How Does Jacobi's Principle Modify Lagrange's Method?

In summary: I really need to brush up on my calculus... Thanks!In summary, the author says that the canonical equation can be reduced to the one that he has written in homework equations, but that in order to solve it, a relation between the various momentum terms must be provided.
  • #1
dRic2
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Homework Statement
Consider the action integral of Jacobi's principle
$$A = \int_{\tau_1}^{\tau_2} \sqrt{2} \sqrt{E-V} \sqrt{ \Sigma a_{ik} q_i' q_k'} d \tau$$
(##q_i' = \frac {dq_i}{d \tau} = \frac {dq_i}{dt} \frac {dt}{d\tau} = \dot q_i t'##).
Show that the ##p_i## associated with this integrand satisfy the following identity:
$$\frac {\Sigma b_{ik} p_i p_k} {2 (E-V)} = 1$$
where ##b_{ik}## are the coefficients of the matrix which is the reciprocal of the original matrix ##a_{ik}##.
Relevant Equations
Symmetrical form of the action integral
$$A = \int_{t_1}^{\tau_2} \sum_i p_i q'_i d \tau$$
I'm studying a chapter on the parametric form of the canonical equation so basically the author says that time is no more an independent variable but it is expressed as a function of an other variable called ##\tau##. In this way the canonical integral is reduced to the one I've written in "Homework Equations". The problem is that in this way I add a variable to my problem so I must also add a condition between the various ##p_i## in order to solve the integral. But I don't know how to derive that relation for the Jacobi's principle.PS: I know this is not strictly related to the exercise, but I never really understood Jacobi's "critique" of Lagrange's method. The book I'm reading said that he used this approach (expressing the time not as independent variable, but as a function of some other parameter) because he criticized that otherwise the process of varying between definite limit was not possible.

Thanks
Ric
 
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  • #2
Hello.

Just to be clear, are you asking how to obtain the identity $$\frac {\Sigma b_{ik} p_i p_k} {2 (E-V)} = 1$$ from $$A = \int_{\tau_1}^{\tau_2} \sqrt{2} \sqrt{E-V} \sqrt{ \Sigma a_{ik} q_i' q_k'} d \tau \,\,\, \rm ?$$
 
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  • #3
Kind of. That integral can't be solved because ##\tau## is a variable arbitrary introduced. I think that relation between ##p_i## must be provided in order to solve it. This is how I interpreted the question. Sounds possible ?
 
  • #4
My interpretation of the homework statement is that from the integrand of the integral in $$A = \int_{\tau_1}^{\tau_2} \sqrt{2} \sqrt{E-V} \sqrt{ \Sigma a_{ik} q_i' q_k'} d \tau$$ you can obtain expressions for the momenta ##p_i##. Using these expressions, you can then prove the identity $$\frac {\Sigma b_{ik} p_i p_k} {2 (E-V)} = 1$$

I'm not sure what you mean when you say,
dRic2 said:
That integral can't be solved because ##\tau## is a variable arbitrary introduced.

By "solving" the integral, do you mean evaluating the integral? There is generally no need to explicitly evaluate the integral, as the integral is used only to obtain the differential equations for the actual path of motion via the variational principle ##\Delta A = 0##. However, if you wanted to, it seems to me that you could (in principle) evaluate the integral for an arbitrary path ##\left\{q_1(\tau), q_2(\tau), ..., q_N(\tau) \right\}## connecting the initial and final points. The value of the integral will not depend on the particular choice of the parameter ##\tau##. The integral depends only on the choice of path and the value of the total energy ##E##.
 
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  • #5
TSny said:
I'm not sure what you mean when you say,

Ok. The integral that can't be solve that I was referring to was $$\int_{\tau_1}^{\tau_2} \sum p_i q_i' d\tau$$
In order to solve the above integral I need a relation between ##p_i## that I can obtain from
$$A = \int_{\tau_1}^{\tau_2} \sqrt{ 2(E-V)} \sqrt{ \sum a_{ik} q_i' q_k' } d \tau$$
So, in my second post, I basically misunderstood my own statement :sorry:
Thanks for pointing that out, now it is definitively clearer.
 
  • #6
So
$$p_i = \frac {\partial L}{ \partial q_i'} = \frac {\sqrt{2(E-V)} \sum_k a_{ik} q_k' } {2 \sqrt{\sum a_{ik} q_i' q_k'}}$$
Now I should find a way to exploit the fact that ##b_{ik}## is the inverse matrix of ##a_{ik}##, am I on the right path ?
 
