4 forms of mechanics












2












$begingroup$


There are 4 main functions in mechanics. L(q,q',t), H(p,q,t), K(p',q',t), G(p,p',t). First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to first two, but we don't use them. Every where they write that using them can cause problems. But why?






classical-mechanics lagrangian-formalism hamiltonian-formalism







share|cite|improve this question







New contributor




Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 1




    $begingroup$
    You may find useful to have a look at this related question and answers: physics.stackexchange.com/q/415775
    $endgroup$
    – GiorgioP
    1 hour ago












  • $begingroup$
    Hm, my question is different, I want to know, why is L and H and more preferable for us, while solving tasks
    $endgroup$
    – Semen Yurchenko
    1 hour ago










  • $begingroup$
    See point 3 in Qmechanic's answer.
    $endgroup$
    – GiorgioP
    56 mins ago
















2












$begingroup$


There are 4 main functions in mechanics. L(q,q',t), H(p,q,t), K(p',q',t), G(p,p',t). First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to first two, but we don't use them. Every where they write that using them can cause problems. But why?






classical-mechanics lagrangian-formalism hamiltonian-formalism







share|cite|improve this question







New contributor




Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 1




    $begingroup$
    You may find useful to have a look at this related question and answers: physics.stackexchange.com/q/415775
    $endgroup$
    – GiorgioP
    1 hour ago












  • $begingroup$
    Hm, my question is different, I want to know, why is L and H and more preferable for us, while solving tasks
    $endgroup$
    – Semen Yurchenko
    1 hour ago










  • $begingroup$
    See point 3 in Qmechanic's answer.
    $endgroup$
    – GiorgioP
    56 mins ago














2












2








2





$begingroup$


There are 4 main functions in mechanics. L(q,q',t), H(p,q,t), K(p',q',t), G(p,p',t). First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to first two, but we don't use them. Every where they write that using them can cause problems. But why?






classical-mechanics lagrangian-formalism hamiltonian-formalism







share|cite|improve this question







New contributor




Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




There are 4 main functions in mechanics. L(q,q',t), H(p,q,t), K(p',q',t), G(p,p',t). First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to first two, but we don't use them. Every where they write that using them can cause problems. But why?






classical-mechanics lagrangian-formalism hamiltonian-formalism




classical-mechanics lagrangian-formalism hamiltonian-formalism






share|cite|improve this question







New contributor




Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







New contributor




Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question






New contributor




Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 2 hours ago









Semen YurchenkoSemen Yurchenko

183




183




New contributor




Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Semen Yurchenko is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    $begingroup$
    You may find useful to have a look at this related question and answers: physics.stackexchange.com/q/415775
    $endgroup$
    – GiorgioP
    1 hour ago












  • $begingroup$
    Hm, my question is different, I want to know, why is L and H and more preferable for us, while solving tasks
    $endgroup$
    – Semen Yurchenko
    1 hour ago










  • $begingroup$
    See point 3 in Qmechanic's answer.
    $endgroup$
    – GiorgioP
    56 mins ago














  • 1




    $begingroup$
    You may find useful to have a look at this related question and answers: physics.stackexchange.com/q/415775
    $endgroup$
    – GiorgioP
    1 hour ago












  • $begingroup$
    Hm, my question is different, I want to know, why is L and H and more preferable for us, while solving tasks
    $endgroup$
    – Semen Yurchenko
    1 hour ago










  • $begingroup$
    See point 3 in Qmechanic's answer.
    $endgroup$
    – GiorgioP
    56 mins ago








1




1




$begingroup$
You may find useful to have a look at this related question and answers: physics.stackexchange.com/q/415775
$endgroup$
– GiorgioP
1 hour ago






$begingroup$
You may find useful to have a look at this related question and answers: physics.stackexchange.com/q/415775
$endgroup$
– GiorgioP
1 hour ago














$begingroup$
Hm, my question is different, I want to know, why is L and H and more preferable for us, while solving tasks
$endgroup$
– Semen Yurchenko
1 hour ago




