Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

In the Lagrangian formalism, a coordinate qi is ignorable if L/qi=0; that is, if L is independent of qi. This guarantees that the momentum pi is constant. In the Hamiltonian approach, we say that qi is ignorable if H is independent of qi, and this too guarantees pi is constant. These two conditions must be the same, since the result " pi= const" is the same either way. Prove directly that this is so, as follows: (a) For a system with one degree of freedom, prove that H/q=L/q starting from the expression (13.14) for the Hamiltonian. This establishes that H/q=0 if and only if L/q=0. (b) For a system with n degrees of freedom, prove that H/qi=L/qi starting from the expression (13.24).

Short Answer

Expert verified
The conditions H/qi=0 and L/qi=0 are equivalent.

Step by step solution

01

Recall the Hamiltonian Definition

The Hamiltonian H is defined in terms of the Lagrangian L by the equation H=ipiq˙iL, where pi=Lq˙i. This expression is crucial for deriving the relationship between the partial derivatives of H and L.
02

Single Degree of Freedom (a)

For a system with one degree of freedom, the Hamiltonian can be expressed as H=pq˙L. Taking the partial derivative of H with respect to q, we get:Hq=q(pq˙)Lq.Since p is defined as Lq˙, and assuming q˙ is independent of q, the first term is zero, leading to:Hq=Lq.
03

Generalize to n Degrees of Freedom (b)

For n degrees of freedom, the expression for Hamiltonian becomes H=i=1npiq˙iL. The partial derivative with respect to one coordinate qi is:Hqi=j=1nqi(pjq˙j)Lqi.Since pj=Lq˙j does not depend on qi when L is independent of it, each term in the sum involving jeqi vanishes, and we are left with:Hqi=Lqi.
04

Conclusion

We have shown through direct differentiation that Hqi=Lqi holds for both single and multiple degrees of freedom situations. Consequently, Hqi=0 implies Lqi=0, and vice versa, for all coordinates. This proves the equivalence conditions for ignorable coordinates in the Lagrangian and Hamiltonian formalisms.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lagrangian Formalism
The Lagrangian formalism is a fundamental concept in classical mechanics, introduced by Joseph Louis Lagrange. It elegantly describes the dynamics of a system by using the Lagrangian function, denoted by L. This function is the difference between the kinetic and potential energies of the system: L=TV, where T is the kinetic energy and V is the potential energy.

One of the major advantages of the Lagrangian formalism is its ability to handle complex systems, especially those with constraints. A coordinate qi is considered ignorable if the Lagrangian L does not depend on it. When a coordinate is ignorable, it implies that the partial derivative Lqi=0. Consequently, the canonical momentum pi associated with this coordinate remains constant.

Key characteristics of the Lagrangian formalism include:
  • Focuses on energy differences (TV) rather than just energetic states.
  • Incorporates constraints naturally using generalized coordinates.
  • A powerful tool in deriving equations of motion through the Euler-Lagrange equations.
Hamiltonian Formalism
The Hamiltonian formalism is another pivotal framework in classical and modern physics. It translates the energy dynamics of a system into a different yet insightful perspective using the Hamiltonian function, H. The Hamiltonian is defined by the Legendre transformation of the Lagrangian, and in many cases, it represents the total energy of the system: H=ipiq˙iL.

In this formalism, a coordinate qi is ignorable if the Hamiltonian H does not depend on it, which leads to the conclusion that the momentum pi is conserved or constant over time. Hamiltonian formalism is particularly useful for systems where energy conservation and symmetries are important.

Notable features of the Hamiltonian formalism involve:
  • Focuses on phase space, comprising coordinates and momenta.
  • Provides a holistic view of both classical and quantum mechanics.
  • Emphasizes the conservation laws and symmetries via Poisson brackets and Hamilton's equations.
Degrees of Freedom
Degrees of Freedom in a physical system refer to the number of independent variables that uniquely define the state of the system. Each degree of freedom corresponds to a coordinate qi that can vary independently. Understanding the degrees of freedom is crucial when analyzing the dynamics of systems in classical mechanics.

