Physics
Lagrangian Density
The Lagrangian density is a mathematical function used in physics to describe the dynamics of a system. It is defined as the difference between the kinetic and potential energy of a system, and is used to derive the equations of motion for the system. The Lagrangian density is a fundamental concept in classical mechanics and quantum field theory.
Written by Perlego with AI-assistance
Related key terms
1 of 5
7 Key excerpts on "Lagrangian Density"
eBook - PDF- Michael T. Vaughn(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
There are some subtleties in that extension, especially when dealing with fields such as the electromagnetic field with its gauge invariance. However, we will not be concerned with such points here, as we simply present the basic ideas of classical Lagrangian field theory. Consider first a real scalar field φ ( x, t ) , whose dynamics is to be described by a Lagrangian Density L that depends on φ and its first derivatives, which we denote by φ t for the time derivative ∂φ/∂t and by φ k for the spatial derivatives ∂φ/∂x k . The dynamics is based on an extension of Hamilton’s principle (see Section 3.5.3) to require that the action S [ φ ( x )] = L ( φ, φ k , φ t ) d n x dt (8.A1) be an extremum relative to nearby fields ( n is the number of space dimensions). This leads to a generalization of the Euler–Lagrange equations of motion, ∂ ∂t ∂ L ∂φ t + ∂ ∂x k ∂ L ∂φ k − ∂ L ∂φ = 0 (8.A2) (summation over k understood) that provides equations of motion for the fields. Canonical momenta for the field can be introduced by π ( x, t ) ≡ ∂ L ∂φ t ( x, t ) (8.A3) The Hamiltonian density H defined by H = π ( x, t ) ∂ L ∂φ t ( x, t ) − L ( φ, φ k , φ t ) (8.A4) can often be identified as an energy density of the field. The total field energy is then given by H = E [ φ ] = π ( x, t ) ∂ L ∂φ t ( x, t ) − L ( φ, φ k , φ t ) d n x (8.A5) The wave equation for the scalar field φ ( x, t ) can be derived from the Lagrangian Density L = 1 2 1 c 2 φ 2 t − φ 2 k (8.A6) (again with implied summation over k ), The canonical momentum density is then given by π ( x, t ) = φ t ( x, t ) and the Hamiltonian density is H = 1 2 [ π ( x, t )] 2 + [ φ k ( x, t )] 2 (8.A7) A Lagrangian Field Theory 385 Symmetry principles can be enforced by constraining the Lagrangian Density to be in-variant under the desired symmetry transformations. This is especially useful in constructing theories of elementary particles that are supposed to possess certain symmetries, as mentioned in Chapter 10.
No longer available |Learn moreRelativistic Quantum Mechanics
An Introduction to Relativistic Quantum Fields
- Luciano Maiani, Omar Benhar(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
The function L is given the name Lagrangian Density or, more simply for brevity, the Lagrangian and depends on the fields (the dynamic variables) and their derivatives. The time derivatives are the generalisation of velocity, while the dependence of the Lagrangian on spatial derivatives allows to couple the degrees of freedom among nearby points in space. We have considered a possible explicit dependence of the Lagrangian on the space-time coordinates, to allow for the effect of possible agents external to the system of fields. For an isolated system, this dependence cannot be present and the Lagrangian depends on the coordinates only through the fields and their derivatives. In terms of L : S = V 4 d 4 x L ( φ, φ μ , x ) (3.4) where V 4 is the region of space-time limited by the hypersurfaces Γ 1 : t = t 1 and Γ 2 : t = t 2 . We assume the values of the fields are fixed on Γ 1 , 2 . The principle of least action states that: • the evolution of the field between these values is given by the function φ ( x ) = ¯ φ ( x , t ) which minimises S , with boundary conditions fixed on Γ 1 , 2 . We note that the Lagrangian Density is not unique. Because the principle of least action stipulates that the fields have defined values on the boundaries of V 4 , we can add to the Lagrangian the divergence of any 4-vector without changing the minimum of the action or, hence, the equations of motion. To derive the differential equations which determine the evolution of the field, we set: φ ( x ) = ¯ φ ( x ) + δφ ( x ); δφ ( x , t 1 ) = δφ ( x , t 2 ) = 0 . (3.5) The condition of least action becomes the equation: δS = 0 = d 4 x [ ∂ L ∂φ δφ + ∂ L ∂∂ μ φ δ ( ∂ μ ) φ ] (3.6) = d 4 x [ ∂ L ∂φ δφ + ∂ L ∂∂ μ φ ∂ μ δφ ] = d 4 x [ ∂ L ∂φ − ∂ μ ( ∂ L ∂∂ μ φ )] δφ + d 4 x ∂ μ ( ∂ L ∂∂ μ φ δφ ) THE LAGRANGIAN THEORY OF FIELDS 21 because δ ( ∂ μ ) φ = ∂ μ δφ . Given that the field variations vanish at the edges of the region of integra-tion, the final term in (3.6) is zero.