$begingroup$
Hm, my question is different, I want to know, why is L and H and more preferable for us, while solving tasks
$endgroup$
– Semen Yurchenko
1 hour ago












$begingroup$
See point 3 in Qmechanic's answer.
$endgroup$
– GiorgioP
56 mins ago




$begingroup$
See point 3 in Qmechanic's answer.
$endgroup$
– GiorgioP
56 mins ago










1 Answer
1






active

oldest

votes


















3












$begingroup$

Here is one argument:




  1. Starting from Newton's 2nd law, the Lagrangian $L(q,v,t)$ is just one step away.


  2. A Legendre transformation $vleftrightarrow p$ to the Hamiltonian $H(q,p,t)$ is well-defined for a wide class of systems because there is typically a bijective relation between velocity $v$ and momentum $p$.


  3. On the other hand, there is seldomly a bijective relation between position $q$ and force $f$ (although Hooke's law is a notable exception). Therefore the Legendre transforms $K(f,v,t)$ and $G(f,p,t)$ are often ill-defined.







share|cite|improve this answer









$endgroup$













  • $begingroup$
    Sounds reasonable, thank you very much!
    $endgroup$
    – Semen Yurchenko
    58 mins ago











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "151"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});






Semen Yurchenko is a new contributor. Be nice, and check out our Code of Conduct.










draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f455135%2f4-forms-of-mechanics%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

Here is one argument:




  1. Starting from Newton's 2nd law, the Lagrangian $L(q,v,t)$ is just one step away.


  2. A Legendre transformation $vleftrightarrow p$ to the Hamiltonian $H(q,p,t)$ is well-defined for a wide class of systems because there is typically a bijective relation between velocity $v$ and momentum $p$.


  3. On the other hand, there is seldomly a bijective relation between position $q$ and force $f$ (although Hooke's law is a notable exception). Therefore the Legendre transforms $K(f,v,t)$ and $G(f,p,t)$ are often ill-defined.







share|cite|improve this answer









$endgroup$













  • $begingroup$
    Sounds reasonable, thank you very much!
    $endgroup$
    – Semen Yurchenko
    58 mins ago
















3












$begingroup$

Here is one argument:




  1. Starting from Newton's 2nd law, the Lagrangian $L(q,v,t)$ is just one step away.


  2. A Legendre transformation $vleftrightarrow p$ to the Hamiltonian $H(q,p,t)$ is well-defined for a wide class of systems because there is typically a bijective relation between velocity $v$ and momentum $p$.


  3. On the other hand, there is seldomly a bijective relation between position $q$ and force $f$ (although Hooke's law is a notable exception). Therefore the Legendre transforms $K(f,v,t)$ and $G(f,p,t)$ are often ill-defined.







share|cite|improve this answer









$endgroup$













  • $begingroup$
    Sounds reasonable, thank you very much!
    $endgroup$
    – Semen Yurchenko
    58 mins ago














3












3








3





$begingroup$

Here is one argument:




  1. Starting from Newton's 2nd law, the Lagrangian $L(q,v,t)$ is just one step away.


  2. A Legendre transformation $vleftrightarrow p$ to the Hamiltonian $H(q,p,t)$ is well-defined for a wide class of systems because there is typically a bijective relation between velocity $v$ and momentum $p$.


  3. On the other hand, there is seldomly a bijective relation between position $q$ and force $f$ (although Hooke's law is a notable exception). Therefore the Legendre transforms $K(f,v,t)$ and $G(f,p,t)$ are often ill-defined.







share|cite|improve this answer









$endgroup$



Here is one argument:




  1. Starting from Newton's 2nd law, the Lagrangian $L(q,v,t)$ is just one step away.


  2. A Legendre transformation $vleftrightarrow p$ to the Hamiltonian $H(q,p,t)$ is well-defined for a wide class of systems because there is typically a bijective relation between velocity $v$ and momentum $p$.