For systems with multiple degrees of freedom (n), each can contribute to kinetic and potential energy in the Lagrangian or Hamiltonian approaches. By taking into account all degrees of freedom, you can fully describe the system’s mechanics.

Key insights into degrees of freedom include:
  • Indicate the number of ways a system can move or be configured.
  • In Lagrangian systems, they correspond to the number of generalized coordinates involved.
  • In Hamiltonian mechanics, degrees of freedom map to coordinates and canonical momenta.
Partial Derivatives
Partial Derivatives are invaluable in the study of complex systems, especially those encountered in classical mechanics, where multiple variables are at play. A partial derivative denotes the rate at which a function changes as one of the variables changes, keeping all other variables constant.

In the context of Lagrangian and Hamiltonian mechanics, partial derivatives are used to determine dependencies between variables. For instance, if the partial derivative of the Lagrangian with respect to a coordinate Lqi=0, it indicates that the system's behavior does not rely on this specific coordinate, marking it as ignorable.

Essential aspects of partial derivatives include:
  • Partial derivatives help uncover relationships between different variables in multi-dimensional functions.
  • Used extensively to derive equations of motion in both Lagrangian and Hamiltonian frameworks.
  • Crucial for determining conservation laws, such as conserved momenta, through simplifications in functional dependencies.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Here is a simple example of a canonical transformation that illustrates how the Hamiltonian formalism lets one mix up the q s and the p 's. Consider a system with one degree of freedom and Hamiltonian H=H(q,p). The equations of motion are, of course, the usual Hamiltonian equations q˙=H/p and p˙=H/q. Now consider new coordinates in phase space defined as Q=p and P=q. Show that the equations of motion for the new coordinates Q and P are Q˙=H/P and P˙=H/Q; that is, the Hamiltonian formalism applies equally to the new choice of coordinates where we have exchanged the roles of position and momentum.

The general proof of the divergence theorem SnvdA=VvdV is fairly complicated and not especially illuminating. However, there are a few special cases where it is reasonably simple and quite instructive. Here is one: Consider a rectangular region bounded by the six planes x=X and X+A,y=Y and Y+B, and z=Z and Z+C, with total volume V=ABC. The surface S of this region is made up of six rectangles that we can call S1 (in the plane x=X ), S2 (in the plane x=X+A ), and so on. The surface integral on the left of (13.63) is then the sum of six integrals, one over each of the rectangles S1,S2, and so forth. (a) Consider the first two of these integrals and show that S1nvdA+S2nvdA=YY+BdyZZ+Cdz[vx(X+A,y,z)vx(X,y,z)] (b) Show that the integrand on the right can be rewritten as an integral of vx/x over x running from x=X to x=X+A.( c) Substitute the result of part (b) into part (a), and write down the corresponding results for the two remaining pairs of faces. Add these results to prove the divergence theorem (13.63).

Consider a mass m confined to the x axis and subject to a force Fx=kx where k>0. (a) Write down and sketch the potential energy U(x) and describe the possible motions of the mass. (Distinguish between the cases that E>0 and E<0.) (b) Write down the Hamiltonian H(x,p), and describe the possible phase-space orbits for the two cases E>0 and E<0. (Remember that the function H(x,p) must equal the constant energy E. ) Explain your answers to part (b) in terms of those to part (a).

A roller coaster of mass m moves along a frictionless track that lies in the xy plane (x horizontal and y vertically up). The height of the track above the ground is given by y=h(x). (a) Using x as your generalized coordinate, write down the Lagrangian, the generalized momentum p, and the Hamiltonian H=px˙L (as a function of x and p ). (b) Find Hamilton's equations and show that they agree with what you would get from the Newtonian approach. [Hint: You know from Section 4.7 that Newton's second law takes the form Ftang =ms¨, where s is the distance measured along the track. Rewrite this as an equation for x¨ and show that you get the same result from Hamilton's equations.]

Set up the Hamiltonian and Hamilton's equations for a projectile of mass m, moving in a vertical plane and subject to gravity but no air resistance. Use as your coordinates x measured horizontally and y measured vertically up. Comment on each of the four equations of motion.

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free