UConn Physics Department, Physics 5201 / Yelin

Physics 5201
Theoretical Mechanics I
Susanne Yelin
Fall Semester, 2010

Syllabus

Grades

Date
Title
Content
HWs/Sols/Reading

Aug. 31 Survey of classical mechanics: single particles [pdf] Short intro and history of mechanics. Newtonian equations of motion for single particles. Chapter 1

Sep. 2 Ensembles of particles. Idea of constraints. [pdf] Real objects can be described as ensembles of particles. This leads to a convenient and often intuitive description in terms of center-of-mass (COM) and relative variables. In particular, total momentum equals COM momentum, total angular momentum the sum of angular momenta of the COM and around the COM, the same is true for kinetic energy. The total potential energy is the sum of the single-particle external potentials, and one half the sum over all two-particle internal potentials.

Sep. 7 Generalized coordinates and constraints [pdf] If there are constraints present in the system, the usual coordinate variables are not independent anymore. d'Alembert's principle (and later Hamilton's principle, both leading to a Lagrange formulation of the equations of motion) gives a formal answer as to how to deal with constraints and (independent) "generalized coordinates." Homework 1
Solutions

Sep. 9 Lagrange's equations [pdf] Lagrange's equations are derived either from d'Alembert's principle (and thus from Newton's second law) or from Hamilton's principle, i.e., the principle of stationary action. They constitute the equations of motion in any set of coordinates. Constraints can then be taken into account by adding Lagrange multipliers, leading to modified Lagrange equations.
Chapter 2.2 - Calculus of variations

Sep. 14 Lagrange's equations with constraints [pdf] It is possible to consider both holonomic contstraints (that is, those depending only on coordinates and time, not on time derivatives of coordinates) and semi-holonomic ones (that also can depend on time derivatives of coordinates). In addition, velocity-dependent potentials and even a certain class of dissipation can be accomodated with Lagrange's formalism. Using the Lagrange formalism, it is very straightforward to link symmetries and conservation properties, especially those for linear and angular momentum. Noether's theorem, a more generalized form of the connection between continuous symmetry and constants of motion, can be derived along the same lines. all of chapter 2

Sep. 16 Energy conservation. The Central Force Problem [pdf] While the conservation of the conjugate momentum is very simple in the Lagrangian formalism, energy conservation needs extra consideration. It will lead to the conservation of the so-called “energy function” which later will be known as the Hamiltonian.
The problem of two masspoints with a “central” force between them is interesting from many aspects: It has, first of all, and enormous impact on basically all areas of physics. Second, much, e.g., symmetries, degrees of freedom, type of motion, and more, can be read of a proper formulation of the problem without actually solving all the math. Chapter 3

Sep. 21 The central force problem: equations of motion [pdf] With all symmetries (i.e., conservation of the COM momentum and of the total angular momentum) leads to the fact that the central force problem is really only two-dimensional and can be formulated as a problem in simple 2D polar coordinates. Treating the centrifugal part as part of the potential leads even to the formulation as an equivalent 1D problem. Using this, the type of motion (e.g. bounded or not, bounded where) can be qualitatively found from the form of the potential without any further calculations.

Sep. 23 Closed orbits under a central force [pdf] Closed orbits have two necessary conditions: They have to be bound, and they have to be periodical. Only a small class of central forces leads to both, in particular, inverse-square (such as gravity) and linear (such as Hooke's) dependencies. Homework 2
Solution

Sep. 28 Kepler's problem and scattering [pdf] In a graviational field, the bodies follow special orbits, in particular, hyperbolic, parabolic, elliptic, or circular.

Sep. 30 Scattering and three-body problems, coordinate transformations [pdf] Scattering follows the same rules as planetary motion, but the quantity one is interested in is the so-called scattering cross-section. If there are three rather than two bodies involved, the equation can in a general case not be solved analytically, but there are special solutions.
A rigid body has six degrees of freedom, three translational and three rotational. Ways of how to understand and describe rotational coordinate systems are described. Chapter 4

Oct. 5 Euler's theorem and rigid body motion [pdf] Euler's theorem states that any transformation in the 3D real space which has at least one fixed point can be described as a simple rotation around one axis.