  3. On the other hand, there is seldomly a bijective relation between position $q$ and force $f$ (although Hooke's law is a notable exception). Therefore the Legendre transforms $K(f,v,t)$ and $G(f,p,t)$ are often ill-defined.








share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 1 hour ago









Qmechanic♦Qmechanic

103k121841172




103k121841172












  • $begingroup$
    Sounds reasonable, thank you very much!
    $endgroup$
    – Semen Yurchenko
    58 mins ago


















  • $begingroup$
    Sounds reasonable, thank you very much!
    $endgroup$
    – Semen Yurchenko
    58 mins ago
















$begingroup$
Sounds reasonable, thank you very much!
$endgroup$
– Semen Yurchenko
58 mins ago




$begingroup$
Sounds reasonable, thank you very much!
$endgroup$
– Semen Yurchenko
58 mins ago










Semen Yurchenko is a new contributor. Be nice, and check out our Code of Conduct.










draft saved

draft discarded


















Semen Yurchenko is a new contributor. Be nice, and check out our Code of Conduct.













Semen Yurchenko is a new contributor. Be nice, and check out our Code of Conduct.












Semen Yurchenko is a new contributor. Be nice, and check out our Code of Conduct.
















Thanks for contributing an answer to Physics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f455135%2f4-forms-of-mechanics%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

What other Star Trek series did the main TNG cast show up in?

Image
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 4 I'm referring to Picard, Riker, Data, Worf, Geordi, Dr. Crusher, Wesley Crusher, Chief O'Brien and family, Deanna Troi, Guinan. I know Picard and Worf show up on DS9, though I haven't seen it myself. I'm curious what happened to the characters after that last episode of TNG? star-trek star-trek-tng share | improve this question edited 10 hours ago Ham Sandwich 6,051 2 25 75 ...
Read more

Berlina muro

Image
La berlina muro ĉe Bethaniendamm, 1986 La berlina muro aŭ simple la muro [1] , nomita ankaŭ "kontraŭfaŝisma remparo" en propaganda lingvo de la iama Germana Demokratia Respubliko, estis parto de la interngermana limo, kiu dividis de la 13-a de aŭgusto 1961 ĝis la 9-a de novembro 1989 okcidentan Berlinon de la orienta parto de la urbo kaj de la ĉirkaŭa tereno de la GDR. Ĝi estis unu el la plej famaj simboloj de la malvarma milito kaj de la disdivido de Germanio. Pli ol simpla muro, la berlina muro estis fakte tuta defendokonstruaĵo kun du muroj, ena malplena strio, rondirvojo, gardturoj kaj alarmsistemoj. Dum la provo transiri la severe garditan limon al okcidenta Berlino multaj homoj estis mortigitaj. La preciza nombro de viktimoj estas pridisputita kaj ne certa; la nombroj anoncataj varias inter 86 kaj 238 mortintoj. La malplifortiĝo de Sovetunio kaj la politiko de liberigo gvidita de Miĥail Gorbaĉov ebligis al la orientgermanoj faligi la 9-an de novembro 1989 l...
Read more

Berlina aerponto

Image
Flugkoridoroj dum la blokado de Berlino La Berlina aerponto estas la provizigo de okcidenta Berlino per aviadiloj fare de la aliancanoj inter la 23-a de junio 1948 kaj la 12-a de majo 1949 dum la blokado de Berlino fare de la sovetia armeo. Enhavo 1 Historio 1.1 La aerkoridoroj 1.2 Unuaj flugoj 1.3 Plibonigo de la aerponto 1.4 Fino de la aerponto 1.5 Rozinbombiloj 1.6 Bilanco 2 Flughavenoj 2.1 Berlino 2.2 Okcidenta Germanio 3 Aviadiloj de la berlina aerponto 4 15-a datreveno de la aerponto 5 Bibliografio 6 Referencoj 7 Eksteraj ligiloj Historio | General Lucius D. Clay, militguberniestro de la usona okupadzono Jam antaŭ la berlina aerponto estiĝis la malgranda aerponto . La estro de la sovetia militadministracio en Germanio, Vasili Daniloviĉ Sokolovski, ordonis la 1-an de aprilo 1948, kiel reago al la konferenco de Londono inter britoj, francoj kaj reprezentantoj de Benelukso pri aligo de Germa...
Read more