About Classical Mechanics
Not a very good book, an exceptional book, really!
It goes quite deep but you are helped all along with the margin notes: looks like the author reads our mind and he answers in these notes.
Also repeating an equation from previous pages is really convenient.
I am ordering right away the Quantum Mechanics book from the same author
Compared to Susskind Theoretical Minimum, i think this books goes into more details and is easier to follow.
Guest
Griff
Hi Jakob, Thanks for writing these great books. They really help with getting the big picture of these subjects. I found a typo in the CM book, on page 157 in the derivation, in the step labelled ‘expanding the products,’ the term $ \frac{\partial \bar F}{\partial Q}\frac{\partial Q}{\partial p}\frac{\partial \bar H}{\partial Q}\frac{\partial Q}{\partial q}$ appears twice, but one of them should be $ \frac{\partial \bar F}{\partial Q}\frac{\partial Q}{\partial p}\frac{\partial \bar H}{\partial P}\frac{\partial P}{\partial q}$. The incorrect one is then retained in the cancellation in the next step. Do you have somewhere online where you post errata? That could help so… Read more »
Guest
Griff
Oh, I thought the LaTeX would compile automagically :/
Guest
Griff
On page 161, equation 7.88, you also have a $\frac{\partial}{\dot q}$ where it should be $\frac{\partial}{\partial \dot q}$.
thanks a lot for reporting these typos. I will correct them as soon as possible. Currently, I have no errata list online but it’s a great idea and will try to put one online as soon as I find the time.
Best,
Jakob
Guest
Griff
Hi Jakob,
On p.97, in the derivation of Eq.5.10, on the last line, you have $L(q,p)$ within the partial on the RHS. Shouldn’t this be $\tilde{L} (q,p)$? The same thing happens in the derivation of 5.11 and in your definition of the Hamiltonian 5.12, and in 5.13.
Yes you’re right! I will correct these typos in the next edition.
Guest
Phil
Hello there. On p 84 the boxed comment states “All terms first order in epsilon must vanish.” I’m not really sure what the author means by this. I can see that if x = a is the lowest point, then for any epsilon (positive or negative), f(a) < f(a + epsilon). However, I don't really understand what the author means by "vanish". I suspect it's something quite straightforward that I'm not seeing, but if someone could flesh this out for me a bit, I would appreciate it.
Hi Phil, the idea is that if we consider a minimum $x=a$, it necessarily goes up if we move away from there. Therefore, if we inspect the neighborhood of the minimum, $x= a + \epsilon$, we know that all terms proportional to $\epsilon$ are necessarily zero if $a$ is indeed a minimum. (But all terms containing $\epsilon^2$ are not problematic and can be neglected if $\epsilon$ is infinitesimal.) If this wouldn’t be true, we could chose $\epsilon$ negative and find a location this way that corresponds to an ever lower value of the function. Here’s how Richard Feynman explains this… Read more »
Guest
Phil
Many thanks, Jakob. That helps a lot.
Guest
Lucas
Hi Jakob! First of all, thank you VERY MUCH for creating these wonderful books. Reading them is a joy. I just wanted to mention that I, too, have found this section to be the first part of the book I had trouble understanding and had to look for an explanation elsewhere. Thank you again for these great books.
The concept is straightforward, but possibly the notation isn’t. If epsilon (e) vanishes then e=0, but 3x2ae =0. That is 0=0 Rather factor e out, e(3x2a +1)=0 ,etc then a=-1/6. This makes more sense to me.
Guest
Ronnie Webb
Hello and thank you for writing an amazing book on Classical Mechanics that even a self learner like myself can finally learn from! I have a question about the derivations of the q and p transformations via the Poisson Bracket (page 171). In the q derivation the bracket is {q, G}, but in the p derivation the bracket is {G, p}. The Poisson Bracket calculation on the next line (for the p transformation) under this portrays a different calculation i.e. it looks like it should be {p, G}. But, above all of this the equations say q+eG o q and… Read more »
Hi Jakob, At the bottom of page 164 to arrive at (7.98) you cancel a partial derivative of Q with a total derivative of Q. Is that allowed? Although final answer is correct it seems that is because the partial (p) of F/pQ x Qdot sums to zero with its equivalent in the parentheses. Your thoughts? Thank you. BTW loving the book.
