Step from Mass Point Mechanics to Field Theory

In summary, the author's problem with classical field theory is that there is a conceptual problem with the step from Mass Point Mechanics to Field Theory, and that the equation that is supposed to be used in this transition is nonsense.
  • #1
Geometry_dude
112
20
At the moment I am trying to understand classical field theory and there's a conceptual problem I encountered, which bothers me a lot and I don't seem to be able to resolve the issue. When making the step to classical field theory, many texts start as follows:
First they recall the/a action in mass point mechanics and write something like this
$$S = \int L \, d t \, ,$$
then they say that we don't want to look at particles anymore but at (say scalar) fields ##\phi## and introduce the Lagrangian density ##\mathcal L## as a function of the "field variables" ##\phi## and its derivatives, say ##\partial_t \phi, \partial_x \phi## as follows
$$L = \int_{- \infty}^{\infty} \mathcal L \, d x \, . $$
Now here's my problem: This equation is rubbish, plain mathematical nonsense.
As Ben Niehoff already clarified in a similar question I asked some time ago ( https://www.physicsforums.com/showthread.php?t=751858 ),
there's an action for curves ##\gamma## and an action for fields ##\phi## and the two are, at least on a mathematical level, two separate concepts. If we want to be a bit more rigorous, we thus write the mass point action like
$$S(\gamma, \tau_1, \tau_2) = \int_{\tau_1}^{\tau_2} L ( \gamma, \dot \gamma) \, d \tau$$
and the field action of a (real) scalar field ##\phi## in analogy
$$\mathcal S (\phi, \phi_\text{boundary}) = \int_{\text{interior spacetime}} \mathcal L(\phi, d \phi)$$
where ##\mathcal L(\phi, d \phi)## is now an actual density on the spacetime ( http://en.wikipedia.org/wiki/Density_on_a_manifold ).
This clarifies why the equation I was talking about is nonsense: ##L## and ##\mathcal L## have two distinct inputs, i.e. live on different spaces and the equation is not invariant under coordinate change.
So my questions are:
1) What is the correct step/transition from Mass Point Mechanics to Field Theory?
2) Why do we only consider first order derivatives of the field in the Lagrangian density?
3) How do I get to the Hamiltonian formalism in the field theory in a coordinate independent manner?
 
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  • #2
I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
 
  • #3
I think my problem stems from the fact that I did not distinguish between the Newtonian and relativistic case carefully enough as I thought they could be cast in a unified coordinate-independent language, but this only works for point particles.

Let us consider the example in pp. 12 of http://www.staff.science.uu.nl/~aruty101/CFT.pdf, where they explain the step to field theory by considering a homogenous string in one Euclidean dimension, modelling it as a collection of equal point masses connected by equal massless strings and then letting the distance between the masses go to zero while keeping the mass density and force strengths fixed.
In the discrete case they get
$$\ddot \phi_i + \frac{k}{m} (2 \phi_i - \phi_{i+1} - \phi_{i-1})= 0 \, ,$$
where the ##\phi_i##s give the value of the displacement of the ##i##th point mass of mass ##m## from the equilibrium position and ##k## is the spring constant. In the continuum case, they get
$$\ddot \phi_i - \frac{Y}{\mu} \frac{\partial^2 \phi}{\partial x ^2}= 0$$
where ##\mu## and ##Y## are the continuum analogues of ##k## and ##m##. How they are obtained is explained in the document.

Now, in the Newtonian case it makes sense to consider an "external time parameter" ##\tau## and then the last equation can be rewritten into a coordinate independent form, also suggesting an obvious generalization to ##3## or ##n## dimensions:
$$\ddot \phi - \frac{Y}{\mu} \Delta \phi = 0 \, .$$
The dot denotes, as usual, time derivatives and the ##\Delta## is the Laplace-Beltrami operator for the Riemannian manifold ##(\mathbb R ^n, \delta = \delta_{ij} \, d x^i \otimes d x^j)##, i.e. the ordinary Laplacian. The bottom line is that this is a perfectly legitimate equation and that we can get it out of an action of the form
$$\mathcal S ( \phi, \phi_{\text{boundary}}(\tau_0), \phi_{\text{boundary}}(\tau_1), \tau_0, \tau_1 ) =\int_{\tau_0}^{\tau_1} \int_{\text{interior space}} \mathcal L (\phi, d \phi, \dot \phi,\tau) \, d \tau \, , $$
where ##\mathcal L## is a density on ##(\mathbb R ^n, \delta)##.

In the relativistic case, it is not meaningful to introduce ##\tau##, that is the proper time, for fields, making the above example useless in this case. So how does one get around that?
 
Last edited:
  • #4
Is this the source of the problem of time?
 
  • #5


I can provide some insights into your questions about the transition from mass point mechanics to field theory.

1) The correct step from mass point mechanics to field theory is to recognize that classical mechanics is a special case of classical field theory. In classical mechanics, we deal with point particles that have definite positions and momenta. In classical field theory, we deal with fields that are defined at every point in space and time. The field value at each point can be thought of as the "position" of the field at that point, and the time derivative of the field can be thought of as the "momentum" of the field at that point. In this way, we can see that classical mechanics is just a special case of classical field theory where the field is concentrated at a single point.

2) The reason we only consider first order derivatives of the field in the Lagrangian density is because this is the simplest way to describe the dynamics of the field. Just like in classical mechanics, we only need to consider the first derivative of the position to fully describe the dynamics of a point particle. In classical field theory, the first derivative of the field is enough to describe the dynamics of the field at each point in space and time.

3) To get to the Hamiltonian formalism in field theory, we need to use a coordinate-independent approach. This can be achieved by using the Hamiltonian density, which is defined as the Legendre transform of the Lagrangian density with respect to the time derivative of the field. The Hamiltonian density is a function of the field and its conjugate momentum, which is the time derivative of the field. Just like in classical mechanics, the Hamiltonian density can be used to derive the equations of motion for the field, and it is also useful for studying the symmetries and conservation laws of the field.
 

1. What is mass point mechanics?

Mass point mechanics is a branch of classical mechanics that describes the motion of particles or objects based on their mass, position, velocity, and acceleration. It is also known as particle mechanics or point mechanics.

2. How does mass point mechanics differ from field theory?

Mass point mechanics focuses on the behavior of individual particles, while field theory studies the interactions between particles and the surrounding field. In mass point mechanics, particles are treated as point masses with no size, whereas in field theory, particles have a finite size and interact through fields such as electromagnetic or gravitational fields.

3. What is the relationship between mass point mechanics and Newton's laws of motion?

Mass point mechanics is based on Newton's laws of motion, specifically the second law which states that the force acting on a particle is equal to its mass multiplied by its acceleration. Mass point mechanics uses this law to predict the motion of particles in a given system.

4. How does field theory incorporate relativity?

Field theory is based on the principles of relativity, which state that the laws of physics should be the same for all observers in uniform motion. In field theory, the interactions between particles and fields are described using a four-dimensional spacetime framework, which takes into account the effects of motion and gravity.

5. What are some real-life applications of mass point mechanics and field theory?

Mass point mechanics is used in various fields such as engineering, physics, and astronomy to study the motion of objects and predict their behavior. Field theory has many applications in modern physics, including the study of electromagnetism, quantum mechanics, and general relativity. It is also used in practical applications such as designing electronic devices and studying the behavior of materials.

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