- #1

dRic2

Gold Member

- 890

- 225

- 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

Thanks

Ric