Principles of Dynamics Flashcards
What is index notation used for?
To express and manipulate multi-dimensional equations
Index notation simplifies mathematical expressions involving vectors and tensors.
Define a scalar.
A magnitude that does not change with a rotation of axes.
Define a vector.
Associates a scalar with a direction.
Define a tensor.
Associates a vector (or tensor) with a direction.
What does the unit vector in index notation represent?
The unit vector along the th direction in Cartesian coordinates.
What is a free index in index notation?
Labels the component/element of the equation and must appear on both sides of the equation.
What is a dummy index in index notation?
Summed over, usually from 1 to 3, and does not have to appear on both sides of the equation.
What is the Kronecker delta?
A rank 2 tensor defined as δ_ij, which is the identity matrix in matrix form.
What does the Levi-Civita tensor represent?
A rank 3 tensor that is equal to 1 for even permutations and -1 for odd permutations.
How is the dot product expressed in index notation?
a_i * b_i = a · b.
How is the cross product expressed in index notation?
a × b = ε_ijk a_j b_k e_i.
What is a symmetric tensor?
A tensor that is equal to its transpose.
What is an anti-symmetric tensor?
A tensor that is equal to the negative of its transpose.
What are the three laws of Newton?
- N1: An object will stay at rest or in constant motion unless an external force is applied.
- N2: Force is directly proportional to the rate of change of momentum.
- N3: Every action has an equal and opposite reaction.
What are Kepler’s three laws?
- K1: The motion of a planet is an ellipse with the sun at the focus.
- K2: Each planet sweeps out equal areas in equal time intervals.
- K3: The square of the time period is proportional to the semi-major axis cubed.
What does the conservation of energy derive from?
Newton’s Second Law (N2L) and dotting both sides with the velocity vector.
What does the conservation of angular momentum derive from?
Crossing both sides of Newton’s Second Law (N2L) with the position vector.
What is the Lagrangian?
A function that describes the dynamics of a system, depending on time, position, and velocity.
What is Hamilton’s Principle?
The correct path of motion corresponds to a stationary path of the action.
What is the two-body problem in dynamics?
A system where two bodies interact through gravitational force, with the center of mass as an important coordinate.
What is a rigid body?
A system of particles whose relative positions to one another are fixed.
State the Perpendicular Axis Theorem.
The moment of inertia about an axis perpendicular to a lamina is the sum of the moments of inertia about two perpendicular axes in the plane of the lamina.
What is the angular momentum of a particle?
L = r × p, where r is the position vector and p is the linear momentum.
Define kinetic energy for a rigid body.
The sum of the kinetic energies of all constituent particles.
What is Chasles’ Theorem?
The most general rigid body displacement can be produced by a translation along a line followed by a rotation about an axis parallel to that line.
What is the axis of the moment of inertia through?
The pivot and perpendicular to the plane of the lamina.
What does the Euler-Lagrange Equation describe?
The motion of a dynamical system.
Chasles’ theorem states that the most general rigid body displacement can be produced by what?
A translation along a line followed by a rotation about an axis parallel to that line.
How can the kinetic energy of a rigid body be split up?
Into the KE of the centre of mass and KE of the rotations relative to the centre of mass.
What are holonomic constraints?
Relations between position variables that can be expressed as equations.
What is an example of a holonomic constraint in a simple pendulum?
A fixed string length introduces a tension in the rope as a constraint force.
How are holonomic constraints introduced into the Lagrangian?
By introducing new variables called Lagrange Multipliers.
What does the LHS of the full Euler-Lagrange equation describe?
The motion of the unconstrained system.
What can be done in a constrained system regarding the coordinate system?
A coordinate system can be chosen to avoid constraints in the Lagrangian.
What is Hamilton’s Principle?
The path of least action is found by solving the Euler Lagrange equations.
Name two types of conserved quantities.
- Conserved momentum
- Conserved energy
What is defined as an ignorable coordinate?
A coordinate for which the Lagrangian is independent.
What does conserved momentum depend on?
Whether the Lagrangian is independent of a particular coordinate.
What is required for conserved energy in terms of the Lagrangian?
The Lagrangian must not explicitly depend on time.
What is the Lorentz Force related to?
The motion of a charged particle in an electromagnetic field.
What is the Hamiltonian function?
A function that describes the total energy of a system.
How do Hamilton’s equations differ from Euler-Lagrange equations?
Hamilton’s equations form a set of 2n 1st order differential equations.
Under what condition is energy conserved in Hamiltonian systems?
When the kinetic energy is explicitly independent of time.
What does a spinning top exhibit in terms of motion?
Rotating about two axes.
What is the potential energy of the spinning top given by?
A specific function of its parameters.
What is the significance of ignorable coordinates in the Lagrangian of a spinning top?
They indicate that the Lagrangian is independent of them.
What does it mean for a system to be reduced to 1 degree of freedom?
The Hamiltonian is independent of certain ignorable coordinates.
What is the equation for the motion of a system derived from Hamilton’s equations?
A differential equation that describes the dynamics.