Yes, $\frac{\partial F }{\partial Q} \dot Q$ cancels with $\frac{\partial F }{\partial Q} \frac{d Q }{d t}$ in the parentheses. Hope this helps! (And if not, just ask again since I’m not entirely sure I understood your question correctly.)
Guest
Tamir
P.77 – “the ball cannot stay at the top indefinitely since we consider paths between a fixed initial position at fixed initial time and a fixed final position at a fixed final time.”
I dont get it. don’t thake into account the posibility of a constant function q(t)? Is there no stable equilibrium in the world?
sure I can see why it’s not even an equilibrium from newton perspective, but as for the lagrangian formulation it’s not clear…
Hi Tamir, thanks for your question! In the situation describes on page 77, we throw the ball into the air from fixed position and want to find its trajectory if it returns after a given number of seconds to its starting position. In other words, we are dealing with a concrete situation which is specified by appropriate initial conditions. In this case, the initial conditions are given by an initial location and initial velocity. As a result of the nonzero initial velocity (since we throw the ball into the air), a constant q(t) is not a valid possibility. I hope… Read more »
Guest
Robert Herman
Great book for me! I am trying to relearn CM after decades, and the pace and clarity fit me well.
I’m pretty sure I know, but on page 34, (2.12) has force = to the first derivative of momentum, p,
or F= m*a.
Am I correct in assuming that when we take the first derivative of p, m just carries over, since the exponent of v time m = m?
I find this better than the Theoretical Minimum too, and plan on buying Mr. Schwichtenberg’s books!
Yes, in many situations the mass $m$ is constant and therefore $\dot p = \frac{dp}{dt} = \frac{d (mv)}{dt}= m\frac{d v}{dt} = m a = m \dot v$. I hope this answers your question and please let me know if you have any further questions!
Guest
Roy
Hello, I will be using John Taylor’s CM next semester in a class. I am hoping this will allow me to get ahead instead of being behind.
Feel free to provide any advice.
I hope so too! Let me know if you have any questions or find something confusing in the book.
Guest
Roy
I purchased all three of your books.
Guest
Lucas
Jakob, on page 370 it seems that in order to obtain equation D.16 you had to assume that the partial derivative of (dx/dt)2 with respect to x and the partial derivative of x2 with respect to dx/dt are both zero. This makes sense if we assume that x and dx/dt are independent variables, but in page 95, when deriving Hamilton’s equations, you state clearly that it is “not possible” to treat x and dx/dt as “completely independent variables” “because the velocity is always simply the rate of change of the location”. If this is so, why can we assume that… Read more »
Our goal in the Lagrangian formalism is to figure out the correct path in configuration space between two fixed locations. A path is characterized by a location and velocity at each point in time. We want to stay as general as possible and consider really all possible paths. In particular, this means that we consider all possible pairings of location and velocity functions. Now the physical path that we are eventually interested in is special for two reasons: it’s a solution of the Euler-Lagrange equation (= extremum of the action) the locations and velocities at each moment in time are… Read more »
Guest
Lucas
Beautiful! Thank you so much.
Guest
Lucas
Jakob, what is the order you recommend when reading your books? I intend to do Classical Mechanics, Electrodynamics, Quantum Mechanics and then Physics from Symmetry? Would you agree?
Congratulations to an excellent book! If you don’t mind, I have a question about the classical Lagrangian T-V. In Section 12.3.6 you show how the gravitational potential affects the proper time and thus gives rise to the –V term. However, my understanding is that the classical Lagrangian T-V hold for any type of potential, gravitational or non-gravitational. So, the following question arises: Do all kinds of potential, e.g., the electrostatic potential, affect the proper time? I know from your other book (Physics from Symmetry) that the Lagrangian of electromagnetic interactions can be obtained by making the U(1) symmetry of the… Read more »
That’s an excellent question and something I haven’t thought about so far. But I think that the electrostatic potential affects the proper time in a similar way as any other form of energy.
While the electromagnetic Lagrangian can be understood using the U(1) gauge symmetry, it also holds in the classical case. However, you’re probably right that there is some deeper insight hidden in the connection between the quantum mechanical derivation and the classical derivation. But I’m not sure if anyone has understood it yet.
Guest
Edi
Thanks for the response Jakob! I thought some more about the -V term. Do you think the following line of reasoning is valid? (I am trying to make an argument similar to what you did in “Physics from Symmetry” sections 7.1.1-7.1.3, but for classical mechanics.) The classical free Lagrangian L = T = 1/2 m (dx/dt)^2 has the global symmetry L’ = L + a. Is this also a local symmetry: L’ = L + a(x)? No. Can we make it a local symmetry? Yes, if we add a term to the free Lagrangian, L = T – V(x), and… Read more »
That’s a really cool argument I haven’t seen anywhere before. And so far, I don’t see any flaw in it.
