the treatise mentioned earlier that I'm working on. For a Hamiltonian, you need an additional prerequisite: a time-like vector flow, and the ability to do Lie derivatives on the fields in question. I think Lecture Notes in Physics 107 is the edition that covers the general issues surrounding all the variational pictures. For field theories, Legendre transforms are not unique, since you can choose any of the infinity of time-like vector fields to do the variation on. |
| Mar31-05, 07:13 PM | #12 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>whopkins@csd.uwm.edu wrote in message news:<1112229351.979482.196260@g14g2000cwa.googlegroups.com>...\n > Neil Price wrote:\n> > The prevailing mathematical point of view is that Lagrangian\n> mechanics\n> > takes place on the tangent bundle of the configuration space, whereas\n> > the Hamiltonian formalism is dealt with on the cotangent bundle.\n>\n> The velocity v = dq/dt resides on the tangent bundle TQ of the manifold\n> Q where q lives. But the state (q,v) lives on the first jet J^1(Q),\n> which is the manifold whose local coordinates are just (q,v)\n> themselves, with v in T_q(Q).\n\nSome caution is in order here. The coordinates on the jet bundle are\n(t, q, v); the first jet bundle is isomorphic to TM x R (but not by a\nnatural map). Furthermore, for autonomous problems the tangent\nbundle will do just fine.\n\n> > As to which formulation to use, I think it mostly depends (in the\n> > regular case) on the formulation of the problem. Sometimes one is\n> > preferred over the other: in the Lagrangian picture one has a\n> > "variational principle"; i.e. the equations of motion are derived by\n> > extremizing a certain functional.\n>\n> The Lagrangian is the one that\'s fundamental.\n(...)\n> For a Hamiltonian, you need an additional prerequisite: a time-like\n> vector flow, and the ability to do Lie derivatives on the fields in\n> question. I think Lecture Notes in Physics 107 is the edition that\n> covers the general issues surrounding all the variational pictures.\n> For field theories, Legendre transforms are not unique, since you can\n> choose any of the infinity of time-like vector fields to do the\n> variation on.\n\nCould you give more information about this LNP 107?\n\nThere are ways of formulating what you could call a "covariant\nHamiltonian formalism". In the case of time-dependent mechanics for\nexample, the Hamiltonian becomes a map from J^1(E)^* to T*E. This\nconcept can be generalized to field theories as well. It becomes\nequivalent to the normal Hamiltonian formalism when choosing a\ntimelike vectorfield and "integrating" this covariant Hamiltonian over\na space-like hypersurface. There also exists a unique Legendre\ntransform, which can be shown to yield the normal Legendre map after\nbreaking covariance.\n\nThere are two unpublished articles by Gotay, Marsden and Isenberg\nfloating around where they treat this formalism in greater detail.\n\nGranted, even with this formalism the Hamiltonian side loses much of\nits appeal, but the original poster talked about mechanics; in this\ncase there really is something to gain from the Hamiltonian formalism.\nThink Hamilton-Jacobi, the Liouville-Arnold integrability theorem,\nthat sort of thing.\n\n\nN.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>whopkins@csd.uwm.edu wrote in message news:<1112229351.979482.196260@g14g2...groups.com>...
> Neil Price wrote: > > The prevailing mathematical point of view is that Lagrangian > mechanics > > takes place on the tangent bundle of the configuration space, whereas > > the Hamiltonian formalism is dealt with on the cotangent bundle. > > The velocity [itex]v = dq/dt[/itex] resides on the tangent bundle TQ of the manifold > Q where q lives. But the state (q,v) lives on the first jet [itex]J^1(Q),[/itex] > which is the manifold whose local coordinates are just (q,v) > themselves, with v in [itex]T_q(Q)[/itex]. Some caution is in order here. The coordinates on the jet bundle are (t, q, v); the first jet bundle is isomorphic to TM x R (but not by a natural map). Furthermore, for autonomous problems the tangent bundle will do just fine. > > As to which formulation to use, I think it mostly depends (in the > > regular case) on the formulation of the problem. Sometimes one is > > preferred over the other: in the Lagrangian picture one has a > > "variational principle"; i.e. the equations of motion are derived by > > extremizing a certain functional. > > The Lagrangian is the one that's fundamental. (...) > For a Hamiltonian, you need an additional prerequisite: a time-like > vector flow, and the ability to do Lie derivatives on the fields in > question. I think Lecture Notes in Physics 107 is the edition that > covers the general issues surrounding all the variational pictures. > For field theories, Legendre transforms are not unique, since you can > choose any of the infinity of time-like vector fields to do the > variation on. Could you give more information about this LNP 107? There are ways of formulating what you could call a "covariant Hamiltonian formalism". In the case of time-dependent mechanics for example, the Hamiltonian becomes a map from [itex]J^1(E)^*[/itex] to T*E. This concept can be generalized to field theories as well. It becomes equivalent to the normal Hamiltonian formalism when choosing a timelike vectorfield and "integrating" this covariant Hamiltonian over a space-like hypersurface. There also exists a unique Legendre transform, which can be shown to yield the normal Legendre map after breaking covariance. There are two unpublished articles by Gotay, Marsden and Isenberg floating around where they treat this formalism in greater detail. Granted, even with this formalism the Hamiltonian side loses much of its appeal, but the original poster talked about mechanics; in this case there really is something to gain from the Hamiltonian formalism. Think Hamilton-Jacobi, the Liouville-Arnold integrability theorem, that sort of thing. N. |