Oct. 7 Equations of motion for moving system [pdf] The consequences of Euler's theorem lead to relatively simple rules about the role of a moving or accelerated coordinate system as a rotating one. The Coriolis effect is one example of equations of motion in a moving reference system. Homework 3
Solution
Chapter 5

Oct. 12 The moment-of-inertia tensor [pdf] The kinetic energy due to rotation can be expressed elegantly and simply. In general, all rotational quantities in a body-fixed system can be calculated easily if the moment of inertia is known. It is a quantity comparable to the mass in linear motion.

Oct. 15 The inertia ellipsoid and Euler's eqs of motion and general torque-free motion [pdf] In a body-fixed system the moment of inertia is constant for a rigid body and does depend both on the direction and location of the turning axis. The moment-of-inertia tensor gives the moments of inertia for all posssible axis directions through a fixed point relative to the body-system-of-coordinates. It can be visualized as a three-dimensional ellipsoid.

Oct. 19 Questions [pdf] You have the opportunity to ask questions regarding practice for the midterm. If there are no questions (anymore), I'll go ahead with the regular class.

Oct. 21 Midterm Midterm is 10:30 am - 12:30 pm Midterm
Solution

Oct. 26 Euler's equations of motion and Heavy top [pdf] Euler's equations of motion describe the general motion of a rigid body in the body-fixed system. In the torque-free (in particular, gravity-free) case, the body can move like the inertia-ellipsoid rolling on a (space-fixed) plane with the COM in constant height. In the case of asymmetric bodies, the motion in a torque-free space can only be steady (i.e., with constant angular velocity) around one of the three principal axes, and stable only about the axes with minimum or maximum moment of inertia. Symmetric bodies, on the other hand, can always be described exactly as precession around the one non-symmetric axis. In the case of a heavy top, torque is present due to gravity. This case is easier described using a space-fixed Lagrangian formalism. In particular, it turns out that this simplifies to a single-variable equation of motion, dependent only on the tipping angle between the figure axis and the vertical. There are a lot of special cases for the heavy top, whose equation of motion follows a third-order polynomial form. Three common cases, however, involve a minimum and a maximum tipping angle.

Oct. 28 Heavy top and Earth's precession [pdf] It turns out that the motion of Earth with respect to the elliptic follows very similar rules, and also an effective precession takes place, albeit only every 26,000 years. The reason for this precession is that the slightly oblate form of the planet induces a direction-dependent potential, similar to the case of the heavy top. Homework 4
Solution
Chapter 6

Nov. 2 Oscillations [pdf] Oscillations happen around a point of stable equilibrium in the system. In most cases, it is appropriate to make asmall-deviations approximation which then leads to a simple standard form of the Lagrangian, and thus a formulation in terms of an eigenvalue problem. Chapter 6

Nov. 4 Oscillations - Formalism and example [pdf] The eigenvalue problem of multi-dimensional oscillations follows special orthonormalization rules, with, in particular, a special definition of the “scalar product.” It will result in a basis of eigenvectors (or “principal axes”). What is the Lagrangian, what the eigenfrequencies, the eigenvectors, the solution?

Nov. 9 Degenerate eigenvalues and normal form. [pdf] Transforming the regular generalized coordinates into the system of principal axes results in the so-called "normal coordinates." A "normal mode" is the function that the system occupies if it oscillates with any of the eigenfrequencies.

Nov. 11 Forced and damped oscillations. [pdf] Forced oscillation leads to resonances, while dissipation terms lead to damping and shifted frequencies. Chapter 8
Homework 5
Solution

Nov. 16 Hamiltonian formulation [pdf] The Hamiltonian equations of motion are equivalent to the Lagrangian ones, but the Hamiltonian is a function of coordinates and their canonical momenta. This allows for a more symmetric and in most cases more intuitive formulation, since often the Hamiltonian is equal to the total energy. Legendre transformations lead from the Lagrangian to the Hamiltonian (and can also be used in other cases, where a change of vaviables is needed, e.g., in thermodynamics).