Guest
Edi
Thanks, Jakob! I have another question, if you don’t mind: On p.175, Eq.(7.129) you give an example of a generating function that flips the p’s and q’s. To get a better understanding, I tried to construct another example: as far as I can tell Q=k*q, P=p/k with parameter k is a canonical transformation, but I cannot construct the corresponding generating function F(q,Q). It seems to me that only certain canonical transformations (maybe only the ones that mix p’s and q’s) can be expressed in terms of generating functions. If this is so, what are the conditions? In contrast, I could… Read more »
Compliments for your book and your extraordinary ability to write about difficult concepts in a very direct way. However, I have a question , the answer to which, I did not find in your book. Here is my question: When we derive Euler-Lagrange equations in classical mechanics following the Lagrangian approach we introduce Boundary conditions at the starting- and end-points of the path in the configuration space. Usually,though not necessarily, one requires that q(tinitial)=q(tfinal)=0, see section 4.3 of your book. But the standard problem in classical mechanics is to instead assume Initial (not Boundary) conditions, for instance it is assumed… Read more »
And please let me know if it doesn’t answer your question!
Guest
Ioannis Papadopoulos
Thanks for your answer and for the link to physics.stackexchange. I visited the site and studied the discussion provided there. I am affraid it does not answer my question. However, in the meantime I found a paper where the problem is fully addressed. The paper is The Classical mechanics for non-conservative systems by Galley and is published in arXiv:1210.2745v2 [gr-qc] 12 Jun 2013. Unfortunately, my current knowledge of theoretical mechanics does not allow me to understand it. I will wait until someone like you, able to make difficult things accessible to a wider audience, read this paper and explain it.… Read more »
About Classical Mechanics
Not a very good book, an exceptional book, really!
It goes quite deep but you are helped all along with the margin notes: looks like the author reads our mind and he answers in these notes.
Also repeating an equation from previous pages is really convenient.
I am ordering right away the Quantum Mechanics book from the same author
Compared to Susskind Theoretical Minimum, i think this books goes into more details and is easier to follow.
Hi Jakob, Thanks for writing these great books. They really help with getting the big picture of these subjects. I found a typo in the CM book, on page 157 in the derivation, in the step labelled ‘expanding the products,’ the term $ \frac{\partial \bar F}{\partial Q}\frac{\partial Q}{\partial p}\frac{\partial \bar H}{\partial Q}\frac{\partial Q}{\partial q}$ appears twice, but one of them should be $ \frac{\partial \bar F}{\partial Q}\frac{\partial Q}{\partial p}\frac{\partial \bar H}{\partial P}\frac{\partial P}{\partial q}$. The incorrect one is then retained in the cancellation in the next step. Do you have somewhere online where you post errata? That could help so… Read more »
Oh, I thought the LaTeX would compile automagically :/
On page 161, equation 7.88, you also have a $\frac{\partial}{\dot q}$ where it should be $\frac{\partial}{\partial \dot q}$.
Hi Griff,
thanks a lot for reporting these typos. I will correct them as soon as possible. Currently, I have no errata list online but it’s a great idea and will try to put one online as soon as I find the time.
Best,
Jakob
Hi Jakob,
On p.97, in the derivation of Eq.5.10, on the last line, you have $L(q,p)$ within the partial on the RHS. Shouldn’t this be $\tilde{L} (q,p)$? The same thing happens in the derivation of 5.11 and in your definition of the Hamiltonian 5.12, and in 5.13.
Cheers,
Griff
Yes you’re right! I will correct these typos in the next edition.
Hello there. On p 84 the boxed comment states “All terms first order in epsilon must vanish.” I’m not really sure what the author means by this. I can see that if x = a is the lowest point, then for any epsilon (positive or negative), f(a) < f(a + epsilon). However, I don't really understand what the author means by "vanish". I suspect it's something quite straightforward that I'm not seeing, but if someone could flesh this out for me a bit, I would appreciate it.
