1
vote
1answer
57 views

Hamilton's equations from the action with boundary conditions involving position and momentum

Generally, when you are given the action $$ S=\int_{t_1}^{t_2}\mathrm dt (p\dot q - \mathcal H )$$ the boundary conditions are $q(t_1)=q_1$ and $q(t_2)=q_2$. This is useful because to calculate ...
3
votes
0answers
358 views

Derivation of Euler-Lagrange equations for Lagrangian with dependence on second order derivatives

Suppose we have a Lagrangian that depends on second-order derivatives: $$L = L(q, \dot{q}, \ddot{q})$$ If we're working on the variational problem for this Lagrangian, then I know that we'll wind up ...
2
votes
3answers
356 views

Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu ...
1
vote
1answer
72 views

Maximum aging and path of rock

When a rock falls from a ledge, why does it head to the surface and not up to where time runs faster? If a rock, free from forces, follows a worldline of maximum aging, why would that rock approach ...
18
votes
4answers
868 views

Is the principle of least action a boundary value or initial condition problem?

Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating: In analytic (Lagrangian) mechanics, the derivation of the ...