Nov. 18 Canonical transformations [pdf] Canonical transformations lead from a Hamiltonian formulation depending on one set of canonical variables to another one. One example where this is useful is the harmonic oscillator where a canonical transformation can be found to make the Hamiltonian cyclic (i.e., all coordinates in the Hamiltonian are cyclic). Chapter 9

Nov. 30 Symplectic form and properties of canonical transformations [pdf] The “symplectic” notation allows to treat the set of all q's and p's as one single set of equal-footed variables. Infinitesimal canonical transformations (ICTs) provide the basic building block for transformations that depend on continuous parameters, such as an angle or time. Poisson brackets are introduced as an example of canonical invariants, quantities that are independent of the particular choice of canonical variables. Homework 6
Solution

Dec. 2 Poisson brackets [pdf] Poisson brackets serve as an alternative formulation to much of Hamiltonian dynamics. This allows, for example, to rewrite formally the Hamiltonian equations of motion independent of the particular choice of phase space variables. In addition, new constants of motion can be found, and a more general formulation of the connection between symmetries and conservation quantities is found.

Dec. 7 Poisson brackets and Hamilton-Jacoby theory [pdf] Yet another way to utilize canonical transformations is the Hamilton-Jacobi theory. The goal is here to directly obtain the coordinate and canonical momentum variables as a function of their boundary values and time. In this particular case, a canonical transformation is constructed such that the new Hamiltonian vanishes, thus all new coordinates and canonical momenta are constant. The particular generating function in this case is called “Hamilton's principal function.” Chapter 10

Dec. 9 Separability and action-angle variables [pdf] Hamilton-Jacobi theory is particularly useful if the problem can be completely separated by variables. In this case, as the example of the Kepler problem shows, the equations are particularly straightforward and easy to solve. A special type of problems consists of periodic motion. By smartly choosing the coordinates as the phases, the frequencies of the motion can be found without solving the whole dynamics. In this case, the canonic momenta are actions.

Dec. 14 Final Final is 10:30 am - 1:30 pm in P121 and you can bring along your Goldstein book! Final
Preliminary Solution

Date		Title		Content		HWs/Sols/Reading

Aug. 31	Survey of classical mechanics: single particles [pdf]	Short intro and history of mechanics. Newtonian equations of motion for single particles.	Chapter 1
Sep. 2	Ensembles of particles. Idea of constraints. [pdf]	Real objects can be described as ensembles of particles. This leads to a convenient and often intuitive description in terms of center-of-mass (COM) and relative variables. In particular, total momentum equals COM momentum, total angular momentum the sum of angular momenta of the COM and around the COM, the same is true for kinetic energy. The total potential energy is the sum of the single-particle external potentials, and one half the sum over all two-particle internal potentials.

Sep. 7	Generalized coordinates and constraints [pdf]	If there are constraints present in the system, the usual coordinate variables are not independent anymore. d'Alembert's principle (and later Hamilton's principle, both leading to a Lagrange formulation of the equations of motion) gives a formal answer as to how to deal with constraints and (independent) "generalized coordinates."	Homework 1 Solutions
Sep. 9	Lagrange's equations [pdf]	Lagrange's equations are derived either from d'Alembert's principle (and thus from Newton's second law) or from Hamilton's principle, i.e., the principle of stationary action. They constitute the equations of motion in any set of coordinates. Constraints can then be taken into account by adding Lagrange multipliers, leading to modified Lagrange equations.	Chapter 2.2 - Calculus of variations

Sep. 14	Lagrange's equations with constraints [pdf]	It is possible to consider both holonomic contstraints (that is, those depending only on coordinates and time, not on time derivatives of coordinates) and semi-holonomic ones (that also can depend on time derivatives of coordinates). In addition, velocity-dependent potentials and even a certain class of dissipation can be accomodated with Lagrange's formalism. Using the Lagrange formalism, it is very straightforward to link symmetries and conservation properties, especially those for linear and angular momentum. Noether's theorem, a more generalized form of the connection between continuous symmetry and constants of motion, can be derived along the same lines.	all of chapter 2
Sep. 16	Energy conservation. The Central Force Problem [pdf]	While the conservation of the conjugate momentum is very simple in the Lagrangian formalism, energy conservation needs extra consideration. It will lead to the conservation of the so-called “energy function” which later will be known as the Hamiltonian. The problem of two masspoints with a “central” force between them is interesting from many aspects: It has, first of all, and enormous impact on basically all areas of physics. Second, much, e.g., symmetries, degrees of freedom, type of motion, and more, can be read of a proper formulation of the problem without actually solving all the math.	Chapter 3