Hi Phil, the idea is that if we consider a minimum $x=a$, it necessarily goes up if we move away from there. Therefore, if we inspect the neighborhood of the minimum, $x= a + \epsilon$, we know that all terms proportional to $\epsilon$ are necessarily zero if $a$ is indeed a minimum. (But all terms containing $\epsilon^2$ are not problematic and can be neglected if $\epsilon$ is infinitesimal.) If this wouldn’t be true, we could chose $\epsilon$ negative and find a location this way that corresponds to an ever lower value of the function. Here’s how Richard Feynman explains this… Read more »
Many thanks, Jakob. That helps a lot.
Hi Jakob! First of all, thank you VERY MUCH for creating these wonderful books. Reading them is a joy. I just wanted to mention that I, too, have found this section to be the first part of the book I had trouble understanding and had to look for an explanation elsewhere. Thank you again for these great books.
Thanks for your feedback! I will try to improve the explanation in future editions of the book.
The concept is straightforward, but possibly the notation isn’t. If epsilon (e) vanishes then e=0, but 3x2ae =0. That is 0=0 Rather factor e out, e(3x2a +1)=0 ,etc then a=-1/6. This makes more sense to me.
Hello and thank you for writing an amazing book on Classical Mechanics that even a self learner like myself can finally learn from! I have a question about the derivations of the q and p transformations via the Poisson Bracket (page 171). In the q derivation the bracket is {q, G}, but in the p derivation the bracket is {G, p}. The Poisson Bracket calculation on the next line (for the p transformation) under this portrays a different calculation i.e. it looks like it should be {p, G}. But, above all of this the equations say q+eG o q and… Read more »
Hi Ronnie,
yes you’re right that’s a typo. It should read ${p,G}$ in the third line. I will fix it as as soon as possible.
Best,
Jakob
Hi Jakob, At the bottom of page 164 to arrive at (7.98) you cancel a partial derivative of Q with a total derivative of Q. Is that allowed? Although final answer is correct it seems that is because the partial (p) of F/pQ x Qdot sums to zero with its equivalent in the parentheses. Your thoughts? Thank you. BTW loving the book.
Yes, $\frac{\partial F }{\partial Q} \dot Q$ cancels with $\frac{\partial F }{\partial Q} \frac{d Q }{d t}$ in the parentheses. Hope this helps! (And if not, just ask again since I’m not entirely sure I understood your question correctly.)
P.77 – “the ball cannot stay at the top indefinitely since we consider paths between a fixed initial position at fixed initial time and a fixed final position at a fixed final time.”
I dont get it. don’t thake into account the posibility of a constant function q(t)? Is there no stable equilibrium in the world?
sure I can see why it’s not even an equilibrium from newton perspective, but as for the lagrangian formulation it’s not clear…
Hi Tamir, thanks for your question! In the situation describes on page 77, we throw the ball into the air from fixed position and want to find its trajectory if it returns after a given number of seconds to its starting position. In other words, we are dealing with a concrete situation which is specified by appropriate initial conditions. In this case, the initial conditions are given by an initial location and initial velocity. As a result of the nonzero initial velocity (since we throw the ball into the air), a constant q(t) is not a valid possibility. I hope… Read more »
Great book for me! I am trying to relearn CM after decades, and the pace and clarity fit me well.
I’m pretty sure I know, but on page 34, (2.12) has force = to the first derivative of momentum, p,
or F= m*a.
Am I correct in assuming that when we take the first derivative of p, m just carries over, since the exponent of v time m = m?
I find this better than the Theoretical Minimum too, and plan on buying Mr. Schwichtenberg’s books!
Yes, in many situations the mass $m$ is constant and therefore $\dot p = \frac{dp}{dt} = \frac{d (mv)}{dt}= m\frac{d v}{dt} = m a = m \dot v$. I hope this answers your question and please let me know if you have any further questions!
Hello, I will be using John Taylor’s CM next semester in a class. I am hoping this will allow me to get ahead instead of being behind.
Feel free to provide any advice.
I hope so too! Let me know if you have any questions or find something confusing in the book.
I purchased all three of your books.
