equation of motion Flashcards
law of conservation of energy
total energy is conserved, E is independent of t
implicit function theorem
U,V open
F is C1
det(partial derivatives at z) is not 0//
there exists nbhd W of x and C1 mapping g(x)=y
F(x,g(x))=0 for all x in W
conservative vector field
grad φ for some scalar function
characterisation of conservative forces with work
work along path depends on endpoints and not shape of path
work along closed path is 0
characterisation of conservative forces with first derivatives
partial derivatives symmetry
converse is true if O is simply connected
simply connected
path connected, trivial fundamental group
kelvin-stokes theorem
integral over boundary of f.dr = integral over S of curl f.dS
conservative system
system with n degrees of freedom such that f is conservative
law of conservation of energy
E(x(t),x.(t)) is independent of t
central vector field
defined on R^n\0 and invariant wrt orthogonal group
poisson bracket
Σdf/dq_j dg/dp_j - df/dp_j dg/dq_j
Lie algebra
vector space equipped with a bilinear skew-symmetric operation which satisfies the Jacobi identity
lie sub algebra
lie bracket of any two elements belongs to it
morphism of Lie algebras
linear and respects the lie bracket
hamiltonian vector field
X_f(g)={g,f}