| Apr4-05, 04:35 AM | #13 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In V. Arnol\'d \'s highly respected Classical Mechanics text (in the\nwonderful Springer-Verlag Graduate Outline series) he mandates that\nNewtonian Mechanics is properly contained in Lagrangian Mechanics which\nin turn is properly contained in Hamiltonian Mechanics; the latter\ntherefore being the most general. Do you then disagree with this\nstructural assertion?\n\nVonny N.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In V. Arnol'd 's highly respected Classical Mechanics text (in the
wonderful Springer-Verlag Graduate Outline series) he mandates that Newtonian Mechanics is properly contained in Lagrangian Mechanics which in turn is properly contained in Hamiltonian Mechanics; the latter therefore being the most general. Do you then disagree with this structural assertion? Vonny N. |
| Apr4-05, 08:45 AM | #14 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>vonnyn@hotmail.com wrote:\n\n> In V. Arnol\'d \'s highly respected Classical Mechanics text (in the\n> wonderful Springer-Verlag Graduate Outline series) he mandates that\n> Newtonian Mechanics is properly contained in Lagrangian Mechanics which\n> in turn is properly contained in Hamiltonian Mechanics; the latter\n> therefore being the most general.\n\nCan you quote the context and outline his argument?\nI can\'t believe that he wrote that.\n\n\n> Do you then disagree with this\n> structural assertion?\n\nYes. One can convert any Hamiltonian system into a Lagrangian system\nin extended phase space. But one can convert a Lagrangian system into\na Hamiltonian one only if d^2L/dq^2 is nonsingular.\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>vonnyn@hotmail.com wrote:
> In V. Arnol'd 's highly respected Classical Mechanics text (in the > wonderful Springer-Verlag Graduate Outline series) he mandates that > Newtonian Mechanics is properly contained in Lagrangian Mechanics which > in turn is properly contained in Hamiltonian Mechanics; the latter > therefore being the most general. Can you quote the context and outline his argument? I can't believe that he wrote that. > Do you then disagree with this > structural assertion? Yes. One can convert any Hamiltonian system into a Lagrangian system in extended phase space. But one can convert a Lagrangian system into a Hamiltonian one only if [itex]d^{2L}/dq^2[/itex] is nonsingular. Arnold Neumaier |
| Apr6-05, 03:23 AM | #15 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>> Yes. One can convert any Hamiltonian system into a Lagrangian system\n> in extended phase space. But one can convert a Lagrangian system into\n> a Hamiltonian one only if d^2L/dq^2 is nonsingular.\n>\n\nThat is only correct if you ignore the possibility to work with\nconstraints. Without constraints you can\'t even handle simple\nrelativistic systems though, so this is usually assumed to be part of\nthe Hamiltonian method. See the appendix of gr-qc/0110034 or Diracs\nLecture on QM (Here you\'ll find the original ideas that allowed GR to\nbe cast into Hamiltonian form).\n\nf\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>> Yes. One can convert any Hamiltonian system into a Lagrangian system
> in extended phase space. But one can convert a Lagrangian system into > a Hamiltonian one only if [itex]d^{2L}/dq^2[/itex] is nonsingular. > That is only correct if you ignore the possibility to work with constraints. Without constraints you can't even handle simple relativistic systems though, so this is usually assumed to be part of the Hamiltonian method. See the appendix of http://www.arxiv.org/abs/gr-qc/0110034 or Diracs Lecture on QM (Here you'll find the original ideas that allowed GR to be cast into Hamiltonian form). f |
| Apr6-05, 09:18 AM | #16 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Frank Hellmann wrote:\n>>Yes. One can convert any Hamiltonian system into a Lagrangian system\n>>in extended phase space. But one can convert a Lagrangian system into\n>>a Hamiltonian one only if d^2L/dq^2 is nonsingular.\n>\n> That is only correct if you ignore the possibility to work with\n> constraints. Without constraints you can\'t even handle simple\n> relativistic systems though, so this is usually assumed to be part of\n> the Hamiltonian method. See the appendix of gr-qc/0110034 or Diracs\n> Lecture on QM (Here you\'ll find the original ideas that allowed GR to\n> be cast into Hamiltonian form).\n\nTrue. But one would usually refer to this as a \'constrained Hamiltonian\nsystem\' and not just as a \'Hamiltonian system\'.\n\nConstrained Hamiltonian systems and Lagrnaigian systems are almost\nequivalent. One still needs a constant rank assumtpion to go from\na Lagraingian to a Hamiltonian.\n\n\nArnold Neumaier\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Frank Hellmann wrote:
>>Yes. One can convert any Hamiltonian system into a Lagrangian system >>in extended phase space. But one can convert a Lagrangian system into >>a Hamiltonian one only if [itex]d^{2L}/dq^2[/itex] is nonsingular. > > That is only correct if you ignore the possibility to work with > constraints. Without constraints you can't even handle simple > relativistic systems though, so this is usually assumed to be part of > the Hamiltonian method. See the appendix of http://www.arxiv.org/abs/gr-qc/0110034 or Diracs > Lecture on QM (Here you'll find the original ideas that allowed GR to > be cast into Hamiltonian form). True. But one would usually refer to this as a 'constrained Hamiltonian system' and not just as a 'Hamiltonian system'. Constrained Hamiltonian systems and Lagrnaigian systems are almost equivalent. One still needs a constant rank assumtpion to go from a Lagraingian to a Hamiltonian. Arnold Neumaier |
| Apr6-05, 09:52 AM | #17 |
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Daniel. |
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