Sep. 21	The central force problem: equations of motion [pdf]	With all symmetries (i.e., conservation of the COM momentum and of the total angular momentum) leads to the fact that the central force problem is really only two-dimensional and can be formulated as a problem in simple 2D polar coordinates. Treating the centrifugal part as part of the potential leads even to the formulation as an equivalent 1D problem. Using this, the type of motion (e.g. bounded or not, bounded where) can be qualitatively found from the form of the potential without any further calculations.
Sep. 23	Closed orbits under a central force [pdf]	Closed orbits have two necessary conditions: They have to be bound, and they have to be periodical. Only a small class of central forces leads to both, in particular, inverse-square (such as gravity) and linear (such as Hooke's) dependencies.	Homework 2 Solution

Sep. 28	Kepler's problem and scattering [pdf]	In a graviational field, the bodies follow special orbits, in particular, hyperbolic, parabolic, elliptic, or circular.
Sep. 30	Scattering and three-body problems, coordinate transformations [pdf]	Scattering follows the same rules as planetary motion, but the quantity one is interested in is the so-called scattering cross-section. If there are three rather than two bodies involved, the equation can in a general case not be solved analytically, but there are special solutions. A rigid body has six degrees of freedom, three translational and three rotational. Ways of how to understand and describe rotational coordinate systems are described.	Chapter 4

Oct. 5	Euler's theorem and rigid body motion [pdf]	Euler's theorem states that any transformation in the 3D real space which has at least one fixed point can be described as a simple rotation around one axis.
Oct. 7	Equations of motion for moving system [pdf]	The consequences of Euler's theorem lead to relatively simple rules about the role of a moving or accelerated coordinate system as a rotating one. The Coriolis effect is one example of equations of motion in a moving reference system.	Homework 3 Solution Chapter 5

Oct. 12	The moment-of-inertia tensor [pdf]	The kinetic energy due to rotation can be expressed elegantly and simply. In general, all rotational quantities in a body-fixed system can be calculated easily if the moment of inertia is known. It is a quantity comparable to the mass in linear motion.
Oct. 15	The inertia ellipsoid and Euler's eqs of motion and general torque-free motion [pdf]	In a body-fixed system the moment of inertia is constant for a rigid body and does depend both on the direction and location of the turning axis. The moment-of-inertia tensor gives the moments of inertia for all posssible axis directions through a fixed point relative to the body-system-of-coordinates. It can be visualized as a three-dimensional ellipsoid.

Oct. 19	Questions [pdf]	You have the opportunity to ask questions regarding practice for the midterm. If there are no questions (anymore), I'll go ahead with the regular class.
Oct. 21	Midterm	Midterm is 10:30 am - 12:30 pm	Midterm Solution

Oct. 26	Euler's equations of motion and Heavy top [pdf]	Euler's equations of motion describe the general motion of a rigid body in the body-fixed system. In the torque-free (in particular, gravity-free) case, the body can move like the inertia-ellipsoid rolling on a (space-fixed) plane with the COM in constant height. In the case of asymmetric bodies, the motion in a torque-free space can only be steady (i.e., with constant angular velocity) around one of the three principal axes, and stable only about the axes with minimum or maximum moment of inertia. Symmetric bodies, on the other hand, can always be described exactly as precession around the one non-symmetric axis. In the case of a heavy top, torque is present due to gravity. This case is easier described using a space-fixed Lagrangian formalism. In particular, it turns out that this simplifies to a single-variable equation of motion, dependent only on the tipping angle between the figure axis and the vertical. There are a lot of special cases for the heavy top, whose equation of motion follows a third-order polynomial form. Three common cases, however, involve a minimum and a maximum tipping angle.
Oct. 28	Heavy top and Earth's precession [pdf]	It turns out that the motion of Earth with respect to the elliptic follows very similar rules, and also an effective precession takes place, albeit only every 26,000 years. The reason for this precession is that the slightly oblate form of the planet induces a direction-dependent potential, similar to the case of the heavy top.	Homework 4 Solution Chapter 6