Jakob, on page 370 it seems that in order to obtain equation D.16 you had to assume that the partial derivative of (dx/dt)2 with respect to x and the partial derivative of x2 with respect to dx/dt are both zero. This makes sense if we assume that x and dx/dt are independent variables, but in page 95, when deriving Hamilton’s equations, you state clearly that it is “not possible” to treat x and dx/dt as “completely independent variables” “because the velocity is always simply the rate of change of the location”. If this is so, why can we assume that… Read more »
Our goal in the Lagrangian formalism is to figure out the correct path in configuration space between two fixed locations. A path is characterized by a location and velocity at each point in time. We want to stay as general as possible and consider really all possible paths. In particular, this means that we consider all possible pairings of location and velocity functions. Now the physical path that we are eventually interested in is special for two reasons: it’s a solution of the Euler-Lagrange equation (= extremum of the action) the locations and velocities at each moment in time are… Read more »
Beautiful! Thank you so much.
Jakob, what is the order you recommend when reading your books? I intend to do Classical Mechanics, Electrodynamics, Quantum Mechanics and then Physics from Symmetry? Would you agree?
Yes this order would be perfect.
Congratulations to an excellent book! If you don’t mind, I have a question about the classical Lagrangian T-V. In Section 12.3.6 you show how the gravitational potential affects the proper time and thus gives rise to the –V term. However, my understanding is that the classical Lagrangian T-V hold for any type of potential, gravitational or non-gravitational. So, the following question arises: Do all kinds of potential, e.g., the electrostatic potential, affect the proper time? I know from your other book (Physics from Symmetry) that the Lagrangian of electromagnetic interactions can be obtained by making the U(1) symmetry of the… Read more »
Thanks!
That’s an excellent question and something I haven’t thought about so far. But I think that the electrostatic potential affects the proper time in a similar way as any other form of energy.
While the electromagnetic Lagrangian can be understood using the U(1) gauge symmetry, it also holds in the classical case. However, you’re probably right that there is some deeper insight hidden in the connection between the quantum mechanical derivation and the classical derivation. But I’m not sure if anyone has understood it yet.
Thanks for the response Jakob! I thought some more about the -V term. Do you think the following line of reasoning is valid? (I am trying to make an argument similar to what you did in “Physics from Symmetry” sections 7.1.1-7.1.3, but for classical mechanics.) The classical free Lagrangian L = T = 1/2 m (dx/dt)^2 has the global symmetry L’ = L + a. Is this also a local symmetry: L’ = L + a(x)? No. Can we make it a local symmetry? Yes, if we add a term to the free Lagrangian, L = T – V(x), and… Read more »
That’s a really cool argument I haven’t seen anywhere before. And so far, I don’t see any flaw in it.
Thanks, Jakob! I have another question, if you don’t mind: On p.175, Eq.(7.129) you give an example of a generating function that flips the p’s and q’s. To get a better understanding, I tried to construct another example: as far as I can tell Q=k*q, P=p/k with parameter k is a canonical transformation, but I cannot construct the corresponding generating function F(q,Q). It seems to me that only certain canonical transformations (maybe only the ones that mix p’s and q’s) can be expressed in terms of generating functions. If this is so, what are the conditions? In contrast, I could… Read more »
I think this answer might exactly what you’re looking for: https://physics.stackexchange.com/a/286682/37286 But in genera, not all canonical transformations can be expressed in terms of generating functions. You can find mathematical discussion here https://physics.stackexchange.com/questions/391216/can-any-symplectomorphism-1-definition-of-canonical-transformation-be-represen
Compliments for your book and your extraordinary ability to write about difficult concepts in a very direct way. However, I have a question , the answer to which, I did not find in your book. Here is my question: When we derive Euler-Lagrange equations in classical mechanics following the Lagrangian approach we introduce Boundary conditions at the starting- and end-points of the path in the configuration space. Usually,though not necessarily, one requires that q(tinitial)=q(tfinal)=0, see section 4.3 of your book. But the standard problem in classical mechanics is to instead assume Initial (not Boundary) conditions, for instance it is assumed… Read more »
You might find the discussion here: https://physics.stackexchange.com/questions/38348/is-the-principle-of-least-action-a-boundary-value-or-initial-condition-problem/
And please let me know if it doesn’t answer your question!
Thanks for your answer and for the link to physics.stackexchange. I visited the site and studied the discussion provided there. I am affraid it does not answer my question. However, in the meantime I found a paper where the problem is fully addressed. The paper is The Classical mechanics for non-conservative systems by Galley and is published in arXiv:1210.2745v2 [gr-qc] 12 Jun 2013. Unfortunately, my current knowledge of theoretical mechanics does not allow me to understand it. I will wait until someone like you, able to make difficult things accessible to a wider audience, read this paper and explain it.… Read more »
The paper looks awesome. Thanks for sharing! I will give it a read.