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.