eBook - PDF- André Rubbia(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
A tensor field describes a quantity with two Lorentz indices, e.g., µν , and will therefore transform as: F µν (x) → F ′µν (x) = Λ µ ρ Λ ν σ F ρσ (Λ −1 x) (6.18) 3 The reason for considering complex numbers is implicit in the fact that one wants to apply the theory to quantum systems. 4 We use the notation φ(x) when the field must be complex, to distinguish from the general case ϕ(x). 187 6.3 Lagrangian Density in Field Theory x ′ y ′ y x A ′ (x) = R A(R −1 x) Figure 6.1 Illustration of the transformation of a vector field under the rotation R. 6.3 Lagrangian Density in Field Theory In the classical case, the Lagrangian function depends on the generalized coordinates and generalized velocities. In the previous section, we have replaced them by a relativistic field and its first derivatives. Now we seek to generalize the Lagrangian, which should lead to the equations of motion of the relativistic field. The Lagrangian function should only depend on the field and its first derivatives at a given x. Stated differently, in order to be able to transform it from one observer to another, it must be local. Consequently, it is introduced as a “density” at each point rather than the absolute quantity L. Let us begin with the case of a real scalar field, and then we will discuss the complex case. • Real scalar field. We define the Lagrangian Density L as a function of the real field ϕ and its four-derivatives: L≡L(ϕ,∂ µ ϕ) (6.19) The action becomes: S ≡ Ldt = dt d 3 xL = d 4 xL(ϕ,∂ µ ϕ) (6.20) where we note that the result should be expressed as a covariant integral. Since the four-dimensional volume element d 4 x is an invariant of the Lorentz transformation, we seek Lagrangian densities that are scalar and also invariant under Lorentz transformations. The invariance of the Lagrange density is going to be a fundamental condition for the covariance of the theory, as will be discussed in the following.
eBook - PDF- Barton Zwiebach(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
There are several reasons for this. First, it is needed for Lorentz invariance. The two options lead to contributions of opposite signs in the Lagrangian, and only one sign is consistent with Lorentz invariance. Second, kinetic energy is always associated with time derivatives. Third, the calculation of the total energy vindicates the correctness of the choice. Indeed, with this additional term the Lagrangian Density becomes L = T − V − V = 1 2 ∂ 0 φ ∂ 0 φ − 1 2 ∂ i φ ∂ i φ − 1 2 m 2 φ 2 , (10.5) where the repeated spatial index i denotes summation. The relative sign between the first two terms on the right-hand side allows us to rewrite them as a single term which uses the Minkowski metric η μν : L = − 1 2 η μν ∂ μ φ ∂ ν φ − 1 2 m 2 φ 2 . (10.6) Since all the indices are matched, the Lagrangian Density is Lorentz invariant. The associated action is S = d D x − 1 2 η μν ∂ μ φ∂ ν φ − 1 2 m 2 φ 2 , (10.7) where d D x = dx 0 dx 1 . . . dx d , and D = d + 1, is the number of spacetime dimensions. This is the action for a free scalar field with mass m . A field is said to be free when its equations of motion are linear. If each term in the action is quadratic in the field, as is the case in ( 10.7 ), the equations of motion will be linear in the field. A field that is not free is said to be interacting, in which case the action contains terms of order three or higher in the field. To find the energy density in this field we calculate the Hamiltonian density H . The momentum conjugate to the field is given by ≡ ∂ L ∂(∂ 0 φ) = ∂ 0 φ , (10.8) where we used ( 10.5 ) to evaluate the derivative. The Hamiltonian density is then con-structed as H = ∂ 0 φ − L . (10.9) 197 10.3 Classical plane-wave solutions Quick calculation 10.1 Show that the Hamiltonian density takes the form H = 1 2 2 + 1 2 ( ∇ φ) 2 + 1 2 m 2 φ 2 . (10.10) The three terms in H are identified as T , V , and V , respectively. This is what we expected physically for the energy density.
