- #1
Hypersphere
- 191
- 8
Hi,
I'm trying to clear up a confusing point in the book by José and Saletan, concerning equivalent Lagrangians (in the sense that they give you the same dynamics). It is clear that if
[itex] L_1 - L_2 = \frac{d\phi ( q,t )}{dt},[/itex]
then [itex]L_1[/itex] and [itex]L_2[/itex] will have the same equations of motion. However, what about the inverse problem?
It seems that it is sometimes assumed that two Lagrangians giving the same equations of motion must differ by such a total time derivative (Jose and Saletan do this in problem 2.4 and in the beginning of section 2.2.2). However, as they write later in that section, the two Lagrangians
[itex]L_1 = \dot{q}_1 \dot{q}_2 - \omega^2 q_1 q_2[/itex]
[itex]L_2 = \frac{\dot{q}_1^2}{2}+\frac{\dot{q}_2^2}{2} - \frac{\omega^2}{2} q_1^2 - \frac{\omega^2}{2} q_2^2[/itex]
quite clearly give the same equations of motion, but aren't related by a total time derivative of a function of position and time.
Is this simply because [itex]L_1[/itex] is non-local? And does the inverse property then always hold for local Lagrangians on the standard form T-V?
I'm trying to clear up a confusing point in the book by José and Saletan, concerning equivalent Lagrangians (in the sense that they give you the same dynamics). It is clear that if
[itex] L_1 - L_2 = \frac{d\phi ( q,t )}{dt},[/itex]
then [itex]L_1[/itex] and [itex]L_2[/itex] will have the same equations of motion. However, what about the inverse problem?
It seems that it is sometimes assumed that two Lagrangians giving the same equations of motion must differ by such a total time derivative (Jose and Saletan do this in problem 2.4 and in the beginning of section 2.2.2). However, as they write later in that section, the two Lagrangians
[itex]L_1 = \dot{q}_1 \dot{q}_2 - \omega^2 q_1 q_2[/itex]
[itex]L_2 = \frac{\dot{q}_1^2}{2}+\frac{\dot{q}_2^2}{2} - \frac{\omega^2}{2} q_1^2 - \frac{\omega^2}{2} q_2^2[/itex]
quite clearly give the same equations of motion, but aren't related by a total time derivative of a function of position and time.
Is this simply because [itex]L_1[/itex] is non-local? And does the inverse property then always hold for local Lagrangians on the standard form T-V?