Linked Questions

102
votes
7answers
18k views

Why are there only derivatives to the first order in the Lagrangian?

Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded? Is there a good reason for ...
73
votes
10answers
8k views

Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? If someone on the street would flat out ask me ...
23
votes
5answers
5k views

Is there a proof from the first principle that the Lagrangian L = T - V?

Is there a proof from the first principle that for the Lagrangian $L$, $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$ in classical mechanics? Assume that Cartesian coordinates are ...
10
votes
5answers
4k views

Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?

All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before. Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
5
votes
3answers
4k 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 A^\...
7
votes
1answer
1k views

Can we find the boundary conditions of fields from the stationary action principle?

First principle of stationary action Consider a real Klein-Gordon scalar field $\phi$ living in a $D$ dimensional flat spacetime. The field is considered off shell (the on shell condition is defined ...
8
votes
2answers
542 views

How to determine the Lagrangian's “true” explicit dependence on time?

If your Lagrangian satisfies $$ \frac{\partial \mathcal L}{\partial t} = 0 $$ then you're happy, energy is conserved, etc. However, if the above doesn't hold, that doesn't necessarily mean energy ...
5
votes
2answers
1k views

Why my 4-divergence term added to a Lagrangian modifies the equation of motion?

I take this Lagrangian: $$\mathcal{L}=\mathcal{L}_0+\partial_\alpha f(\phi, \partial_\mu \phi).$$ In this topic Does a four-divergence extra term in a Lagrangian density matter to the field ...
2
votes
4answers
597 views

What are the boundary conditions associated to this lagrangian?

Suppose that $L(q^i, \dot{q}^i)$ is a standard and well behaved lagrangian associated to some Dirichlet boundary conditions : $q^i(t_1) = q_1^i$ and $q^i(t_2) = q_2^i$. Now I have this new lagrangian ...
2
votes
2answers
971 views

Detailed conditions for symmetries of Lagrangian

Edit: To clarify the question, I am asking why we are justified in calling a continuous symmetry a symmetry of a system when it changes the Lagrangian by a total derivative of a function of $t, q(t)$ ...
0
votes
1answer
808 views

Proving invariance of Lagrange's equations under transformation of the form $L' = L + \dfrac{d}{dt}g(\mathbf{q},t)$

A problem in my classical mechanics textbook is stated as follows: Show that if the Lagrangian $L(\mathbf{q},\dot{\mathbf{q}},t)$ is modified to $L'$ by any transformation of the form $$ L' = L + ...
1
vote
1answer
631 views

When can one omit a total time derivative in the Lagrangian formulation?

I am studying Lagrangian and Hamiltonian mechanics and i am using Landau & Lifshitz and Goldstein books. Both of them state that a modified lagrangian $$L'=L+\frac{df}{dt}$$ gives the same ...