eBook - PDFAdvanced Modern Physics
Theoretical Foundations
- John Dirk Walecka(Author)
- 2010(Publication Date)
- WSPC(Publisher)
Chapter 5 Lagrangian Field Theory It was argued in Vol. I that a consistent relativistic quantum mechanics presents from the outset a many-body problem, involving the creation and destruction of particles. The appropriate framework for dealing with this problem is quantum field theory, as introduced in chapter 12 of Vol. I. In order to have a consistent dynamical framework for quantum field theory, we turn to the lagrangian formulation of continuum mechanics. 1 There are several reasons for this: • One can easily pass from a lagrangian to a hamiltonian, through which the Schr¨odinger equation i planckover2pi1 ∂/∂t | Ψ( t ) angbracketright = ˆ H | Ψ( t ) angbracketright is formulated; • We know how to quantize with, for example, [ˆ p, ˆ q ] = planckover2pi1 /i in discrete mechanics, and the canonical momentum is obtained directly from the lagrangian; • Lagrange’s equations turn out to be manifestly covariant with a Lorentz-invariant Lagrangian Density; • The stress tensor, which provides the energy and momentum of the system, is obtained directly from the lagrangian; • Symmetry properties of the lagrangian immediately lead to conserved currents (Noether’s theorem); • This approach provides a framework for introducing new fields and new interactions. So far, there is no other completely consistent mechanics framework for dealing with the quantum problem of many interacting relativistic particles. To see how this all works, let us first review the results obtained in Vol. I for the classical lagrangian mechanics of a point particle of mass m . 1 See, for example, [Fetter and Walecka (2003)]. 95 96 Advanced Modern Physics 5.1 Particle Mechanics Classical lagrangian mechanics for a point particle of mass m was developed in ProbIs. 10.3–10.5. Here we summarize the results from those problems. Suppose one has a system with n degrees of freedom and n generalized coordinates q i with i = 1 , 2 , ··· ,n .
eBook - PDFHydrodynamics
Concepts and Experiments
- Harry Edmar Schulz(Author)
- 2015(Publication Date)
- IntechOpen(Publisher)
This is, in extremely few words, why I have chosen to treat this subject in the present Chapter. In this Chapter a special care is devoted to examining how the results about dissipation, usually presented in the Eulerian framework, appear in the LF. Processes involved in dissipation are the ones keeping trace of the granular nature of matter, and a very natural way of seeing this is to describe the fluid in the Lagrangian Formulation. Actually, the postulates of the LF must be criticized right in vision of the granularity of matter, as reported in § 6. For the moment being, let’s just consider the LF as non-in-conflict with matter granularity. In § 2 the fundamental tools of the LF are proposed, together with the physical sense of the idea of fluid parcel, and the relationship of this with the particles of matters forming the continuum. In this § the fluid geometry, kinematics and mass conservation are sketched. Figure 2. Some mechanical waves propagating along the surface of a fluid. The description of such modes of the con‐ tinuum is traditionally (and better) done in the Eulerian Formalism, where the physical quantities locally describing the motion of a fluid are conceived as local space properties, i.e. classical fields. In waves, indeed, matter does not propagate itself, and prevalently linear terms appear in the equations of motion. Picture by “hamad M”, on Flickr, at the webpage https://www.flickr.com/photos/meshal/. Lagrangian Hydrodynamics, Entropy and Dissipation http://dx.doi.org/10.5772/59319 119 The application of the LF tools to fluids without dissipation is then given in § 3: in the absence of friction the fluid is treated as a Hamiltonian system, and here the dynamics appears in LF as inherited from that of point particles. The parcel variables of the LF are regarded as a centre-of-mass versus relative variables decomposition of the discrete analytical mechanics.
eBook - PDF- Nivaldo A. Lemos(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
1 Lagrangian Dynamics Lagrange has perhaps done more than any other analyst by showing that the most varied consequences respecting the motion of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem. William Rowan Hamilton, On a General Method in Dynamics Mechanical systems subject to restrictions (constraints) of a geometric or kinematic nature occur very often. In such situations the Newtonian formulation of dynamics turns out to be inconvenient and wasteful, since it not only requires the use of redundant variables but also the explicit appearance of the constraint forces in the equations of motion. Lagrange’s powerful and elegant formalism allows one to write down the equations of motion of most physical systems from a single scalar function expressed in terms of arbitrary independent coordinates, with the additional advantage of not involving the constraint forces. 1.1 Principles of Newtonian Mechanics A fair appreciation of the meaning and breadth of the general formulations of classical mechanics demands a brief overview of Newtonian mechanics, with which the reader is assumed to be familiar. Virtually ever since they first appeared in the Principia, Newton’s three laws of motion have been controversial regarding their physical content and logical consistency, giving rise to proposals to cast the traditional version in a new form free from criticism (Eisenbud, 1958; Weinstock, 1961). Although the first and second laws are sometimes interpreted as a definition of force (Marion & Thornton, 1995; Jos´ e & Saletan, 1998; Fasano & Marmi, 2006), we shall adhere to what we believe is the correct viewpoint that regards them as genuine laws and not mere definitions (Feynman, Leighton & Sands, 1963, p. 12–1; Anderson, 1990).
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
