1
$\begingroup$

Newton's laws are invariant under time reversal transformation $$ t \longrightarrow -t $$ for time-independent potentials.

But Hamilton-Jacobi equation is too an equivalent description of classical mechanics which is not invariant under this transformation because of being 1st order in time.

Where is the fallacy?

$\endgroup$
1
$\begingroup$

The Hamilton -Jacobi equation:

$$\frac{\partial S}{\partial t}=-H$$

with:

$$H=\frac{\partial \mathscr L}{\partial \dot{q}}\,\frac{d q}{dt}- \mathscr L$$

you get:

$$\frac{\partial S}{\partial t}=-\frac{\partial \mathscr L}{\partial \dot{q}}\,\frac{d q}{dt}+ \mathscr L\tag 1$$

reverse time $t\mapsto -t$ we obtain for eq. (1)

$$-\frac{\partial S}{\partial t}=+\frac{\partial \mathscr L(-t)}{\partial \dot{q}}\,\frac{d q}{dt}+ \mathscr L (-t)\tag 2$$

so equation (2) is equal equation (1), if $\mathscr L (-t)=\mathscr L (+t)$ this is the case in your for your question.

you get the same results for Euler- Lagrange equations.

$\endgroup$
0
$\begingroup$

Assuming the Hamiltonian is invariant under $T$-transformation, the Hamilton-Jacobi (HJ) equation is also invariant, since Hamilton's principal and characteristic functions change sign.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.