  • #7
dRic2 said:
So
$$p_i = \frac {\partial L}{ \partial q_i'} = \frac {\sqrt{2(E-V)} \sum_k a_{ik} q_k' } {2 \sqrt{\sum a_{ik} q_i' q_k'}}$$
Now I should find a way to exploit the fact that ##b_{ik}## is the inverse matrix of ##a_{ik}##, am I on the right path ?
OK, you're on the right path. But I think you missed a factor of 2 in the numerator, which is easy to do. Try to see that $$\frac {\partial}{ \partial q_j'} \sum_{ik} a_{ik} q_i' q_k'= {\color{brown}{2}} \sum_k a_{jk} q_k'$$
 
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  • #8
(I'll drop the summation sign from now on)

$$p_i = \sqrt{2(E-V)} \frac {a_{ik}q_k'} {\sqrt{a_{ik}q_i'q_k'}}$$

$$b_{ij}p_i p_j = 2(E-V) \frac {b_{ij}a_{ik}q_k'a_{lj}q_l'} {\sqrt{a_{ik}q_i'q_k'} \sqrt{a_{lj}q_l'q_j'} }$$

(I was a little sloppy in choosing the indices of summation)

Now I exploit the fact that ##B## is the inverse of ##A##, so I get ##b_{ij}a_{ik} = \delta_{jk}## and thus ##\delta_{jk}a_{lj} = a_{lk}##. Finally:

$$ (...) = 2(E-V) \frac {a_{lk}q_k'q_l'}{a_{ik} q_i' q_k'} $$

where I use the fact that ##i##, ##k##, ##j## and ##l## are dummy indices to simplify the denominator. The numerator is identical to the denominator in the final equation so the exercise is done.

I can't believe I was stuck because I couldn't remember that ##b_{ij}a_{ik} = \delta_{jk}##... :headbang: :headbang: for inverse matrices... I feel so stupid right now :wideeyed:

BTW do you happen to know the argument between Jacobi ad Lagrange that I mentioned at the end of my original post ?
 
  • #9
Looks good.

dRic2 said:
BTW do you happen to know the argument between Jacobi ad Lagrange that I mentioned at the end of my original post ?
I'm not familiar with this, other than the brief remarks in Lanczos' text near the bottom of page 136:
https://archive.org/details/VariationalPrinciplesOfMechanicsLanczos_201610/page/n155
(similar remark middle of page 134)
 
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  • #10
That is the text I'm reading and the one where i got this exercise from ahahahah
 
  • #11
Interesting. On what page of Lanczos is this exercise given?
 
  • #12
In the link you posted you can find it at pagE 188
 
  • #13
Ah, thanks.
 
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  • #14
Some of the history is given in https://www.jstor.org/stable/27900374?seq=1#metadata_info_tab_contents of 1912, which you can download. I only glanced through it, but section v looks relevant.
 
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Related to How Does Jacobi's Principle Modify Lagrange's Method?

1. What is Jacobi's principle of least action?

Jacobi's principle of least action is a fundamental principle in physics that states that the path of a physical system between two points will follow the path of least action, where action is defined as the integral of a Lagrangian function over time.

2. What is the significance of Jacobi's principle of least action?

Jacobi's principle of least action is significant because it provides a fundamental framework for understanding the behavior of physical systems. It allows for the prediction of the path of a system without having to solve complex differential equations, and it is also a fundamental principle in the formulation of the laws of motion in classical mechanics.

3. How is Jacobi's principle of least action related to the principle of least constraint?

The principle of least constraint is a generalization of Jacobi's principle of least action. It states that the path of a physical system will follow the path of least action, but with additional constraints imposed. These constraints can include forces, boundary conditions, and other physical laws.

4. Can Jacobi's principle of least action be applied to quantum mechanics?

Yes, Jacobi's principle of least action can be applied to quantum mechanics through the use of the path integral formulation. In this formulation, the path of a quantum system is determined by considering all possible paths and calculating the action for each one. The path with the least action is then the most probable path for the system.

5. What are some applications of Jacobi's principle of least action?

Jacobi's principle of least action has many applications in physics, including classical mechanics, quantum mechanics, and electromagnetism. It is also used in fields such as optics, fluid mechanics, and general relativity. It has also been used in the development of variational methods in mathematics and optimization problems in engineering.

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