Nov. 2	Oscillations [pdf]	Oscillations happen around a point of stable equilibrium in the system. In most cases, it is appropriate to make asmall-deviations approximation which then leads to a simple standard form of the Lagrangian, and thus a formulation in terms of an eigenvalue problem.	Chapter 6
Nov. 4	Oscillations - Formalism and example [pdf]	The eigenvalue problem of multi-dimensional oscillations follows special orthonormalization rules, with, in particular, a special definition of the “scalar product.” It will result in a basis of eigenvectors (or “principal axes”). What is the Lagrangian, what the eigenfrequencies, the eigenvectors, the solution?

Nov. 9	Degenerate eigenvalues and normal form. [pdf]	Transforming the regular generalized coordinates into the system of principal axes results in the so-called "normal coordinates." A "normal mode" is the function that the system occupies if it oscillates with any of the eigenfrequencies.
Nov. 11	Forced and damped oscillations. [pdf]	Forced oscillation leads to resonances, while dissipation terms lead to damping and shifted frequencies.	Chapter 8 Homework 5 Solution

Nov. 16	Hamiltonian formulation [pdf]	The Hamiltonian equations of motion are equivalent to the Lagrangian ones, but the Hamiltonian is a function of coordinates and their canonical momenta. This allows for a more symmetric and in most cases more intuitive formulation, since often the Hamiltonian is equal to the total energy. Legendre transformations lead from the Lagrangian to the Hamiltonian (and can also be used in other cases, where a change of vaviables is needed, e.g., in thermodynamics).
Nov. 18	Canonical transformations [pdf]	Canonical transformations lead from a Hamiltonian formulation depending on one set of canonical variables to another one. One example where this is useful is the harmonic oscillator where a canonical transformation can be found to make the Hamiltonian cyclic (i.e., all coordinates in the Hamiltonian are cyclic).	Chapter 9

Nov. 30	Symplectic form and properties of canonical transformations [pdf]	The “symplectic” notation allows to treat the set of all q's and p's as one single set of equal-footed variables. Infinitesimal canonical transformations (ICTs) provide the basic building block for transformations that depend on continuous parameters, such as an angle or time. Poisson brackets are introduced as an example of canonical invariants, quantities that are independent of the particular choice of canonical variables.	Homework 6 Solution
Dec. 2	Poisson brackets [pdf]	Poisson brackets serve as an alternative formulation to much of Hamiltonian dynamics. This allows, for example, to rewrite formally the Hamiltonian equations of motion independent of the particular choice of phase space variables. In addition, new constants of motion can be found, and a more general formulation of the connection between symmetries and conservation quantities is found.

Dec. 7	Poisson brackets and Hamilton-Jacoby theory [pdf]	Yet another way to utilize canonical transformations is the Hamilton-Jacobi theory. The goal is here to directly obtain the coordinate and canonical momentum variables as a function of their boundary values and time. In this particular case, a canonical transformation is constructed such that the new Hamiltonian vanishes, thus all new coordinates and canonical momenta are constant. The particular generating function in this case is called “Hamilton's principal function.”	Chapter 10
Dec. 9	Separability and action-angle variables [pdf]	Hamilton-Jacobi theory is particularly useful if the problem can be completely separated by variables. In this case, as the example of the Kepler problem shows, the equations are particularly straightforward and easy to solve. A special type of problems consists of periodic motion. By smartly choosing the coordinates as the phases, the frequencies of the motion can be found without solving the whole dynamics. In this case, the canonic momenta are actions.

Dec. 14	Final	Final is 10:30 am - 1:30 pm in P121 and you can bring along your Goldstein book!	Final Preliminary Solution

Here is a space where you can leave feedback on the class. This will send me an anonymous email (unless you use your name). Any feedback (positive and negative, especially before the semester is actually over) is very welcome

This page is maintained by S. Yelin susanne.yelin@uconn.edu
last updated December 20, 2010